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Solutions to the Harmonic Oscillator Equation and Hermite Polynomials

  1. Aug 18, 2011 #1
    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?

  2. jcsd
  3. Aug 18, 2011 #2
    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]
  4. Aug 18, 2011 #3


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    Staff: Mentor

    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.
  5. Aug 18, 2011 #4
    Being a solution to the Schrodinger equation and being normalizable are two different things.
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