1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    ˆ
    (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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...