{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T14:04:23+00:00","modifiedTime":"2022-09-22T20:38:33+00:00","timestamp":"2022-09-23T00:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Quantum Physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"},"slug":"quantum-physics","categoryId":33770}],"title":"How to Calculate the Energy Degeneracy of a Hydrogen Atom","strippedTitle":"how to calculate the energy degeneracy of a hydrogen atom","slug":"how-to-calculate-the-energy-degeneracy-of-a-hydrogen-atom-in-terms-of-n-l-and-m","canonicalUrl":"","seo":{"metaDescription":"Learn how to determine how many of quantum states of the hydrogen atom (n, l, m) have the same energy, meaning the energy degeneracy.","noIndex":0,"noFollow":0},"content":"Each quantum state of the hydrogen atom is specified with three quantum numbers: <i>n</i> (the principal quantum number), <i>l</i> (the angular momentum quantum number of the electron), and <i>m</i> (the <i>z</i> component of the electron’s angular momentum,\r\n\r\n<img src=\"https://sg.cdnki.com/what-is-the-orbital-degeneracy-of-the-level-that-has-energy-in-hydrogen-atom-where-rh-is-rydberg-constant-for-the-hydrogen-atom---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM5NzkyNS5pbWFnZTAucG5n.webp\" alt=\"image0.png\" width=\"88\" height=\"27\" />\r\n\r\nHow many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers <i>n</i>, <i>l</i>, and <i>m</i>?\r\n\r\nWell, the actual energy is just dependent on <i>n</i>, as you see in the following equation:\r\n\r\n<img src=\"https://sg.cdnki.com/what-is-the-orbital-degeneracy-of-the-level-that-has-energy-in-hydrogen-atom-where-rh-is-rydberg-constant-for-the-hydrogen-atom---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM5NzkyNi5pbWFnZTEucG5n.webp\" alt=\"image1.png\" width=\"168\" height=\"43\" />\r\n\r\nThat means the E is independent of <i>l</i> and <i>m</i>. So how many states, |<i>n</i>, <i>l</i>, <i>m</i>>, have the same energy for a particular value of <i>n</i>? Well, for a particular value of <i>n</i>, <i>l</i> can range from zero to <i>n</i> – 1. And each <i>l</i> can have different values of <i>m</i>, so the total degeneracy is\r\n\r\n<img src=\"https://sg.cdnki.com/what-is-the-orbital-degeneracy-of-the-level-that-has-energy-in-hydrogen-atom-where-rh-is-rydberg-constant-for-the-hydrogen-atom---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM5NzkyNy5pbWFnZTIucG5n.webp\" alt=\"image2.png\" width=\"235\" height=\"48\" />\r\n\r\nThe degeneracy in <i>m</i> is the number of states with different values of <i>m</i> that have the same value of <i>l</i>. For any particular value of <i>l</i>, you can have <i>m</i> values of –<i>l</i>, –<i>l</i> + 1, ..., 0, ..., <i>l</i> – 1, <i>l</i>. And that’s (2<i>l</i> + 1) possible <i>m</i> states for a particular value of <i>l</i>. So you can plug in (2<i>l </i>+ 1) for the degeneracy in <i>m</i>:\r\n\r\n<img src=\"https://sg.cdnki.com/what-is-the-orbital-degeneracy-of-the-level-that-has-energy-in-hydrogen-atom-where-rh-is-rydberg-constant-for-the-hydrogen-atom---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM5NzkyOC5pbWFnZTMucG5n.webp\" alt=\"image3.png\" width=\"159\" height=\"48\" />\r\n\r\nAnd this series works out to be just <i>n</i><sup>2</sup>.\r\n\r\nSo the degeneracy of the energy levels of the hydrogen atom is <i>n</i><sup>2</sup>. For example, the ground state, <i>n</i> = 1, has degeneracy = <i>n</i><sup>2</sup> = 1 (which makes sense because <i>l</i>, and therefore <i>m</i>, can only equal zero for this state).\r\n\r\nFor <i>n</i> = 2, you have a degeneracy of 4:\r\n\r\n<img src=\"https://sg.cdnki.com/what-is-the-orbital-degeneracy-of-the-level-that-has-energy-in-hydrogen-atom-where-rh-is-rydberg-constant-for-the-hydrogen-atom---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzM5NzkyOS5pbWFnZTQucG5n.webp\" alt=\"image4.png\" width=\"93\" height=\"128\" />\r\n\r\nCool.","description":"Each quantum state of the hydrogen atom is specified with three quantum numbers: <i>n</i> (the principal quantum number), <i>l</i> (the angular momentum quantum number of the electron), and <i>m</i> (the <i>z</i> component of the electron’s angular momentum,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397925.image0.png\" alt=\"image0.png\" width=\"88\" height=\"27\" />\r\n\r\nHow many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers <i>n</i>, <i>l</i>, and <i>m</i>?\r\n\r\nWell, the actual energy is just dependent on <i>n</i>, as you see in the following equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397926.image1.png\" alt=\"image1.png\" width=\"168\" height=\"43\" />\r\n\r\nThat means the E is independent of <i>l</i> and <i>m</i>. So how many states, |<i>n</i>, <i>l</i>, <i>m</i>>, have the same energy for a particular value of <i>n</i>? Well, for a particular value of <i>n</i>, <i>l</i> can range from zero to <i>n</i> – 1. And each <i>l</i> can have different values of <i>m</i>, so the total degeneracy is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397927.image2.png\" alt=\"image2.png\" width=\"235\" height=\"48\" />\r\n\r\nThe degeneracy in <i>m</i> is the number of states with different values of <i>m</i> that have the same value of <i>l</i>. For any particular value of <i>l</i>, you can have <i>m</i> values of –<i>l</i>, –<i>l</i> + 1, ..., 0, ..., <i>l</i> – 1, <i>l</i>. And that’s (2<i>l</i> + 1) possible <i>m</i> states for a particular value of <i>l</i>. So you can plug in (2<i>l </i>+ 1) for the degeneracy in <i>m</i>:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397928.image3.png\" alt=\"image3.png\" width=\"159\" height=\"48\" />\r\n\r\nAnd this series works out to be just <i>n</i><sup>2</sup>.\r\n\r\nSo the degeneracy of the energy levels of the hydrogen atom is <i>n</i><sup>2</sup>. For example, the ground state, <i>n</i> = 1, has degeneracy = <i>n</i><sup>2</sup> = 1 (which makes sense because <i>l</i>, and therefore <i>m</i>, can only equal zero for this state).\r\n\r\nFor <i>n</i> = 2, you have a degeneracy of 4:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/397929.image4.png\" alt=\"image4.png\" width=\"93\" height=\"128\" />\r\n\r\nCool.","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33770,"title":"Quantum Physics","slug":"quantum-physics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33770"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}},{"articleId":161817,"title":"How Spin Operators Resemble Angular Momentum Operators","slug":"how-spin-operators-resemble-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161817"}},{"articleId":161816,"title":"How to Find Angular Momentum Eigenvalues","slug":"how-to-find-angular-momentum-eigenvalues","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161816"}},{"articleId":161814,"title":"Translate the Schrödinger Equation to Three Dimensions","slug":"translate-the-schrdinger-equation-to-three-dimensions","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161814"}}],"fromCategory":[{"articleId":208083,"title":"Quantum Physics For Dummies Cheat Sheet","slug":"quantum-physics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208083"}},{"articleId":194414,"title":"The Laws of Quantum Physics: The Schrödinger Equation","slug":"the-laws-of-quantum-physics-the-schrdinger-equation","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/194414"}},{"articleId":170679,"title":"Spin Operators and Commutation in Quantum Physics","slug":"spin-operators-and-commutation-in-quantum-physics","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/170679"}},{"articleId":161819,"title":"Find the Eigenfunctions of L<i><sub>z</sub></i> in Spherical Coordinates","slug":"find-the-eigenfunctions-of-lz-in-spherical-coordinates","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161819"}},{"articleId":161818,"title":"Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators","slug":"find-the-eigenvalues-of-the-raising-and-lowering-angular-momentum-operators","categoryList":["academics-the-arts","science","quantum-physics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/161818"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282518,"slug":"quantum-physics-for-dummies-revised-edition","isbn":"9781118460825","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"https://www.amazon.com/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118460820-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118460820/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/quantum-physics-for-dummies-revised-edition-cover-9781118460825-203x255.jpg","width":203,"height":255},"title":"Quantum Physics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8967\">Steven Holzner</b> is an award-winning author of technical and science books (like <i>Physics For Dummies</i> and <i>Differential Equations For Dummies</i>). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8967"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{"key":"cat","values":["academics-the-arts","science","quantum-physics"]},{"key":"isbn","values":["9781118460825"]}]\" id=\"du-slot-632cf73e7c3e2\"></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 = 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How many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers n, l, and m? Well, the actual energy is just dependent on n, as you see in the following equation: That means the E is independent of l and m. So how many states, |n, l, m>, have the same energy for a particular value of n? Well, for a particular value of n, l can range from zero to n – 1. And each l can have different values of m, so the total degeneracy is The degeneracy in m is the number of states with different values of m that have the same value of l. For any particular value of l, you can have m values of –l, –l + 1, ..., 0, ..., l – 1, l. And that’s (2l + 1) possible m states for a particular value of l. So you can plug in (2l + 1) for the degeneracy in m: And this series works out to be just n2. So the degeneracy of the energy levels of the hydrogen atom is n2. For example, the ground state, n = 1, has degeneracy = n2 = 1 (which makes sense because l, and therefore m, can only equal zero for this state). For n = 2, you have a degeneracy of 4: Cool. |