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A shortcut to the eigenvalues of the harmonic oscillator and the hydrogen atom.

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Title: A shortcut to the eigenvalues of the harmonic oscillator and the hydrogen atom.
Authors: Zhang, J. M.1,2 (AUTHOR) wdlang06@gmail.com, Lin, R. K.2,3 (AUTHOR) linrongkang413@gmail.com
Source: American Journal of Physics. Jul2026, Vol. 94 Issue 7, p582-584. 3p.
Subjects: Eigenvalues, Harmonic oscillators, Hydrogen atom, Energy levels (Quantum mechanics), Differential operators, Quantum mechanics, Wave functions
Abstract: Textbook solutions for the harmonic oscillator and the hydrogen atom using the Sommerfeld polynomial method often appear tedious and formidable to students. However, the differential operators acting on the reduced wave functions possess a convenient algebraic property: they preserve the subspace of polynomials of degree at most k. This observation makes the derivation of energy eigenvalues transparent and direct, without the need for the full power series recurrence relations. Editor's Note: This Note will delight quantum mechanics instructors, as it points out a simple way to derive the eigenstates and energies of the harmonic oscillator and hydrogen atom potentials. Do not miss it! [ABSTRACT FROM AUTHOR]
Copyright of American Journal of Physics is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 194749461
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
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  Data: A shortcut to the eigenvalues of the harmonic oscillator and the hydrogen atom.
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  Data: <searchLink fieldCode="AR" term="%22Zhang%2C+J%2E+M%2E%22">Zhang, J. M.</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> wdlang06@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Lin%2C+R%2E+K%2E%22">Lin, R. K.</searchLink><relatesTo>2,3</relatesTo> (AUTHOR)<i> linrongkang413@gmail.com</i>
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  Data: <searchLink fieldCode="JN" term="%22American+Journal+of+Physics%22">American Journal of Physics</searchLink>. Jul2026, Vol. 94 Issue 7, p582-584. 3p.
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  Data: <searchLink fieldCode="DE" term="%22Eigenvalues%22">Eigenvalues</searchLink><br /><searchLink fieldCode="DE" term="%22Harmonic+oscillators%22">Harmonic oscillators</searchLink><br /><searchLink fieldCode="DE" term="%22Hydrogen+atom%22">Hydrogen atom</searchLink><br /><searchLink fieldCode="DE" term="%22Energy+levels+%28Quantum+mechanics%29%22">Energy levels (Quantum mechanics)</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+operators%22">Differential operators</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+mechanics%22">Quantum mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Wave+functions%22">Wave functions</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: Textbook solutions for the harmonic oscillator and the hydrogen atom using the Sommerfeld polynomial method often appear tedious and formidable to students. However, the differential operators acting on the reduced wave functions possess a convenient algebraic property: they preserve the subspace of polynomials of degree at most k. This observation makes the derivation of energy eigenvalues transparent and direct, without the need for the full power series recurrence relations. Editor's Note: This Note will delight quantum mechanics instructors, as it points out a simple way to derive the eigenstates and energies of the harmonic oscillator and hydrogen atom potentials. Do not miss it! [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of American Journal of Physics is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1119/5.0332743
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      – Code: eng
        Text: English
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        PageCount: 3
        StartPage: 582
    Subjects:
      – SubjectFull: Eigenvalues
        Type: general
      – SubjectFull: Harmonic oscillators
        Type: general
      – SubjectFull: Hydrogen atom
        Type: general
      – SubjectFull: Energy levels (Quantum mechanics)
        Type: general
      – SubjectFull: Differential operators
        Type: general
      – SubjectFull: Quantum mechanics
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      – SubjectFull: Wave functions
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      – TitleFull: A shortcut to the eigenvalues of the harmonic oscillator and the hydrogen atom.
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              M: 07
              Text: Jul2026
              Type: published
              Y: 2026
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