Solutions to the Harmonic Oscillator Equation and Hermite Polynomials

  • #1

Main Question or Discussion Point

How are Hermite Polynomials related to the solutions to the Schrodinger 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!!!
 

Answers and Replies

  • #2
135
1
Wavefunction of the n-th excited state of the harmonic oscillator is equal to the n-th Hermite polynomial times [itex]{\large e^{-\frac{x^{2}}{4l}}}[/itex] where [itex]l = \sqrt{\frac{\hbar}{2m\omega}}[/itex]
 
  • #3
jtbell
Mentor
15,577
3,555
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.
 
  • #4
166
0
Being a solution to the Schrodinger equation and being normalizable are two different things.
 

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