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Homework Help: Quantum mechanics hermite polynomials

  1. Mar 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that the one-dimensional Schr¨odinger equation
    (p^2/2m) ψ+ 1/2(mw^2)(x)ψ = En ψ
    can be transformed into
    (d^2/d ξ ^2)ψ+ (λn- ξ ^2) ψ= 0 where λn = 2n + 1.
    using hermite polynomials

    2. Relevant equations

    know that dHn(X)/dX= 2nHn(x)

    3. The attempt at a solution
    I don't know how to start this question off at all
  2. jcsd
  3. Mar 19, 2009 #2
    The question is a bit unclear to me. If I understand it correctly you should do the following:
    1. get the first equation to the form
    [tex]$ d^2\psi/dx^2 + 2m/\hbar^2(E-m\omega^2 x^2/2) \psi = 0$[/tex]
    2. substitute [tex]$ \xi = x \sqrt{m\omega/\hbar}$[/tex]
    3. you should get
    [tex]$ d^2\psi/d\xi^2 + (2E/\hbar \omega - \xi^2) \psi = 0$[/tex]
    4. now substitute [tex]$\psi = \varphi(\xi) \exp(-\xi^2/2)$[/tex] to get
    5. [tex]$ d^2\varphi/d\xi^2 - 2\xi d\varphi/d\xi + (2E/\hbar\omega-1)\varphi = 0$[/tex]
    This equation has solutions that diverge at infinity not faster than a polynomial - Hermite polynomials - only when [tex]$ 2E/\hbar\omega-1 = 2n$[/tex], where [tex]$ n = 0,1,2,...$[/tex]

    Hope that helps!
  4. Mar 19, 2009 #3
    Hey thats perfest thanks was stuck on how to get ξ into the equation.
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