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Hermite Polynomials As Part of the Solution to the Harmonic Oscillator

  1. Feb 7, 2009 #1
    1. The problem statement, all variables and given/known data

    When trying to generate solutions to the harmonic oscillator, I'm trying to use hermite polynomials. I understand that there's a recursive relationship between the hermite polynomials but I'm confused in how each hermite polynomial is generated.

    2. Relevant equations
    Hn+1(p)=2*p*Hn(p)-2n*Hn-1(p) to generate each hermite polynomial
    An+1,k = 2An,k-1 -2*n*an-1,k the formula for the coefficients of the hermite polynomials


    3. The attempt at a solution

    The solutions to the harmonic oscillator's energy levels are a combination of hermite polynomials*gaussian functions*normalization constant

    There's a recursive relationship in trying to generate the hermite polynomials but I don't understand how to use the formula for the coefficients of the hermite polynomials. I have that H0=1 and H1=2p as the base cases for the hermite polynomials, but how do I make use of the coefficient formula to generate more hermite polynomials? Can't I just use recursive relationships and have H2 for example depend on H0 and H1? Also, I've read that there are odd and even solutions for hermite polynomials but I don't see how odd n values and even n values have to be treated separately.

    Thanks.
     
  2. jcsd
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