{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2022-05-27T12:31:27+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"}},"_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":102,"items":[{"headers":{"creationTime":"2016-03-27T16:50:13+00:00","modifiedTime":"2022-02-14T15:21:18+00:00","timestamp":"2022-02-24T17:07:31+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":"Quantum Physics For Dummies Cheat Sheet","strippedTitle":"quantum physics for dummies cheat sheet","slug":"quantum-physics-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This Cheat Sheet provides a quick reference to some of the main equations used in the study of quantum physics.","noIndex":0,"noFollow":0},"content":"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.","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-6217bb5362b83\"></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-6217bb53634f7\"></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}},"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":"2022-02-24T16:57:53+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":"","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 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":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-6217b91156d30\"></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-6217b9115770a\"></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}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":194414},{"headers":{"creationTime":"2016-03-26T15:48:26+00:00","modifiedTime":"2016-03-26T15:48:26+00:00","timestamp":"2022-02-24T16:53:11+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":"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":"","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-6217b7f7d0b39\"></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-6217b7f7d15a3\"></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}},"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":"2022-02-24T16:50:57+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":"","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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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, Revised Edition","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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-6217b771e7c7d\"></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-6217b771e860d\"></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}},"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":"2022-02-24T16:50:57+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":"","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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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, Revised Edition","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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-6217b771dfa91\"></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-6217b771e06ab\"></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}},"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":"2022-02-24T16:50:57+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":"","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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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, Revised Edition","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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-6217b771d7545\"></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-6217b771d7ec9\"></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}},"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":"2022-02-24T16:50:57+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":"","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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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, Revised Edition","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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-6217b771cf32f\"></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-6217b771cfca4\"></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}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":161816},{"headers":{"creationTime":"2016-03-26T14:09:36+00:00","modifiedTime":"2016-03-26T14:09:36+00:00","timestamp":"2022-02-24T16:50:57+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":"Translate the Schrödinger Equation to Three Dimensions","strippedTitle":"translate the schrödinger equation to three dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"<p>In quantum physics, you can break the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations to make it easier to solve 3D problems. In one dimension, the time-dependent Schrödinger equation (which lets you find a wave function) looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395134.image0.png\" width=\"281\" height=\"43\" alt=\"image0.png\"/>\n<p>And you can generalize that into three dimensions like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395135.image1.png\" width=\"437\" height=\"51\" alt=\"image1.png\"/>\n<p>Using the Laplacian operator, you can recast this into a more compact form. Here’s what the Laplacian looks like:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395136.image2.png\" width=\"153\" height=\"51\" alt=\"image2.png\"/>\n<p>And here’s the 3D Schrödinger equation using the Laplacian:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395137.image3.png\" width=\"375\" height=\"40\" alt=\"image3.png\"/>\n<p>To solve this equation, when the potential doesn’t vary with time, break out the time-dependent part of the wave function:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395138.image4.png\" width=\"187\" height=\"28\" alt=\"image4.png\"/>\n<p>Here, </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395139.image5.png\" width=\"63\" height=\"27\" alt=\"image5.png\"/>\n<p>is the solution of the time-independent Schrödinger equation, and E is the energy:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395140.image6.png\" width=\"321\" height=\"40\" alt=\"image6.png\"/>\n<p>So far, so good. But now you’ve run into a wall — the expression</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395141.image7.png\" width=\"80\" height=\"28\" alt=\"image7.png\"/>\n<p>is in general very hard to deal with, so the current equation is in general very hard to solve.</p>\n<p>So what should you do? Well, you can focus on the case in which the equation is separable — that is, where you can separate out the <i>x</i>, <i>y</i>, and <i>z</i> dependence and find the solution in each dimension separately. In other words, in separable cases, the potential, V(<i>x</i>, <i>y</i>, <i>z</i>), is actually the sum of the <i>x</i>, <i>y</i>, and <i>z</i> potentials:</p>\n<p>V(<i>x</i>, <i>y</i>, <i>z</i>) = V<i><sub>x</sub></i>(<i>x</i>) + V<i><sub>y</sub></i>(<i>y</i>) + V<i><sub>z</sub></i>(<i>z</i>)</p>\n<p>Now you can break the Hamiltonian in</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395142.image8.png\" width=\"321\" height=\"40\" alt=\"image8.png\"/>\n<p>into three Hamilitonians, H<i><sub>x</sub></i>, H<i><sub>y</sub></i>, and H<i><sub>z</sub></i>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395143.image9.png\" width=\"236\" height=\"29\" alt=\"image9.png\"/>\n<p>where</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395144.image10.png\" width=\"157\" height=\"157\" alt=\"image10.png\"/>\n<p>When you divide up the Hamiltonian as in </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395145.image11.png\" width=\"239\" height=\"29\" alt=\"image11.png\"/>\n<p>you can also divide up the wave function that solves that equation. In particular, you can break the wave function into three parts, one for <i>x</i>, <i>y</i>, and <i>z</i>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395146.image12.png\" width=\"176\" height=\"27\" alt=\"image12.png\"/>\n<p>Where X(<i>x</i>), Y(<i>y</i>), and Z(<i>z</i>) are functions of the coordinates <i>x</i>, <i>y</i>, and <i>z</i> and are not to be confused with the position operators. This separation of the wave function into three parts is going to make life considerably easier, because now you can break the Hamiltonian up into three separate operators added together:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395147.image13.png\" width=\"415\" height=\"51\" alt=\"image13.png\"/>\n<p>E = E<i><sub>x</sub></i> + E<i><sub>y</sub></i> + E<i><sub>z</sub></i></p>\n<p>So you now have three independent Schrödinger equations for the three dimensions:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395148.image14.png\" width=\"251\" height=\"157\" alt=\"image14.png\"/>\n<p>This system of independent differential equations looks a lot easier to solve than </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395149.image15.png\" width=\"239\" height=\"29\" alt=\"image15.png\"/>\n<p>In essence, you’ve broken the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations. That makes solving 3D problems tractable.</p>","description":"<p>In quantum physics, you can break the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations to make it easier to solve 3D problems. In one dimension, the time-dependent Schrödinger equation (which lets you find a wave function) looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395134.image0.png\" width=\"281\" height=\"43\" alt=\"image0.png\"/>\n<p>And you can generalize that into three dimensions like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395135.image1.png\" width=\"437\" height=\"51\" alt=\"image1.png\"/>\n<p>Using the Laplacian operator, you can recast this into a more compact form. Here’s what the Laplacian looks like:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395136.image2.png\" width=\"153\" height=\"51\" alt=\"image2.png\"/>\n<p>And here’s the 3D Schrödinger equation using the Laplacian:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395137.image3.png\" width=\"375\" height=\"40\" alt=\"image3.png\"/>\n<p>To solve this equation, when the potential doesn’t vary with time, break out the time-dependent part of the wave function:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395138.image4.png\" width=\"187\" height=\"28\" alt=\"image4.png\"/>\n<p>Here, </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395139.image5.png\" width=\"63\" height=\"27\" alt=\"image5.png\"/>\n<p>is the solution of the time-independent Schrödinger equation, and E is the energy:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395140.image6.png\" width=\"321\" height=\"40\" alt=\"image6.png\"/>\n<p>So far, so good. But now you’ve run into a wall — the expression</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395141.image7.png\" width=\"80\" height=\"28\" alt=\"image7.png\"/>\n<p>is in general very hard to deal with, so the current equation is in general very hard to solve.</p>\n<p>So what should you do? Well, you can focus on the case in which the equation is separable — that is, where you can separate out the <i>x</i>, <i>y</i>, and <i>z</i> dependence and find the solution in each dimension separately. In other words, in separable cases, the potential, V(<i>x</i>, <i>y</i>, <i>z</i>), is actually the sum of the <i>x</i>, <i>y</i>, and <i>z</i> potentials:</p>\n<p>V(<i>x</i>, <i>y</i>, <i>z</i>) = V<i><sub>x</sub></i>(<i>x</i>) + V<i><sub>y</sub></i>(<i>y</i>) + V<i><sub>z</sub></i>(<i>z</i>)</p>\n<p>Now you can break the Hamiltonian in</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395142.image8.png\" width=\"321\" height=\"40\" alt=\"image8.png\"/>\n<p>into three Hamilitonians, H<i><sub>x</sub></i>, H<i><sub>y</sub></i>, and H<i><sub>z</sub></i>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395143.image9.png\" width=\"236\" height=\"29\" alt=\"image9.png\"/>\n<p>where</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395144.image10.png\" width=\"157\" height=\"157\" alt=\"image10.png\"/>\n<p>When you divide up the Hamiltonian as in </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395145.image11.png\" width=\"239\" height=\"29\" alt=\"image11.png\"/>\n<p>you can also divide up the wave function that solves that equation. In particular, you can break the wave function into three parts, one for <i>x</i>, <i>y</i>, and <i>z</i>:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395146.image12.png\" width=\"176\" height=\"27\" alt=\"image12.png\"/>\n<p>Where X(<i>x</i>), Y(<i>y</i>), and Z(<i>z</i>) are functions of the coordinates <i>x</i>, <i>y</i>, and <i>z</i> and are not to be confused with the position operators. This separation of the wave function into three parts is going to make life considerably easier, because now you can break the Hamiltonian up into three separate operators added together:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395147.image13.png\" width=\"415\" height=\"51\" alt=\"image13.png\"/>\n<p>E = E<i><sub>x</sub></i> + E<i><sub>y</sub></i> + E<i><sub>z</sub></i></p>\n<p>So you now have three independent Schrödinger equations for the three dimensions:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395148.image14.png\" width=\"251\" height=\"157\" alt=\"image14.png\"/>\n<p>This system of independent differential equations looks a lot easier to solve than </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395149.image15.png\" width=\"239\" height=\"29\" alt=\"image15.png\"/>\n<p>In essence, you’ve broken the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations. That makes solving 3D problems tractable.</p>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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":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, Revised Edition","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.","_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-6217b771c7422\"></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-6217b771c7dba\"></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}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":161814},{"headers":{"creationTime":"2016-03-26T14:09:36+00:00","modifiedTime":"2016-03-26T14:09:36+00:00","timestamp":"2022-02-24T16:50:57+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":"Spin One-Half Matrices","strippedTitle":"spin one-half matrices","slug":"spin-one-half-matrices","canonicalUrl":"","seo":{"metaDescription":"","noIndex":0,"noFollow":0},"content":"<p>In quantum physics, when you look at the spin eigenstates and operators for particles of spin 1/2 in terms of matrices, there are only two possible states, spin up and spin down. </p>\n<p>The eigenvalues of the S<sup>2</sup> operator are</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395103.image0.png\" width=\"252\" height=\"37\" alt=\"image0.png\"/>\n<p>and the eigenvalues of the S<sub>z</sub> operator are</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395104.image1.png\" width=\"199\" height=\"37\" alt=\"image1.png\"/>\n<p>You can represent these two equations graphically as shown in the following figure, where the two spin states have different projections along the <i>z</i> axis.</p>\n<div class=\"imageBlock\" style=\"width:350px;\"><img src=\"https://www.dummies.com/wp-content/uploads/395105.image2.jpg\" width=\"350\" height=\"414\" alt=\"Spin magnitude and <i>z</i> projection.\"/><div class=\"imageCaption\">Spin magnitude and <i>z</i> projection.</div></div>\n<p>In the case of spin 1/2 matrices, you first represent the eigenstate </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395106.image3.png\" width=\"61\" height=\"35\" alt=\"image3.png\"/>\n<p>like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395107.image4.png\" width=\"113\" height=\"48\" alt=\"image4.png\"/>\n<p>And the eigenstate </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395108.image5.png\" width=\"71\" height=\"35\" alt=\"image5.png\"/>\n<p>looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395109.image6.png\" width=\"124\" height=\"48\" alt=\"image6.png\"/>\n<p>Now what about spin operators like S<sup>2</sup>? The S<sup>2</sup> operator looks like this in matrix terms:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395110.image7.png\" width=\"377\" height=\"80\" alt=\"image7.png\"/>\n<p>And this works out to be the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395111.image8.png\" width=\"125\" height=\"48\" alt=\"image8.png\"/>\n<p>Similarly, you can represent the S<i><sub>z</sub></i> operator this way:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395112.image9.png\" width=\"376\" height=\"80\" alt=\"image9.png\"/>\n<p>This works out to</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395113.image10.png\" width=\"113\" height=\"48\" alt=\"image10.png\"/>\n<p>Using the matrix version of S<i><sub>z</sub></i>, for example, you can find the <i>z</i> component of the spin of, say, the eigenstate </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395114.image11.png\" width=\"76\" height=\"35\" alt=\"image11.png\"/>\n<p>Finding the <i>z</i> component looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395115.image12.png\" width=\"91\" height=\"35\" alt=\"image12.png\"/>\n<p>Putting this in matrix terms gives you this matrix product:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395116.image13.png\" width=\"120\" height=\"48\" alt=\"image13.png\"/>\n<p>Here’s what you get by performing the matrix multiplication:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395117.image14.png\" width=\"193\" height=\"48\" alt=\"image14.png\"/>\n<p>And putting this back into ket notation, you get the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395118.image15.png\" width=\"191\" height=\"37\" alt=\"image15.png\"/>\n<p>How about the raising and lowering operators S<sub>+</sub> and S<sub>–</sub>? The S<sub>+</sub> operator looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395119.image16.png\" width=\"107\" height=\"48\" alt=\"image16.png\"/>\n<p>And the lowering operator looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395120.image17.png\" width=\"107\" height=\"48\" alt=\"image17.png\"/>\n<p>Here it is in matrix terms:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395121.image18.png\" width=\"113\" height=\"48\" alt=\"image18.png\"/>\n<p>Performing the multiplication gives you this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395122.image19.png\" width=\"176\" height=\"48\" alt=\"image19.png\"/>\n<p>Or in ket form, it’s </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395123.image20.png\" width=\"176\" height=\"35\" alt=\"image20.png\"/>\n<p>Cool.</p>","description":"<p>In quantum physics, when you look at the spin eigenstates and operators for particles of spin 1/2 in terms of matrices, there are only two possible states, spin up and spin down. </p>\n<p>The eigenvalues of the S<sup>2</sup> operator are</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395103.image0.png\" width=\"252\" height=\"37\" alt=\"image0.png\"/>\n<p>and the eigenvalues of the S<sub>z</sub> operator are</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395104.image1.png\" width=\"199\" height=\"37\" alt=\"image1.png\"/>\n<p>You can represent these two equations graphically as shown in the following figure, where the two spin states have different projections along the <i>z</i> axis.</p>\n<div class=\"imageBlock\" style=\"width:350px;\"><img src=\"https://www.dummies.com/wp-content/uploads/395105.image2.jpg\" width=\"350\" height=\"414\" alt=\"Spin magnitude and <i>z</i> projection.\"/><div class=\"imageCaption\">Spin magnitude and <i>z</i> projection.</div></div>\n<p>In the case of spin 1/2 matrices, you first represent the eigenstate </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395106.image3.png\" width=\"61\" height=\"35\" alt=\"image3.png\"/>\n<p>like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395107.image4.png\" width=\"113\" height=\"48\" alt=\"image4.png\"/>\n<p>And the eigenstate </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395108.image5.png\" width=\"71\" height=\"35\" alt=\"image5.png\"/>\n<p>looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395109.image6.png\" width=\"124\" height=\"48\" alt=\"image6.png\"/>\n<p>Now what about spin operators like S<sup>2</sup>? The S<sup>2</sup> operator looks like this in matrix terms:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395110.image7.png\" width=\"377\" height=\"80\" alt=\"image7.png\"/>\n<p>And this works out to be the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395111.image8.png\" width=\"125\" height=\"48\" alt=\"image8.png\"/>\n<p>Similarly, you can represent the S<i><sub>z</sub></i> operator this way:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395112.image9.png\" width=\"376\" height=\"80\" alt=\"image9.png\"/>\n<p>This works out to</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395113.image10.png\" width=\"113\" height=\"48\" alt=\"image10.png\"/>\n<p>Using the matrix version of S<i><sub>z</sub></i>, for example, you can find the <i>z</i> component of the spin of, say, the eigenstate </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395114.image11.png\" width=\"76\" height=\"35\" alt=\"image11.png\"/>\n<p>Finding the <i>z</i> component looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395115.image12.png\" width=\"91\" height=\"35\" alt=\"image12.png\"/>\n<p>Putting this in matrix terms gives you this matrix product:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395116.image13.png\" width=\"120\" height=\"48\" alt=\"image13.png\"/>\n<p>Here’s what you get by performing the matrix multiplication:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395117.image14.png\" width=\"193\" height=\"48\" alt=\"image14.png\"/>\n<p>And putting this back into ket notation, you get the following:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395118.image15.png\" width=\"191\" height=\"37\" alt=\"image15.png\"/>\n<p>How about the raising and lowering operators S<sub>+</sub> and S<sub>–</sub>? The S<sub>+</sub> operator looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395119.image16.png\" width=\"107\" height=\"48\" alt=\"image16.png\"/>\n<p>And the lowering operator looks like this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395120.image17.png\" width=\"107\" height=\"48\" alt=\"image17.png\"/>\n<p>Here it is in matrix terms:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395121.image18.png\" width=\"113\" height=\"48\" alt=\"image18.png\"/>\n<p>Performing the multiplication gives you this:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395122.image19.png\" width=\"176\" height=\"48\" alt=\"image19.png\"/>\n<p>Or in ket form, it’s </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395123.image20.png\" width=\"176\" height=\"35\" alt=\"image20.png\"/>\n<p>Cool.</p>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. 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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":"<strong>Steven Holzner</strong> was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. 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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 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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Quantum Physics Translate the Schrödinger Equation to Three Dimensions

Article / Updated 03-26-2016

In quantum physics, you can break the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations to make it easier to solve 3D problems. In one dimension, the time-dependent Schrödinger equation (which lets you find a wave function) looks like this: And you can generalize that into three dimensions like this: Using the Laplacian operator, you can recast this into a more compact form. Here’s what the Laplacian looks like: And here’s the 3D Schrödinger equation using the Laplacian: To solve this equation, when the potential doesn’t vary with time, break out the time-dependent part of the wave function: Here, is the solution of the time-independent Schrödinger equation, and E is the energy: So far, so good. But now you’ve run into a wall — the expression is in general very hard to deal with, so the current equation is in general very hard to solve. So what should you do? Well, you can focus on the case in which the equation is separable — that is, where you can separate out the x, y, and z dependence and find the solution in each dimension separately. In other words, in separable cases, the potential, V(x, y, z), is actually the sum of the x, y, and z potentials: V(x, y, z) = Vx(x) + Vy(y) + Vz(z) Now you can break the Hamiltonian in into three Hamilitonians, Hx, Hy, and Hz: where When you divide up the Hamiltonian as in you can also divide up the wave function that solves that equation. In particular, you can break the wave function into three parts, one for x, y, and z: Where X(x), Y(y), and Z(z) are functions of the coordinates x, y, and z and are not to be confused with the position operators. This separation of the wave function into three parts is going to make life considerably easier, because now you can break the Hamiltonian up into three separate operators added together: E = Ex + Ey + Ez So you now have three independent Schrödinger equations for the three dimensions: This system of independent differential equations looks a lot easier to solve than In essence, you’ve broken the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations. That makes solving 3D problems tractable.

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Quantum Physics Spin One-Half Matrices

Article / Updated 03-26-2016

In quantum physics, when you look at the spin eigenstates and operators for particles of spin 1/2 in terms of matrices, there are only two possible states, spin up and spin down. The eigenvalues of the S2 operator are and the eigenvalues of the Sz operator are You can represent these two equations graphically as shown in the following figure, where the two spin states have different projections along the z axis. Spin magnitude and z projection. In the case of spin 1/2 matrices, you first represent the eigenstate like this: And the eigenstate looks like this: Now what about spin operators like S2? The S2 operator looks like this in matrix terms: And this works out to be the following: Similarly, you can represent the Sz operator this way: This works out to Using the matrix version of Sz, for example, you can find the z component of the spin of, say, the eigenstate Finding the z component looks like this: Putting this in matrix terms gives you this matrix product: Here’s what you get by performing the matrix multiplication: And putting this back into ket notation, you get the following: How about the raising and lowering operators S+ and S–? The S+ operator looks like this: And the lowering operator looks like this: Here it is in matrix terms: Performing the multiplication gives you this: Or in ket form, it’s Cool.

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Quantum Physics How Pair Production and Pair Annihilation Define Light Particles

Article / Updated 03-26-2016

By observing both pair production and pair annihilation, 20th-century physicists were able to prove that light has the characteristics of a particle. This process of discovery began in 1928, when the physicist Paul Dirac posited the existence of a positively charged anti-electron, the positron. He did this by taking the newly evolving field of quantum physics to new territory by combining relativity with quantum mechanics to create relativistic quantum mechanics — and that was the theory that predicted, through a plus/minus–sign interchange — the existence of the positron. It was a bold prediction — an anti-particle of the electron? But just four years later, physicists actually saw the positron. Today's high-powered elementary particle physics has all kinds of synchrotrons and other particle accelerators to create all the elementary particles they need, but in the early 20th century, this wasn't always so. In those days, physicists relied on cosmic rays — those particles and high-powered photons (called gamma rays) that strike the Earth from outer space — as their source of high-energy particles. They used cloud-chambers, which were filled with vapor from dry ice, to see the trails such particles left. They put their chambers into magnetic fields to be able to measure the momentum of the particles as they curved in those fields. In 1932, a physicist noticed a surprising event. A pair of particles, oppositely charged (which could be determined from the way they curved in the magnetic field) appeared from apparently nowhere. No particle trail led to the origin of the two particles that appeared. That was pair-production — the conversion of a high-powered photon into an electron and positron, which can happen when the photon passes near a heavy atomic nucleus. So experimentally, physicists had now seen a photon turning into a pair of particles. Wow. As if everyone needed more evidence of the particle nature of light. Later on, researchers also saw pair annihilation: the conversion of an electron and positron into pure light. Pair production and annihilation turned out to be governed by Einstein's newly introduced theory of relativity — in particular, his most famous formula, E = mc2, which gives the pure energy equivalent of mass. At this point, there was an abundance of evidence of the particle-like aspects of light.

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