Solutions to the Harmonic Oscillator Equation and Hermite Polynomials

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Discussion Overview

The discussion centers on the relationship between Hermite polynomials and the solutions to the Schrödinger equation for a harmonic oscillator. Participants explore whether Hermite polynomials are the solutions themselves or if they form part of a broader solution that includes an exponential function. The conversation touches on theoretical aspects of quantum mechanics and mathematical properties of the wavefunctions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether Hermite polynomials are the solutions to the Schrödinger equation or if they are part of a solution that includes an exponential function.
  • Another participant asserts that the wavefunction of the n-th excited state of the harmonic oscillator is the product of the n-th Hermite polynomial and an exponential function, specifying the form of the exponential.
  • A different participant argues that Hermite polynomials cannot be solutions of the Schrödinger equation over all x due to their non-normalizability.
  • Another reply clarifies that being a solution to the Schrödinger equation and being normalizable are distinct concepts.

Areas of Agreement / Disagreement

Participants express differing views on the nature of Hermite polynomials as solutions to the Schrödinger equation, with some asserting their role in the wavefunction while others challenge their validity as solutions due to normalizability issues. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations regarding the definitions of normalizability and the conditions under which Hermite polynomials are considered solutions. The discussion does not resolve these aspects.

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How are Hermite Polynomials related to the solutions to the Schrödinger equation for a harmonic oscillator? Are they the solutions themselves, or are the solutions to the equation the product of a Hermite polynomial and an exponential function?

Thanks!
 
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Wavefunction of the n-th excited state of the harmonic oscillator is equal to the n-th Hermite polynomial times {\large e^{-\frac{x^{2}}{4l}}} where l = \sqrt{\frac{\hbar}{2m\omega}}
 
The Hermite polynomials (or any polynomials for that matter) cannot be solutions of the SE over all x because they are not normalizable over all x.
 
Being a solution to the Schrödinger equation and being normalizable are two different things.
 

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