{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T14:08:46+00:00","modifiedTime":"2016-03-26T14:08:46+00:00","timestamp":"2022-09-14T18:04:01+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 the Wave Function of the Ground State of a Quantum Oscillator","strippedTitle":"how to find the wave function of the ground state of a quantum oscillator","slug":"how-to-find-the-wave-function-of-the-ground-state-of-a-quantum-oscillator","canonicalUrl":"","seo":{"metaDescription":"In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a ","noIndex":0,"noFollow":0},"content":"<p>In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve.</p>\n<div class=\"imageBlock\" style=\"width:535px;\"><img src=\"https://www.dummies.com/wp-content/uploads/395332.image0.jpg\" width=\"535\" height=\"169\" alt=\"The ground state of a quantum mechanical harmonic oscillator.\"/><div class=\"imageCaption\">The ground state of a quantum mechanical harmonic oscillator.</div></div>\n<p>As a gaussian curve, the ground state of a quantum oscillator is </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395333.image1.png\" width=\"144\" height=\"53\" alt=\"image1.png\"/>\n<p>How can you figure out A? Wave functions must be normalized, so the following has to be true:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395334.image2.png\" width=\"109\" height=\"51\" alt=\"image2.png\"/>\n<p>Substituting for </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395335.image3.png\" width=\"45\" height=\"27\" alt=\"image3.png\"/>\n<p>gives you this next equation:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395336.image4.png\" width=\"172\" height=\"128\" alt=\"image4.png\"/>\n<p>You can evaluate this integral to be</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395337.image5.png\" width=\"365\" height=\"61\" alt=\"image5.png\"/>\n<p>Therefore,</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395338.image6.png\" width=\"85\" height=\"92\" alt=\"image6.png\"/>\n<p>This means that the wave function for the ground state of a quantum mechanical harmonic oscillator is</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395339.image7.png\" width=\"181\" height=\"53\" alt=\"image7.png\"/>\n<p>Cool. Now you’ve got an exact wave function.</p>","description":"<p>In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve.</p>\n<div class=\"imageBlock\" style=\"width:535px;\"><img src=\"https://www.dummies.com/wp-content/uploads/395332.image0.jpg\" width=\"535\" height=\"169\" alt=\"The ground state of a quantum mechanical harmonic oscillator.\"/><div class=\"imageCaption\">The ground state of a quantum mechanical harmonic oscillator.</div></div>\n<p>As a gaussian curve, the ground state of a quantum oscillator is </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395333.image1.png\" width=\"144\" height=\"53\" alt=\"image1.png\"/>\n<p>How can you figure out A? Wave functions must be normalized, so the following has to be true:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395334.image2.png\" width=\"109\" height=\"51\" alt=\"image2.png\"/>\n<p>Substituting for </p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395335.image3.png\" width=\"45\" height=\"27\" alt=\"image3.png\"/>\n<p>gives you this next equation:</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395336.image4.png\" width=\"172\" height=\"128\" alt=\"image4.png\"/>\n<p>You can evaluate this integral to be</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395337.image5.png\" width=\"365\" height=\"61\" alt=\"image5.png\"/>\n<p>Therefore,</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395338.image6.png\" width=\"85\" height=\"92\" alt=\"image6.png\"/>\n<p>This means that the wave function for the ground state of a quantum mechanical harmonic oscillator is</p>\n<img src=\"https://www.dummies.com/wp-content/uploads/395339.image7.png\" width=\"181\" height=\"53\" alt=\"image7.png\"/>\n<p>Cool. Now you’ve got an exact wave function.</p>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). 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How to Find the Wave Function of the Ground State of a Quantum Oscillator

By: Steven Holzner and

Updated: 03-26-2016

From The Book: Quantum Physics For Dummies

Quantum Physics For Dummies

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In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve.

The ground state of a quantum mechanical harmonic oscillator.

As a gaussian curve, the ground state of a quantum oscillator is

How can you figure out A? Wave functions must be normalized, so the following has to be true:

Substituting for

gives you this next equation:

You can evaluate this integral to be

Therefore,

This means that the wave function for the ground state of a quantum mechanical harmonic oscillator is

Cool. Now you’ve got an exact wave function.

About This Article

This article is from the book:

Quantum Physics For Dummies ,

About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). 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.

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