{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2022-10-18T16:01:27+00:00"},"categoryId":33770,"data":{"title":"Quantum Physics","slug":"quantum-physics","image":{"src":null,"width":0,"height":0},"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"parentCategory":{"categoryId":33756,"title":"Science","slug":"science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"}},"childCategories":[],"description":"You, yes you, can understand the laws of quantum physics and use them to solve those pesky subatomic problems.","relatedArticles":{"self":"https://dummies-api.dummies.com/v2/articles?category=33770&offset=0&size=5"},"hasArticle":true,"hasBook":true,"articleCount":102,"bookCount":3},"_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":102,"items":[{"headers":{"creationTime":"2016-03-26T14:04:17+00:00","modifiedTime":"2022-10-06T20:19:55+00:00","timestamp":"2022-10-06T21:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"Finding x, y, and z Equations for Three-Dimensional Free Particle Problems","strippedTitle":"finding x, y, and z equations for three-dimensional free particle problems","slug":"finding-the-x-y-and-z-equations-for-three-dimensional-free-particle-problems","canonicalUrl":"","seo":{"metaDescription":"In quantum physics, learn how to find the x, y, and z equations for three-dimensional free particle problems.","noIndex":0,"noFollow":0},"content":"At some point, your quantum physics instructor may want you to find the <i>x</i>, <i>y</i>, and <i>z</i> equations for three-dimensional free particle problems. Take a look at the <i>x</i> equation for the free particle,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397870.image0.png\" alt=\"image0.png\" width=\"140\" height=\"43\" />\r\n\r\nYou can write its general solution as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397871.image1.png\" alt=\"image1.png\" width=\"456\" height=\"156\" />\r\n\r\nwhere A<i><sub>y</sub></i> and A<i><sub>z</sub></i> are constants.\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397872.image2.png\" alt=\"image2.png\" width=\"180\" height=\"27\" />\r\n\r\nyou get this for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397873.image3.png\" alt=\"image3.png\" width=\"229\" height=\"100\" />\r\n\r\nwhere A= A<i><sub>x</sub></i> A<i><sub>y</sub></i> A<i><sub>z</sub></i>.\r\n\r\nThe part in the parentheses in the exponent is the dot product of the vectors\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397874.image4.png\" alt=\"image4.png\" width=\"97\" height=\"19\" />\r\n\r\nThat is, if the vector <b><i>a</i></b> = (<i>a</i><i><sub>x</sub></i>, <i>a</i><i><sub>y</sub></i>, <i>a</i><i><sub>z</sub></i>) in terms of components and the vector <b><i>b</i></b> = (<i>b</i><i><sub>x</sub></i>, <i>b</i><i><sub>y</sub></i>, <i>b</i><i><sub>z</sub></i>), then the dot product of <b><i>a</i></b> and <b><i>b</i></b> is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397875.image5.png\" alt=\"image5.png\" width=\"176\" height=\"29\" />\r\n\r\nSo here’s how you can rewrite the\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397876.image6.png\" alt=\"image6.png\" width=\"127\" height=\"63\" />","description":"At some point, your quantum physics instructor may want you to find the <i>x</i>, <i>y</i>, and <i>z</i> equations for three-dimensional free particle problems. Take a look at the <i>x</i> equation for the free particle,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397870.image0.png\" alt=\"image0.png\" width=\"140\" height=\"43\" />\r\n\r\nYou can write its general solution as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397871.image1.png\" alt=\"image1.png\" width=\"456\" height=\"156\" />\r\n\r\nwhere A<i><sub>y</sub></i> and A<i><sub>z</sub></i> are constants.\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397872.image2.png\" alt=\"image2.png\" width=\"180\" height=\"27\" />\r\n\r\nyou get this for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397873.image3.png\" alt=\"image3.png\" width=\"229\" height=\"100\" />\r\n\r\nwhere A= A<i><sub>x</sub></i> A<i><sub>y</sub></i> A<i><sub>z</sub></i>.\r\n\r\nThe part in the parentheses in the exponent is the dot product of the vectors\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397874.image4.png\" alt=\"image4.png\" width=\"97\" height=\"19\" />\r\n\r\nThat is, if the vector <b><i>a</i></b> = (<i>a</i><i><sub>x</sub></i>, <i>a</i><i><sub>y</sub></i>, <i>a</i><i><sub>z</sub></i>) in terms of components and the vector <b><i>b</i></b> = (<i>b</i><i><sub>x</sub></i>, <i>b</i><i><sub>y</sub></i>, <i>b</i><i><sub>z</sub></i>), then the dot product of <b><i>a</i></b> and <b><i>b</i></b> is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397875.image5.png\" alt=\"image5.png\" width=\"176\" height=\"29\" />\r\n\r\nSo here’s how you can rewrite the\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397876.image6.png\" alt=\"image6.png\" width=\"127\" height=\"63\" />","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-633f420e99a5c\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-633f420e9a754\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-10-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":161185},{"headers":{"creationTime":"2016-03-26T14:05:51+00:00","modifiedTime":"2022-10-06T17:29:36+00:00","timestamp":"2022-10-06T18:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"Adding Time Dependence for Three-Dimensional Free Particle Problems","strippedTitle":"adding time dependence for three-dimensional free particle problems","slug":"how-to-add-time-dependence-and-get-a-physical-equation-for-three-dimensional-free-particle-problems","canonicalUrl":"","seo":{"metaDescription":"Learn how to add time dependence and get a physical equation for a three-dimensional free particle problem.","noIndex":0,"noFollow":0},"content":"At some point, your quantum physics instructor may want you to add time dependence and get a physical equation for a three-dimensional free particle problem. You can add time dependence to the solution for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397231.image0.png\" alt=\"image0.png\" width=\"212\" height=\"27\" />\r\n\r\nif you remember that, for a free particle,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397232.image1.png\" alt=\"image1.png\" width=\"185\" height=\"28\" />\r\n\r\nThat equation gives you this form for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397233.image2.png\" alt=\"image2.png\" width=\"161\" height=\"72\" />\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397234.image3.png\" alt=\"image3.png\" width=\"44\" height=\"37\" />\r\n\r\nthe equation turns into\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397235.image4.png\" alt=\"image4.png\" width=\"153\" height=\"31\" />\r\n\r\nIn fact, now that the right side of the equation is in terms of the radius vector <b><i>r</i></b>, you can make the left side match:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397236.image5.png\" alt=\"image5.png\" width=\"128\" height=\"31\" />\r\n\r\nThat’s the solution to the Schrödinger equation, but it’s unphysical. Why? Trying to normalize this equation in three dimensions, for example, gives you the following, where A is a constant:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397237.image6.png\" alt=\"image6.png\" width=\"244\" height=\"51\" />\r\n\r\n(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number. The limits on the integral just mean to integrate over all of space, like this:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397238.image7.png\" alt=\"image7.png\" width=\"191\" height=\"51\" />\r\n\r\nThus, the integral diverges and you can’t normalize\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397239.image8.png\" alt=\"image8.png\" width=\"47\" height=\"27\" />\r\n\r\nas written here. So what do you do here to get a physical particle?\r\n<p class=\"Tip\">The key to solving this problem is realizing that if you have a number of solutions to the Schrödinger equation, then any linear combination of those solutions is also a solution. In other words, you add various wave functions together so that you get a <i>wave packet</i>, which is a collection of wave functions of the form</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397240.image9.png\" alt=\"image9.png\" width=\"36\" height=\"23\" />\r\n\r\nsuch that\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The wave functions interfere constructively at one location.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">They interfere destructively (go to zero) at all other locations.</p>\r\n</li>\r\n</ul>\r\nLook at the time-independent version:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397241.image10.png\" alt=\"image10.png\" width=\"113\" height=\"47\" />\r\n\r\nHowever, for a free particle, the energy states are not separated into distinct bands; the possible energies are continuous, so people write this summation as an integral:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397242.image11.png\" alt=\"image11.png\" width=\"195\" height=\"57\" />\r\n\r\nSo what is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397243.image12.png\" alt=\"image12.png\" width=\"44\" height=\"27\" />\r\n\r\nIt’s the three-dimensional analog of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397244.image13.png\" alt=\"image13.png\" width=\"37\" height=\"27\" />\r\n\r\nThat is, it’s the amplitude of each component wave function. You can find\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397245.image14.png\" alt=\"image14.png\" width=\"35\" height=\"27\" />\r\n\r\nfrom the Fourier transform of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397246.image15.png\" alt=\"image15.png\" width=\"36\" height=\"27\" />\r\n\r\nlike this:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397247.image16.png\" alt=\"image16.png\" width=\"207\" height=\"59\" />\r\n\r\nIn practice, you choose\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397248.image17.png\" alt=\"image17.png\" width=\"35\" height=\"27\" />\r\n\r\nyourself. Look at an example, using the following form for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397249.image18.png\" alt=\"image18.png\" width=\"35\" height=\"27\" />\r\n\r\nwhich is for a Gaussian wave packet (Note: The exponential part is what makes this a Gaussian wave form):\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397250.image19.png\" alt=\"image19.png\" width=\"189\" height=\"45\" />\r\n\r\nwhere <i>a</i> and A are constants. You can begin by normalizing\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397251.image20.png\" alt=\"image20.png\" width=\"35\" height=\"27\" />\r\n\r\nto determine what A is. Here’s how that works:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397252.image21.png\" alt=\"image21.png\" width=\"272\" height=\"51\" />\r\n\r\nOkay. Performing the integral gives you\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397253.image22.png\" alt=\"image22.png\" width=\"101\" height=\"120\" />\r\n\r\nwhich means that the wave function is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397254.image23.png\" alt=\"image23.png\" width=\"317\" height=\"64\" />\r\n\r\nYou can evaluate this equation to give you the following, which is what the time-independent wave function for a Gaussian wave packet looks like in 3D:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397255.image24.png\" alt=\"image24.png\" width=\"187\" height=\"53\" />","description":"At some point, your quantum physics instructor may want you to add time dependence and get a physical equation for a three-dimensional free particle problem. You can add time dependence to the solution for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397231.image0.png\" alt=\"image0.png\" width=\"212\" height=\"27\" />\r\n\r\nif you remember that, for a free particle,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397232.image1.png\" alt=\"image1.png\" width=\"185\" height=\"28\" />\r\n\r\nThat equation gives you this form for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397233.image2.png\" alt=\"image2.png\" width=\"161\" height=\"72\" />\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397234.image3.png\" alt=\"image3.png\" width=\"44\" height=\"37\" />\r\n\r\nthe equation turns into\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397235.image4.png\" alt=\"image4.png\" width=\"153\" height=\"31\" />\r\n\r\nIn fact, now that the right side of the equation is in terms of the radius vector <b><i>r</i></b>, you can make the left side match:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397236.image5.png\" alt=\"image5.png\" width=\"128\" height=\"31\" />\r\n\r\nThat’s the solution to the Schrödinger equation, but it’s unphysical. Why? Trying to normalize this equation in three dimensions, for example, gives you the following, where A is a constant:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397237.image6.png\" alt=\"image6.png\" width=\"244\" height=\"51\" />\r\n\r\n(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number. The limits on the integral just mean to integrate over all of space, like this:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397238.image7.png\" alt=\"image7.png\" width=\"191\" height=\"51\" />\r\n\r\nThus, the integral diverges and you can’t normalize\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397239.image8.png\" alt=\"image8.png\" width=\"47\" height=\"27\" />\r\n\r\nas written here. So what do you do here to get a physical particle?\r\n<p class=\"Tip\">The key to solving this problem is realizing that if you have a number of solutions to the Schrödinger equation, then any linear combination of those solutions is also a solution. In other words, you add various wave functions together so that you get a <i>wave packet</i>, which is a collection of wave functions of the form</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397240.image9.png\" alt=\"image9.png\" width=\"36\" height=\"23\" />\r\n\r\nsuch that\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The wave functions interfere constructively at one location.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">They interfere destructively (go to zero) at all other locations.</p>\r\n</li>\r\n</ul>\r\nLook at the time-independent version:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397241.image10.png\" alt=\"image10.png\" width=\"113\" height=\"47\" />\r\n\r\nHowever, for a free particle, the energy states are not separated into distinct bands; the possible energies are continuous, so people write this summation as an integral:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397242.image11.png\" alt=\"image11.png\" width=\"195\" height=\"57\" />\r\n\r\nSo what is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397243.image12.png\" alt=\"image12.png\" width=\"44\" height=\"27\" />\r\n\r\nIt’s the three-dimensional analog of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397244.image13.png\" alt=\"image13.png\" width=\"37\" height=\"27\" />\r\n\r\nThat is, it’s the amplitude of each component wave function. You can find\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397245.image14.png\" alt=\"image14.png\" width=\"35\" height=\"27\" />\r\n\r\nfrom the Fourier transform of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397246.image15.png\" alt=\"image15.png\" width=\"36\" height=\"27\" />\r\n\r\nlike this:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397247.image16.png\" alt=\"image16.png\" width=\"207\" height=\"59\" />\r\n\r\nIn practice, you choose\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397248.image17.png\" alt=\"image17.png\" width=\"35\" height=\"27\" />\r\n\r\nyourself. Look at an example, using the following form for\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397249.image18.png\" alt=\"image18.png\" width=\"35\" height=\"27\" />\r\n\r\nwhich is for a Gaussian wave packet (Note: The exponential part is what makes this a Gaussian wave form):\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397250.image19.png\" alt=\"image19.png\" width=\"189\" height=\"45\" />\r\n\r\nwhere <i>a</i> and A are constants. You can begin by normalizing\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397251.image20.png\" alt=\"image20.png\" width=\"35\" height=\"27\" />\r\n\r\nto determine what A is. Here’s how that works:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397252.image21.png\" alt=\"image21.png\" width=\"272\" height=\"51\" />\r\n\r\nOkay. Performing the integral gives you\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397253.image22.png\" alt=\"image22.png\" width=\"101\" height=\"120\" />\r\n\r\nwhich means that the wave function is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397254.image23.png\" alt=\"image23.png\" width=\"317\" height=\"64\" />\r\n\r\nYou can evaluate this equation to give you the following, which is what the time-independent wave function for a Gaussian wave packet looks like in 3D:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397255.image24.png\" alt=\"image24.png\" width=\"187\" height=\"53\" />","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-633f17df6c8ac\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-633f17df6cf72\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-10-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":161400},{"headers":{"creationTime":"2016-03-26T14:04:23+00:00","modifiedTime":"2022-09-22T20:38:33+00:00","timestamp":"2022-09-23T00:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"How to Calculate the Energy Degeneracy of a Hydrogen Atom","strippedTitle":"how to calculate the energy degeneracy of a hydrogen atom","slug":"how-to-calculate-the-energy-degeneracy-of-a-hydrogen-atom-in-terms-of-n-l-and-m","canonicalUrl":"","seo":{"metaDescription":"Learn how to determine how many of quantum states of the hydrogen atom (n, l, m) have the same energy, meaning the energy degeneracy.","noIndex":0,"noFollow":0},"content":"Each quantum state of the hydrogen atom is specified with three quantum numbers: <i>n</i> (the principal quantum number), <i>l</i> (the angular momentum quantum number of the electron), and <i>m</i> (the <i>z</i> component of the electron’s angular momentum,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397925.image0.png\" alt=\"image0.png\" width=\"88\" height=\"27\" />\r\n\r\nHow many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers <i>n</i>, <i>l</i>, and <i>m</i>?\r\n\r\nWell, the actual energy is just dependent on <i>n</i>, as you see in the following equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397926.image1.png\" alt=\"image1.png\" width=\"168\" height=\"43\" />\r\n\r\nThat means the E is independent of <i>l</i> and <i>m</i>. So how many states, |<i>n</i>, <i>l</i>, <i>m</i>>, have the same energy for a particular value of <i>n</i>? Well, for a particular value of <i>n</i>, <i>l</i> can range from zero to <i>n</i> – 1. And each <i>l</i> can have different values of <i>m</i>, so the total degeneracy is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397927.image2.png\" alt=\"image2.png\" width=\"235\" height=\"48\" />\r\n\r\nThe degeneracy in <i>m</i> is the number of states with different values of <i>m</i> that have the same value of <i>l</i>. For any particular value of <i>l</i>, you can have <i>m</i> values of –<i>l</i>, –<i>l</i> + 1, ..., 0, ..., <i>l</i> – 1, <i>l</i>. And that’s (2<i>l</i> + 1) possible <i>m</i> states for a particular value of <i>l</i>. So you can plug in (2<i>l </i>+ 1) for the degeneracy in <i>m</i>:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397928.image3.png\" alt=\"image3.png\" width=\"159\" height=\"48\" />\r\n\r\nAnd this series works out to be just <i>n</i><sup>2</sup>.\r\n\r\nSo the degeneracy of the energy levels of the hydrogen atom is <i>n</i><sup>2</sup>. For example, the ground state, <i>n</i> = 1, has degeneracy = <i>n</i><sup>2</sup> = 1 (which makes sense because <i>l</i>, and therefore <i>m</i>, can only equal zero for this state).\r\n\r\nFor <i>n</i> = 2, you have a degeneracy of 4:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397929.image4.png\" alt=\"image4.png\" width=\"93\" height=\"128\" />\r\n\r\nCool.","description":"Each quantum state of the hydrogen atom is specified with three quantum numbers: <i>n</i> (the principal quantum number), <i>l</i> (the angular momentum quantum number of the electron), and <i>m</i> (the <i>z</i> component of the electron’s angular momentum,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397925.image0.png\" alt=\"image0.png\" width=\"88\" height=\"27\" />\r\n\r\nHow many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers <i>n</i>, <i>l</i>, and <i>m</i>?\r\n\r\nWell, the actual energy is just dependent on <i>n</i>, as you see in the following equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397926.image1.png\" alt=\"image1.png\" width=\"168\" height=\"43\" />\r\n\r\nThat means the E is independent of <i>l</i> and <i>m</i>. So how many states, |<i>n</i>, <i>l</i>, <i>m</i>>, have the same energy for a particular value of <i>n</i>? Well, for a particular value of <i>n</i>, <i>l</i> can range from zero to <i>n</i> – 1. And each <i>l</i> can have different values of <i>m</i>, so the total degeneracy is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397927.image2.png\" alt=\"image2.png\" width=\"235\" height=\"48\" />\r\n\r\nThe degeneracy in <i>m</i> is the number of states with different values of <i>m</i> that have the same value of <i>l</i>. For any particular value of <i>l</i>, you can have <i>m</i> values of –<i>l</i>, –<i>l</i> + 1, ..., 0, ..., <i>l</i> – 1, <i>l</i>. And that’s (2<i>l</i> + 1) possible <i>m</i> states for a particular value of <i>l</i>. So you can plug in (2<i>l </i>+ 1) for the degeneracy in <i>m</i>:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397928.image3.png\" alt=\"image3.png\" width=\"159\" height=\"48\" />\r\n\r\nAnd this series works out to be just <i>n</i><sup>2</sup>.\r\n\r\nSo the degeneracy of the energy levels of the hydrogen atom is <i>n</i><sup>2</sup>. For example, the ground state, <i>n</i> = 1, has degeneracy = <i>n</i><sup>2</sup> = 1 (which makes sense because <i>l</i>, and therefore <i>m</i>, can only equal zero for this state).\r\n\r\nFor <i>n</i> = 2, you have a degeneracy of 4:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397929.image4.png\" alt=\"image4.png\" width=\"93\" height=\"128\" />\r\n\r\nCool.","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. 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You also learn about various effects named for people, such as the Hamiltonian, the Heisenberg Uncertainty Principle, the Schrödinger Equation, and the Compton Effect.\r\n\r\nThis Cheat Sheet provides a quick reference to some of the main equations used in quantum physics.","description":"In dabbling in quantum physics, you come across spin operators, commutation relationships, and many formulae and principles. You also learn about various effects named for people, such as the Hamiltonian, the Heisenberg Uncertainty Principle, the Schrödinger Equation, and the Compton Effect.\r\n\r\nThis Cheat Sheet provides a quick reference to some of the main equations used in quantum physics.","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-63221b1a6b965\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-63221b1a6c2cf\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":170680,"title":"Quantum Physics and the Hamiltonian","slug":"quantum-physics-and-the-hamiltonian","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170680"}},{"articleId":170690,"title":"Quantum Physics and the Heisenberg Uncertainty Principle","slug":"quantum-physics-and-the-heisenberg-uncertainty-principle","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170690"}},{"articleId":170689,"title":"Quantum Physics and the Schrödinger Equation","slug":"quantum-physics-and-the-schrdinger-equation","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170689"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":170735,"title":"Quantum Physics and the Compton Effect","slug":"quantum-physics-and-the-compton-effect","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170735"}}],"content":[{"title":"Quantum physics and the Hamiltonian","thumb":null,"image":null,"content":"<p>One of the central problems of quantum mechanics is to calculate the energy levels of a system. The energy operator, called the <i>Hamiltonian</i><i>,</i><i> </i>abbreviated <i>H, </i>gives you the total energy. Finding the energy levels of a system breaks down to finding the eigenvalues of the problem</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354939.image0.png\" alt=\"image0.png\" width=\"84\" height=\"27\" /></p>\n<p>The eigenvalues can be found by solving the equation:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354940.image1.png\" alt=\"image1.png\" width=\"311\" height=\"145\" /></p>\n"},{"title":"The Heisenberg uncertainty principle","thumb":null,"image":null,"content":"<p>In quantum physics, you encounter the Heisenberg uncertainty principle, which says that the better you know the position of a particle, the less you know the momentum, and vice versa. In the <i>x</i> direction, for example, that looks like this:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354942.image0.png\" alt=\"image0.png\" width=\"73\" height=\"41\" /></p>\n<p>where D<i>x</i> is the measurement uncertainty in the particle’s <i>x</i> position, D<i>p</i><i><sub>x</sub></i> is its measurement uncertainty in its momentum in the <i>x</i> direction, and</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354943.image1.png\" alt=\"image1.png\" width=\"52\" height=\"41\" /></p>\n<p>This relation holds for all three dimensions:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354944.image2.png\" alt=\"image2.png\" width=\"76\" height=\"88\" /></p>\n"},{"title":"The Schrödinger equation","thumb":null,"image":null,"content":"<p>When a quantum mechanical state can be described by a wave function,</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354947.image1.jpg\" alt=\"image1.jpg\" width=\"68\" height=\"32\" /></p>\n<p>then this is a solution of the Schrödinger equation, which is written in terms of the potential</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354948.image2.jpg\" alt=\"image2.jpg\" width=\"65\" height=\"32\" /></p>\n<p>and energy</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354949.image3.jpg\" alt=\"image3.jpg\" width=\"15\" height=\"23\" /></p>\n<p>like so:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354950.image4.jpg\" alt=\"image4.jpg\" width=\"368\" height=\"59\" /></p>\n<p>The Schrödinger equation work in three dimensions as well:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354951.image5.jpg\" alt=\"image5.jpg\" width=\"341\" height=\"161\" /></p>\n"},{"title":"Spin operators and commutation in quantum physics","thumb":null,"image":null,"content":"<p>Don’t think quantum physics is devoid of anything but dry science. The fact is that it’s full of relationships, they’re just commutation relationships — which are pretty dry science after all. In any case, among the angular momentum operators L<i><sub>x</sub></i><i>,</i> L<i><sub>y</sub></i><i>, </i>and L<i><sub>z</sub></i>, are these commutation relations:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354953.image0.png\" alt=\"image0.png\" width=\"101\" height=\"107\" /></p>\n<p>All the orbital angular momentum operators, such as L<i><sub>x</sub></i><i>,</i> L<i><sub>y</sub></i><i>,</i> and L<i><sub>z</sub></i><i>,</i> have analogous spin operators: S<i><sub>x</sub></i><i>,</i> S<i><sub>y</sub></i><i>,</i> and S<sub>z</sub>. And the commutation relations work the same way for spin:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354954.image1.png\" alt=\"image1.png\" width=\"107\" height=\"107\" /></p>\n"},{"title":"The Compton effect","thumb":null,"image":null,"content":"<p>In quantum physics, you may deal with the Compton effect of X-ray and gamma ray qualities in matter. To calculate these effects, use the following formula, which assumes that the light is represented by a photon with energy E = <i>h</i>u and that its momentum is <i>p</i> = E/<i>c</i>:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/354956.image0.png\" alt=\"image0.png\" width=\"499\" height=\"189\" /></p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2022-02-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":208083},{"headers":{"creationTime":"2016-03-26T21:48:17+00:00","modifiedTime":"2016-03-26T21:48:17+00:00","timestamp":"2022-09-14T18:10:31+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"The Laws of Quantum Physics: The Schrödinger Equation","strippedTitle":"the laws of quantum physics: the schrödinger equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","canonicalUrl":"","seo":{"metaDescription":"The Schrödinger equation is one of the most basic formulas of quantum physics. With the Schrödinger equation, you can solve for the wave functions of particles,","noIndex":0,"noFollow":0},"content":"<p>The Schrödinger equation is one of the most basic formulas of quantum physics. With the Schrödinger equation, you can solve for the wave functions of particles, and that allows you to say everything you can about the particle — where it is, what its momentum is, and so on. </p>\n<p>In the following version of the Schrödinger equation, the first term represents the kinetic energy and the second term represents the potential energy:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/164843.image0.jpg\" width=\"449\" height=\"34\" alt=\"image0.jpg\"/>\n<p>where</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/164844.image1.jpg\" width=\"157\" height=\"40\" alt=\"image1.jpg\"/>","description":"<p>The Schrödinger equation is one of the most basic formulas of quantum physics. With the Schrödinger equation, you can solve for the wave functions of particles, and that allows you to say everything you can about the particle — where it is, what its momentum is, and so on. </p>\n<p>In the following version of the Schrödinger equation, the first term represents the kinetic energy and the second term represents the potential energy:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/164843.image0.jpg\" width=\"449\" height=\"34\" alt=\"image0.jpg\"/>\n<p>where</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/164844.image1.jpg\" width=\"157\" height=\"40\" alt=\"image1.jpg\"/>","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum 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Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-63221917d8fbb\"></div></div>","rightAd":"<div class=\"du-ad-region row\" 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Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"Spin Operators and Commutation in Quantum Physics","strippedTitle":"spin operators and commutation in quantum physics","slug":"spin-operators-and-commutation-in-quantum-physics","canonicalUrl":"","seo":{"metaDescription":"Don’t think quantum physics is devoid of anything but dry science. The fact is that it’s full of relationships, they’re just commutation relationships — which a","noIndex":0,"noFollow":0},"content":"<p>Don’t think quantum physics is devoid of anything but dry science. The fact is that it’s full of relationships, they’re just commutation relationships — which are pretty dry science after all. In any case, among the angular momentum operators L<i><sub>x</sub></i><i>,</i> L<i><sub>y</sub></i><i>, </i>and L<i><sub>z</sub></i>, are these commutation relations:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/354953.image0.png\" width=\"101\" height=\"107\" alt=\"image0.png\"/>\n<p>All the orbital angular momentum operators, such as L<i><sub>x</sub></i><i>,</i> L<i><sub>y</sub></i><i>,</i> and L<i><sub>z</sub></i><i>,</i> have analogous spin operators: S<i><sub>x</sub></i><i>,</i> S<i><sub>y</sub></i><i>,</i> and S<sub>z</sub>. And the commutation relations work the same way for spin:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/354954.image1.png\" width=\"107\" height=\"107\" alt=\"image1.png\"/>","description":"<p>Don’t think quantum physics is devoid of anything but dry science. The fact is that it’s full of relationships, they’re just commutation relationships — which are pretty dry science after all. In any case, among the angular momentum operators L<i><sub>x</sub></i><i>,</i> L<i><sub>y</sub></i><i>, </i>and L<i><sub>z</sub></i>, are these commutation relations:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/354953.image0.png\" width=\"101\" height=\"107\" alt=\"image0.png\"/>\n<p>All the orbital angular momentum operators, such as L<i><sub>x</sub></i><i>,</i> L<i><sub>y</sub></i><i>,</i> and L<i><sub>z</sub></i><i>,</i> have analogous spin operators: S<i><sub>x</sub></i><i>,</i> S<i><sub>y</sub></i><i>,</i> and S<sub>z</sub>. And the commutation relations work the same way for spin:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/354954.image1.png\" width=\"107\" height=\"107\" alt=\"image1.png\"/>","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6322181179e27\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-632218117a710\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":170679},{"headers":{"creationTime":"2016-03-26T14:09:40+00:00","modifiedTime":"2016-03-26T14:09:40+00:00","timestamp":"2022-09-14T18:04:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","strippedTitle":"find the eigenfunctions of lz in spherical coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","canonicalUrl":"","seo":{"metaDescription":"At some point, your quantum physics instructor may ask you to find the eigenfunctions of L z in spherical coordinates. In spherical coordinates, the L z operato","noIndex":0,"noFollow":0},"content":"<p>At some point, your quantum physics instructor may ask you to find the eigenfunctions of L<i><sub>z</sub></i> in spherical coordinates. In spherical coordinates, the L<sub>z</sub> operator looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395073.image0.png\" width=\"288\" height=\"136\" alt=\"image0.png\"/>\n<p>which is the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395074.image1.png\" width=\"244\" height=\"44\" alt=\"image1.png\"/>\n<p>And because </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395075.image2.png\" width=\"171\" height=\"27\" alt=\"image2.png\"/>\n<p>this equation can be written in this version:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395076.image3.png\" width=\"247\" height=\"44\" alt=\"image3.png\"/>\n<p>Cancelling out terms from the two sides of this equation gives you this differential equation:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395077.image4.png\" width=\"140\" height=\"44\" alt=\"image4.png\"/>\n<p>This looks easy to solve, and the solution is just</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395078.image5.png\" width=\"101\" height=\"28\" alt=\"image5.png\"/>\n<p>where C is a constant of integration. </p>\n<p>You can determine C by insisting that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395079.image6.png\" width=\"48\" height=\"27\" alt=\"image6.png\"/>\n<p>be normalized — that is, that the following hold true:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395080.image7.png\" width=\"145\" height=\"51\" alt=\"image7.png\"/>\n<p>(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)</p>\n<p>So this gives you</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395081.image8.png\" width=\"169\" height=\"161\" alt=\"image8.png\"/>\n<p>You are now able to determine the form of </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395082.image9.png\" width=\"247\" height=\"27\" alt=\"image9.png\"/>\n<p>which equals</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395083.image10.png\" width=\"275\" height=\"53\" alt=\"image10.png\"/>","description":"<p>At some point, your quantum physics instructor may ask you to find the eigenfunctions of L<i><sub>z</sub></i> in spherical coordinates. In spherical coordinates, the L<sub>z</sub> operator looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395073.image0.png\" width=\"288\" height=\"136\" alt=\"image0.png\"/>\n<p>which is the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395074.image1.png\" width=\"244\" height=\"44\" alt=\"image1.png\"/>\n<p>And because </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395075.image2.png\" width=\"171\" height=\"27\" alt=\"image2.png\"/>\n<p>this equation can be written in this version:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395076.image3.png\" width=\"247\" height=\"44\" alt=\"image3.png\"/>\n<p>Cancelling out terms from the two sides of this equation gives you this differential equation:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395077.image4.png\" width=\"140\" height=\"44\" alt=\"image4.png\"/>\n<p>This looks easy to solve, and the solution is just</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395078.image5.png\" width=\"101\" height=\"28\" alt=\"image5.png\"/>\n<p>where C is a constant of integration. </p>\n<p>You can determine C by insisting that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395079.image6.png\" width=\"48\" height=\"27\" alt=\"image6.png\"/>\n<p>be normalized — that is, that the following hold true:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395080.image7.png\" width=\"145\" height=\"51\" alt=\"image7.png\"/>\n<p>(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)</p>\n<p>So this gives you</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395081.image8.png\" width=\"169\" height=\"161\" alt=\"image8.png\"/>\n<p>You are now able to determine the form of </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395082.image9.png\" width=\"247\" height=\"27\" alt=\"image9.png\"/>\n<p>which equals</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395083.image10.png\" width=\"275\" height=\"53\" alt=\"image10.png\"/>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}},{"articleId":161815,"title":"Spin One-Half Matrices","slug":"spin-one-half-matrices","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161815"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-632217929754c\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-6322179297dd7\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":161819},{"headers":{"creationTime":"2016-03-26T14:09:39+00:00","modifiedTime":"2016-03-26T14:09:39+00:00","timestamp":"2022-09-14T18:04:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","strippedTitle":"find the eigenvalues of the raising and lowering angular momentum operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","canonicalUrl":"","seo":{"metaDescription":"In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s z component of angular ","noIndex":0,"noFollow":0},"content":"<p>In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s <i>z</i> component of angular momentum.</p>\n<p>Start by taking a look at L<sub>+</sub>, and plan to solve for <i>c</i>:</p>\n<p>L<sub>+</sub>| <i>l</i>, <i>m</i> > = <i>c</i> | <i>l</i>, <i>m</i> + 1 ></p>\n<p>So L<sub>+</sub> | <i>l</i>, <i>m</i> > gives you a new state, and multiplying that new state by its transpose should give you <i>c</i><sup>2</sup>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395035.image0.png\" width=\"160\" height=\"35\" alt=\"image0.png\"/>\n<p>To see this equation, note that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395036.image1.png\" width=\"304\" height=\"35\" alt=\"image1.png\"/>\n<p>On the other hand, also note that</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395037.image2.png\" width=\"251\" height=\"35\" alt=\"image2.png\"/>\n<p>so you have</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395038.image3.png\" width=\"139\" height=\"28\" alt=\"image3.png\"/>\n<p>What do you do about L<sub>+</sub> L<sub>–</sub>? Well, you assume that the following is true: </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395039.image4.png\" width=\"136\" height=\"27\" alt=\"image4.png\"/>\n<p>So your equation becomes the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395040.image5.png\" width=\"193\" height=\"35\" alt=\"image5.png\"/>\n<p>Great! That means that <i>c</i> is equal to</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395041.image6.png\" width=\"217\" height=\"45\" alt=\"image6.png\"/>\n<p>So what is</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395042.image7.png\" width=\"207\" height=\"43\" alt=\"image7.png\"/>\n<p>Applying the L<sup>2</sup> and L<i><sub>z</sub></i> operators gives you this value for <i>c</i>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395043.image8.png\" width=\"179\" height=\"37\" alt=\"image8.png\"/>\n<p>And that’s the eigenvalue of L<sub>+</sub>, which means you have this relation:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395044.image9.png\" width=\"280\" height=\"37\" alt=\"image9.png\"/>\n<p>Similarly, you can show that L<sub>–</sub> gives you the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395045.image10.png\" width=\"281\" height=\"37\" alt=\"image10.png\"/>","description":"<p>In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s <i>z</i> component of angular momentum.</p>\n<p>Start by taking a look at L<sub>+</sub>, and plan to solve for <i>c</i>:</p>\n<p>L<sub>+</sub>| <i>l</i>, <i>m</i> > = <i>c</i> | <i>l</i>, <i>m</i> + 1 ></p>\n<p>So L<sub>+</sub> | <i>l</i>, <i>m</i> > gives you a new state, and multiplying that new state by its transpose should give you <i>c</i><sup>2</sup>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395035.image0.png\" width=\"160\" height=\"35\" alt=\"image0.png\"/>\n<p>To see this equation, note that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395036.image1.png\" width=\"304\" height=\"35\" alt=\"image1.png\"/>\n<p>On the other hand, also note that</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395037.image2.png\" width=\"251\" height=\"35\" alt=\"image2.png\"/>\n<p>so you have</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395038.image3.png\" width=\"139\" height=\"28\" alt=\"image3.png\"/>\n<p>What do you do about L<sub>+</sub> L<sub>–</sub>? Well, you assume that the following is true: </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395039.image4.png\" width=\"136\" height=\"27\" alt=\"image4.png\"/>\n<p>So your equation becomes the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395040.image5.png\" width=\"193\" height=\"35\" alt=\"image5.png\"/>\n<p>Great! That means that <i>c</i> is equal to</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395041.image6.png\" width=\"217\" height=\"45\" alt=\"image6.png\"/>\n<p>So what is</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395042.image7.png\" width=\"207\" height=\"43\" alt=\"image7.png\"/>\n<p>Applying the L<sup>2</sup> and L<i><sub>z</sub></i> operators gives you this value for <i>c</i>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395043.image8.png\" width=\"179\" height=\"37\" alt=\"image8.png\"/>\n<p>And that’s the eigenvalue of L<sub>+</sub>, which means you have this relation:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395044.image9.png\" width=\"280\" height=\"37\" alt=\"image9.png\"/>\n<p>Similarly, you can show that L<sub>–</sub> gives you the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395045.image10.png\" width=\"281\" height=\"37\" alt=\"image10.png\"/>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}},{"articleId":161815,"title":"Spin One-Half Matrices","slug":"spin-one-half-matrices","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161815"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-632217928f8c1\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-6322179290161\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":161818},{"headers":{"creationTime":"2016-03-26T14:09:38+00:00","modifiedTime":"2016-03-26T14:09:38+00:00","timestamp":"2022-09-14T18:04:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"How Spin Operators Resemble Angular Momentum Operators","strippedTitle":"how spin operators resemble angular momentum operators","slug":"how-spin-operators-resemble-angular-momentum-operators","canonicalUrl":"","seo":{"metaDescription":"Because spin is a type of built-in angular momentum, spin operators have a lot in common with orbital angular momentum operators. As your quantum physics instru","noIndex":0,"noFollow":0},"content":"<p>Because spin is a type of built-in angular momentum, spin operators have a lot in common with orbital angular momentum operators. As your quantum physics instructor will tell you, there are analogous spin operators, S<sup>2</sup> and S<i><sub>z</sub></i>, to orbital angular momentum operators L<sup>2</sup> and L<sub>z</sub>. However, these operators are just operators; they don’t have a differential form like the orbital angular momentum operators do.</p>\n<p>In fact, all the orbital angular momentum operators, such as L<i><sub>x</sub></i>, L<i><sub>y</sub></i>, and L<i><sub>z</sub></i>, have analogs here: S<i><sub>x</sub></i>, S<i><sub>y</sub></i>, and S<i><sub>z</sub></i>. The commutation relations among L<i><sub>x</sub></i>, L<i><sub>y</sub></i>, and L<i><sub>z</sub></i> are the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395093.image0.png\" width=\"113\" height=\"116\" alt=\"image0.png\"/>\n<p>And they work the same way for spin:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395094.image1.png\" width=\"119\" height=\"116\" alt=\"image1.png\"/>\n<p>The L<sup>2</sup> operator gives you the following result when you apply it to an orbital angular momentum eigenstate:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395095.image2.png\" width=\"157\" height=\"28\" alt=\"image2.png\"/>\n<p>And just as you’d expect, the S<sup>2</sup> operator works in an analogous fashion:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395096.image3.png\" width=\"175\" height=\"28\" alt=\"image3.png\"/>\n<p>The L<i><sub>z</sub></i> operator gives you this result when you apply it to an orbital angular momentum eigenstate:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395097.image4.png\" width=\"120\" height=\"27\" alt=\"image4.png\"/>\n<p>And by analogy, the S<i><sub>z</sub></i> operator works this way:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395098.image5.png\" width=\"128\" height=\"27\" alt=\"image5.png\"/>\n<p>What about the raising and lowering operators, L<sub>+</sub> and L<sub>–</sub>? Are there analogs for spin? In angular momentum terms, L<sub>+</sub> and L<sub>–</sub> work like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395099.image6.png\" width=\"281\" height=\"83\" alt=\"image6.png\"/>\n<p>There are spin raising and lowering operators as well, S<sub>+</sub> and S<sub>–</sub>, and they work like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395100.image7.png\" width=\"295\" height=\"83\" alt=\"image7.png\"/>","description":"<p>Because spin is a type of built-in angular momentum, spin operators have a lot in common with orbital angular momentum operators. As your quantum physics instructor will tell you, there are analogous spin operators, S<sup>2</sup> and S<i><sub>z</sub></i>, to orbital angular momentum operators L<sup>2</sup> and L<sub>z</sub>. However, these operators are just operators; they don’t have a differential form like the orbital angular momentum operators do.</p>\n<p>In fact, all the orbital angular momentum operators, such as L<i><sub>x</sub></i>, L<i><sub>y</sub></i>, and L<i><sub>z</sub></i>, have analogs here: S<i><sub>x</sub></i>, S<i><sub>y</sub></i>, and S<i><sub>z</sub></i>. The commutation relations among L<i><sub>x</sub></i>, L<i><sub>y</sub></i>, and L<i><sub>z</sub></i> are the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395093.image0.png\" width=\"113\" height=\"116\" alt=\"image0.png\"/>\n<p>And they work the same way for spin:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395094.image1.png\" width=\"119\" height=\"116\" alt=\"image1.png\"/>\n<p>The L<sup>2</sup> operator gives you the following result when you apply it to an orbital angular momentum eigenstate:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395095.image2.png\" width=\"157\" height=\"28\" alt=\"image2.png\"/>\n<p>And just as you’d expect, the S<sup>2</sup> operator works in an analogous fashion:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395096.image3.png\" width=\"175\" height=\"28\" alt=\"image3.png\"/>\n<p>The L<i><sub>z</sub></i> operator gives you this result when you apply it to an orbital angular momentum eigenstate:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395097.image4.png\" width=\"120\" height=\"27\" alt=\"image4.png\"/>\n<p>And by analogy, the S<i><sub>z</sub></i> operator works this way:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395098.image5.png\" width=\"128\" height=\"27\" alt=\"image5.png\"/>\n<p>What about the raising and lowering operators, L<sub>+</sub> and L<sub>–</sub>? Are there analogs for spin? In angular momentum terms, L<sub>+</sub> and L<sub>–</sub> work like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395099.image6.png\" width=\"281\" height=\"83\" alt=\"image6.png\"/>\n<p>There are spin raising and lowering operators as well, S<sub>+</sub> and S<sub>–</sub>, and they work like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395100.image7.png\" width=\"295\" height=\"83\" alt=\"image7.png\"/>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}},{"articleId":161815,"title":"Spin One-Half Matrices","slug":"spin-one-half-matrices","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161815"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-6322179287bd1\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;quantum-physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118460825&quot;]}]\" id=\"du-slot-632217928846b\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":161817},{"headers":{"creationTime":"2016-03-26T14:09:37+00:00","modifiedTime":"2016-03-26T14:09:37+00:00","timestamp":"2022-09-14T18:04:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"How to Find Angular Momentum Eigenvalues","strippedTitle":"how to find angular momentum eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","canonicalUrl":"","seo":{"metaDescription":"When you have the eigenvalues of angular momentum states in quantum mechanics, you can solve the Hamiltonian and get the allowed energy levels of an object with","noIndex":0,"noFollow":0},"content":"<p>When you have the eigenvalues of angular momentum states in quantum mechanics, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum. The eigenvalues of the angular momentum are the possible values the angular momentum can take. </p>\n<p>Here’s how to derive eigenstate equations with </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394986.image0.png\" width=\"96\" height=\"24\" alt=\"image0.png\"/>\n<p>Note that L<sup>2</sup> – L<i><sub>z</sub></i><sup>2</sup> = L<i><sub>x</sub></i><sup>2</sup> + L<i><sub>y</sub></i><sup>2</sup>, which is a positive number, so </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394987.image1.png\" width=\"77\" height=\"27\" alt=\"image1.png\"/>\n<p>That means that</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394988.image2.png\" width=\"155\" height=\"35\" alt=\"image2.png\"/>\n<p>And substituting in </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394989.image3.png\" width=\"303\" height=\"28\" alt=\"image3.png\"/>\n<p>and using the fact that the eigenstates are normalized, gives you this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394990.image4.png\" width=\"240\" height=\"67\" alt=\"image4.png\"/>\n<p>So there’s a maximum possible value of </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394991.image5.png\" width=\"16\" height=\"21\" alt=\"image5.png\"/>\n<p>which you can call </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394992.image6.png\" width=\"37\" height=\"24\" alt=\"image6.png\"/>\n<p>You can be clever now, because there has to be a state </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394993.image7.png\" width=\"65\" height=\"27\" alt=\"image7.png\"/>\n<p>such that you can’t raise </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394994.image8.png\" width=\"15\" height=\"21\" alt=\"image8.png\"/>\n<p>any more. Thus, if you apply the raising operator, you get zero:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394995.image9.png\" width=\"103\" height=\"27\" alt=\"image9.png\"/>\n<p>Applying the lowering operator to this also gives you zero:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394996.image10.png\" width=\"120\" height=\"27\" alt=\"image10.png\"/>\n<p>And because </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394997.image11.png\" width=\"136\" height=\"27\" alt=\"image11.png\"/>\n<p>that means the following is true:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394998.image12.png\" width=\"176\" height=\"32\" alt=\"image12.png\"/>\n<p>Putting in </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394999.image13.png\" width=\"339\" height=\"28\" alt=\"image13.png\"/>\n<p>gives you this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395000.image14.png\" width=\"200\" height=\"111\" alt=\"image14.png\"/>\n<p>At this point, it’s usual to rename </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395001.image15.png\" width=\"348\" height=\"97\" alt=\"image15.png\"/>\n<p>You can say even more. In addition to a </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395002.image16.png\" width=\"37\" height=\"24\" alt=\"image16.png\"/>\n<p>there must also be a </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395003.image17.png\" width=\"31\" height=\"24\" alt=\"image17.png\"/>\n<p>such that when you apply the lowering operator, L<sub>–</sub>, you get zero, because you can’t go any lower than </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395004.image18.png\" width=\"93\" height=\"59\" alt=\"image18.png\"/>\n<p>And you can apply L<sub>+</sub> on this as well:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395005.image19.png\" width=\"112\" height=\"27\" alt=\"image19.png\"/>\n<p>From </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395006.image20.png\" width=\"135\" height=\"27\" alt=\"image20.png\"/>\n<p>you know that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395007.image21.png\" width=\"172\" height=\"32\" alt=\"image21.png\"/>\n<p>which gives you the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395008.image22.png\" width=\"220\" height=\"133\" alt=\"image22.png\"/>\n<p>And comparing this equation to </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395009.image23.png\" width=\"145\" height=\"27\" alt=\"image23.png\"/>\n<p>gives you</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395010.image24.png\" width=\"85\" height=\"24\" alt=\"image24.png\"/>\n<p>Note that because you reach </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395011.image25.png\" width=\"61\" height=\"27\" alt=\"image25.png\"/>\n<p>by <i>n</i> successive applications of </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395012.image26.png\" width=\"112\" height=\"27\" alt=\"image26.png\"/>\n<p>you get the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395013.image27.png\" width=\"99\" height=\"24\" alt=\"image27.png\"/>\n<p>Coupling these two equations gives you</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395014.image28.png\" width=\"59\" height=\"37\" alt=\"image28.png\"/>\n<p>Therefore, </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395015.image29.png\" width=\"33\" height=\"24\" alt=\"image29.png\"/>\n<p>can be either an integer or half an integer (depending on whether <i>n</i> is even or odd).</p>\n<p class=\"Remember\">Because </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395016.image30.png\" width=\"103\" height=\"24\" alt=\"image30.png\"/>\n<p>and <i>n</i> is a positive number, you can find that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395017.image31.png\" width=\"68\" height=\"16\" alt=\"image31.png\"/>\n<p>So now you have it:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">The eigenstates are | <i>l</i>, <i>m</i> ><i>.</i></p>\n </li>\n <li><p class=\"first-para\">The quantum number of the total angular momentum is <i>l</i>.</p>\n </li>\n <li><p class=\"first-para\">The quantum number of the angular momentum along the <i>z</i> axis is <i>m</i>.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395018.image32.png\" width=\"349\" height=\"89\" alt=\"image32.png\"/>\n </li>\n</ul>\n<p>For each <i>l</i>, there are 2<i>l</i> + 1 values of <i>m</i>. For example, if <i>l</i> = 2, then <i>m</i> can equal –2, –1, 0, 1, or 2. </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395019.image33.png\" width=\"431\" height=\"32\" alt=\"image33.png\"/>\n<p>You can see a representative L and L<i><sub>z</sub></i> in the figure. </p>\n<div class=\"imageBlock\" style=\"width:500px;\"><img src=\"https://www.dummies.com/wp-content/uploads/395020.image34.jpg\" width=\"500\" height=\"456\" alt=\"L and L<i><sub>z</sub></i>.\"/><div class=\"imageCaption\">L and L<i><sub>z</sub></i>. </div></div>\n<p>L is the total angular momentum and L<i><sub>z</sub></i> is the projection of that total angular momentum on the <i>z</i> axis.</p>","description":"<p>When you have the eigenvalues of angular momentum states in quantum mechanics, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum. The eigenvalues of the angular momentum are the possible values the angular momentum can take. </p>\n<p>Here’s how to derive eigenstate equations with </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394986.image0.png\" width=\"96\" height=\"24\" alt=\"image0.png\"/>\n<p>Note that L<sup>2</sup> – L<i><sub>z</sub></i><sup>2</sup> = L<i><sub>x</sub></i><sup>2</sup> + L<i><sub>y</sub></i><sup>2</sup>, which is a positive number, so </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394987.image1.png\" width=\"77\" height=\"27\" alt=\"image1.png\"/>\n<p>That means that</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394988.image2.png\" width=\"155\" height=\"35\" alt=\"image2.png\"/>\n<p>And substituting in </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394989.image3.png\" width=\"303\" height=\"28\" alt=\"image3.png\"/>\n<p>and using the fact that the eigenstates are normalized, gives you this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394990.image4.png\" width=\"240\" height=\"67\" alt=\"image4.png\"/>\n<p>So there’s a maximum possible value of </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394991.image5.png\" width=\"16\" height=\"21\" alt=\"image5.png\"/>\n<p>which you can call </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394992.image6.png\" width=\"37\" height=\"24\" alt=\"image6.png\"/>\n<p>You can be clever now, because there has to be a state </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394993.image7.png\" width=\"65\" height=\"27\" alt=\"image7.png\"/>\n<p>such that you can’t raise </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394994.image8.png\" width=\"15\" height=\"21\" alt=\"image8.png\"/>\n<p>any more. Thus, if you apply the raising operator, you get zero:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394995.image9.png\" width=\"103\" height=\"27\" alt=\"image9.png\"/>\n<p>Applying the lowering operator to this also gives you zero:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394996.image10.png\" width=\"120\" height=\"27\" alt=\"image10.png\"/>\n<p>And because </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394997.image11.png\" width=\"136\" height=\"27\" alt=\"image11.png\"/>\n<p>that means the following is true:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394998.image12.png\" width=\"176\" height=\"32\" alt=\"image12.png\"/>\n<p>Putting in </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/394999.image13.png\" width=\"339\" height=\"28\" alt=\"image13.png\"/>\n<p>gives you this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395000.image14.png\" width=\"200\" height=\"111\" alt=\"image14.png\"/>\n<p>At this point, it’s usual to rename </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395001.image15.png\" width=\"348\" height=\"97\" alt=\"image15.png\"/>\n<p>You can say even more. In addition to a </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395002.image16.png\" width=\"37\" height=\"24\" alt=\"image16.png\"/>\n<p>there must also be a </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395003.image17.png\" width=\"31\" height=\"24\" alt=\"image17.png\"/>\n<p>such that when you apply the lowering operator, L<sub>–</sub>, you get zero, because you can’t go any lower than </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395004.image18.png\" width=\"93\" height=\"59\" alt=\"image18.png\"/>\n<p>And you can apply L<sub>+</sub> on this as well:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395005.image19.png\" width=\"112\" height=\"27\" alt=\"image19.png\"/>\n<p>From </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395006.image20.png\" width=\"135\" height=\"27\" alt=\"image20.png\"/>\n<p>you know that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395007.image21.png\" width=\"172\" height=\"32\" alt=\"image21.png\"/>\n<p>which gives you the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395008.image22.png\" width=\"220\" height=\"133\" alt=\"image22.png\"/>\n<p>And comparing this equation to </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395009.image23.png\" width=\"145\" height=\"27\" alt=\"image23.png\"/>\n<p>gives you</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395010.image24.png\" width=\"85\" height=\"24\" alt=\"image24.png\"/>\n<p>Note that because you reach </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395011.image25.png\" width=\"61\" height=\"27\" alt=\"image25.png\"/>\n<p>by <i>n</i> successive applications of </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395012.image26.png\" width=\"112\" height=\"27\" alt=\"image26.png\"/>\n<p>you get the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395013.image27.png\" width=\"99\" height=\"24\" alt=\"image27.png\"/>\n<p>Coupling these two equations gives you</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395014.image28.png\" width=\"59\" height=\"37\" alt=\"image28.png\"/>\n<p>Therefore, </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395015.image29.png\" width=\"33\" height=\"24\" alt=\"image29.png\"/>\n<p>can be either an integer or half an integer (depending on whether <i>n</i> is even or odd).</p>\n<p class=\"Remember\">Because </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395016.image30.png\" width=\"103\" height=\"24\" alt=\"image30.png\"/>\n<p>and <i>n</i> is a positive number, you can find that </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395017.image31.png\" width=\"68\" height=\"16\" alt=\"image31.png\"/>\n<p>So now you have it:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">The eigenstates are | <i>l</i>, <i>m</i> ><i>.</i></p>\n </li>\n <li><p class=\"first-para\">The quantum number of the total angular momentum is <i>l</i>.</p>\n </li>\n <li><p class=\"first-para\">The quantum number of the angular momentum along the <i>z</i> axis is <i>m</i>.</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395018.image32.png\" width=\"349\" height=\"89\" alt=\"image32.png\"/>\n </li>\n</ul>\n<p>For each <i>l</i>, there are 2<i>l</i> + 1 values of <i>m</i>. For example, if <i>l</i> = 2, then <i>m</i> can equal –2, –1, 0, 1, or 2. </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395019.image33.png\" width=\"431\" height=\"32\" alt=\"image33.png\"/>\n<p>You can see a representative L and L<i><sub>z</sub></i> in the figure. </p>\n<div class=\"imageBlock\" style=\"width:500px;\"><img src=\"https://www.dummies.com/wp-content/uploads/395020.image34.jpg\" width=\"500\" height=\"456\" alt=\"L and L<i><sub>z</sub></i>.\"/><div class=\"imageCaption\">L and L<i><sub>z</sub></i>. </div></div>\n<p>L is the total angular momentum and L<i><sub>z</sub></i> is the projection of that total angular momentum on the <i>z</i> axis.</p>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}},{"articleId":161815,"title":"Spin One-Half Matrices","slug":"spin-one-half-matrices","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161815"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). 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Quantum Physics Articles

You, yes you, can understand the laws of quantum physics and use them to solve those pesky subatomic problems.

Articles From Quantum Physics

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Quantum Physics Finding x, y, and z Equations for Three-Dimensional Free Particle Problems

Article / Updated 10-06-2022

At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle, You can write its general solution as where Ay and Az are constants. Because you get this for where A= Ax Ay Az. The part in the parentheses in the exponent is the dot product of the vectors That is, if the vector a = (ax, ay, az) in terms of components and the vector b = (bx, by, bz), then the dot product of a and b is So here’s how you can rewrite the

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Quantum Physics Adding Time Dependence for Three-Dimensional Free Particle Problems

Article / Updated 10-06-2022

At some point, your quantum physics instructor may want you to add time dependence and get a physical equation for a three-dimensional free particle problem. You can add time dependence to the solution for if you remember that, for a free particle, That equation gives you this form for Because the equation turns into In fact, now that the right side of the equation is in terms of the radius vector r, you can make the left side match: That’s the solution to the Schrödinger equation, but it’s unphysical. Why? Trying to normalize this equation in three dimensions, for example, gives you the following, where A is a constant: (Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number. The limits on the integral just mean to integrate over all of space, like this: Thus, the integral diverges and you can’t normalize as written here. So what do you do here to get a physical particle? The key to solving this problem is realizing that if you have a number of solutions to the Schrödinger equation, then any linear combination of those solutions is also a solution. In other words, you add various wave functions together so that you get a wave packet, which is a collection of wave functions of the form such that The wave functions interfere constructively at one location. They interfere destructively (go to zero) at all other locations. Look at the time-independent version: However, for a free particle, the energy states are not separated into distinct bands; the possible energies are continuous, so people write this summation as an integral: So what is It’s the three-dimensional analog of That is, it’s the amplitude of each component wave function. You can find from the Fourier transform of like this: In practice, you choose yourself. Look at an example, using the following form for which is for a Gaussian wave packet (Note: The exponential part is what makes this a Gaussian wave form): where a and A are constants. You can begin by normalizing to determine what A is. Here’s how that works: Okay. Performing the integral gives you which means that the wave function is You can evaluate this equation to give you the following, which is what the time-independent wave function for a Gaussian wave packet looks like in 3D:

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Quantum Physics How to Calculate the Energy Degeneracy of a Hydrogen Atom

Article / Updated 09-22-2022

Each quantum state of the hydrogen atom is specified with three quantum numbers: n (the principal quantum number), l (the angular momentum quantum number of the electron), and m (the z component of the electron’s angular momentum, How many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers n, l, and m? Well, the actual energy is just dependent on n, as you see in the following equation: That means the E is independent of l and m. So how many states, |n, l, m>, have the same energy for a particular value of n? Well, for a particular value of n, l can range from zero to n – 1. And each l can have different values of m, so the total degeneracy is The degeneracy in m is the number of states with different values of m that have the same value of l. For any particular value of l, you can have m values of –l, –l + 1, ..., 0, ..., l – 1, l. And that’s (2l + 1) possible m states for a particular value of l. So you can plug in (2l + 1) for the degeneracy in m: And this series works out to be just n2. So the degeneracy of the energy levels of the hydrogen atom is n2. For example, the ground state, n = 1, has degeneracy = n2 = 1 (which makes sense because l, and therefore m, can only equal zero for this state). For n = 2, you have a degeneracy of 4: Cool.

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Quantum Physics Quantum Physics For Dummies Cheat Sheet

Cheat Sheet / Updated 02-14-2022

In dabbling in quantum physics, you come across spin operators, commutation relationships, and many formulae and principles. You also learn about various effects named for people, such as the Hamiltonian, the Heisenberg Uncertainty Principle, the Schrödinger Equation, and the Compton Effect. This Cheat Sheet provides a quick reference to some of the main equations used in quantum physics.

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Quantum Physics The Laws of Quantum Physics: The Schrödinger Equation

Article / Updated 03-26-2016

The Schrödinger equation is one of the most basic formulas of quantum physics. With the Schrödinger equation, you can solve for the wave functions of particles, and that allows you to say everything you can about the particle — where it is, what its momentum is, and so on. In the following version of the Schrödinger equation, the first term represents the kinetic energy and the second term represents the potential energy: where

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Quantum Physics Spin Operators and Commutation in Quantum Physics

Article / Updated 03-26-2016

Don’t think quantum physics is devoid of anything but dry science. The fact is that it’s full of relationships, they’re just commutation relationships — which are pretty dry science after all. In any case, among the angular momentum operators Lx, Ly, and Lz, are these commutation relations: All the orbital angular momentum operators, such as Lx, Ly, and Lz, have analogous spin operators: Sx, Sy, and Sz. And the commutation relations work the same way for spin:

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Quantum Physics Find the Eigenfunctions of Lz in Spherical Coordinates

Article / Updated 03-26-2016

At some point, your quantum physics instructor may ask you to find the eigenfunctions of Lz in spherical coordinates. In spherical coordinates, the Lz operator looks like this: which is the following: And because this equation can be written in this version: Cancelling out terms from the two sides of this equation gives you this differential equation: This looks easy to solve, and the solution is just where C is a constant of integration. You can determine C by insisting that be normalized — that is, that the following hold true: (Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.) So this gives you You are now able to determine the form of which equals

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Quantum Physics Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators

Article / Updated 03-26-2016

In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s z component of angular momentum. Start by taking a look at L+, and plan to solve for c: L+| l, m > = c | l, m + 1 > So L+ | l, m > gives you a new state, and multiplying that new state by its transpose should give you c2: To see this equation, note that On the other hand, also note that so you have What do you do about L+ L–? Well, you assume that the following is true: So your equation becomes the following: Great! That means that c is equal to So what is Applying the L2 and Lz operators gives you this value for c: And that’s the eigenvalue of L+, which means you have this relation: Similarly, you can show that L– gives you the following:

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Quantum Physics How Spin Operators Resemble Angular Momentum Operators

Article / Updated 03-26-2016

Because spin is a type of built-in angular momentum, spin operators have a lot in common with orbital angular momentum operators. As your quantum physics instructor will tell you, there are analogous spin operators, S2 and Sz, to orbital angular momentum operators L2 and Lz. However, these operators are just operators; they don’t have a differential form like the orbital angular momentum operators do. In fact, all the orbital angular momentum operators, such as Lx, Ly, and Lz, have analogs here: Sx, Sy, and Sz. The commutation relations among Lx, Ly, and Lz are the following: And they work the same way for spin: The L2 operator gives you the following result when you apply it to an orbital angular momentum eigenstate: And just as you’d expect, the S2 operator works in an analogous fashion: The Lz operator gives you this result when you apply it to an orbital angular momentum eigenstate: And by analogy, the Sz operator works this way: What about the raising and lowering operators, L+ and L–? Are there analogs for spin? In angular momentum terms, L+ and L– work like this: There are spin raising and lowering operators as well, S+ and S–, and they work like this:

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Quantum Physics How to Find Angular Momentum Eigenvalues

Article / Updated 03-26-2016

When you have the eigenvalues of angular momentum states in quantum mechanics, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum. The eigenvalues of the angular momentum are the possible values the angular momentum can take. Here’s how to derive eigenstate equations with Note that L2 – Lz2 = Lx2 + Ly2, which is a positive number, so That means that And substituting in and using the fact that the eigenstates are normalized, gives you this: So there’s a maximum possible value of which you can call You can be clever now, because there has to be a state such that you can’t raise any more. Thus, if you apply the raising operator, you get zero: Applying the lowering operator to this also gives you zero: And because that means the following is true: Putting in gives you this: At this point, it’s usual to rename You can say even more. In addition to a there must also be a such that when you apply the lowering operator, L–, you get zero, because you can’t go any lower than And you can apply L+ on this as well: From you know that which gives you the following: And comparing this equation to gives you Note that because you reach by n successive applications of you get the following: Coupling these two equations gives you Therefore, can be either an integer or half an integer (depending on whether n is even or odd). Because and n is a positive number, you can find that So now you have it: The eigenstates are | l, m >. The quantum number of the total angular momentum is l. The quantum number of the angular momentum along the z axis is m. For each l, there are 2l + 1 values of m. For example, if l = 2, then m can equal –2, –1, 0, 1, or 2. You can see a representative L and Lz in the figure. L and Lz. L is the total angular momentum and Lz is the projection of that total angular momentum on the z axis.

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