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!

Eigenvalues/functions of a mixed potential

  1. Oct 6, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the first two energy eigenfunctions and eigenvalues for a particle in a potential

    [tex]
    V(x)=\frac{1}{2}m\omega_0^2\left(x^2-2cx\right)
    [/tex]



    2. Relevant equations
    [tex]
    H=\frac{p^2}{2m}+V(x)=\frac{p^2}{2m}+\frac{1}{2}m\omega_0^2\left(x^2-2cx\right)
    [/tex]

    [tex]
    -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+\frac{1}{2}m\omega_0^2x^2\psi(x)-m\omega_0^2cx\psi(x)=E\psi(x)
    [/tex]


    3. The attempt at a solution

    I think what is confusing me is having the mixed potential, which is the whole point of the problem. I have tried using a completing the square and making the substitution of

    [tex]
    \frac{1}{2}m\omega_0^2\left(x^2-2cx\right)\rightarrow\frac{1}{2}m\omega_0^2\left(u^2-c^2\right)
    [/tex]

    but not really sure where this leads me (thought I might've ended up with a Bessel function or. I also tried solving it without substitution but end up with a quadratic in x differential equation

    [tex]
    y''+(ax^2-bx-c)y=0
    [/tex]

    but unsure of where to start with this as wikipedia says this is the Weber-Hermite function, but I don't know what to do with this, at least not from what wiki says.

    Any pointers as to where I should go from either starting point? (Or perhaps a starting point not considered that would be a better option)
     
  2. jcsd
  3. Oct 7, 2009 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Hi jdwood 983, welcome to PF!:smile:

    I haven't tackled the problem myself yet, but you might try solving it in the momentum basis instead and see if that leaves you with something less complicated...
     
  4. Oct 7, 2009 #3
    The professor said he wants this done in coordinate space, so that's a no go :(

    Thanks for the welcome!
     
  5. Oct 7, 2009 #4

    kuruman

    User Avatar
    Homework Helper
    Gold Member

    This leads you to

    [tex]-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi(u)}{du^{2}}+\frac{1}{2}m\omega^{2}u^{2}\psi(u)=(E+\frac{1}{2}m\omega^{2}c^{2}})\psi(u)[/tex]

    It is ye olde harmonicke oscillator equation with the origin shifted by c and the "zero of energy" shifted by -(1/2)mω2c2. A parabola is a parabola is a parabola.
     
  6. Oct 7, 2009 #5
    Wow, I was too busy looking at special differential functions that I didn't even see that one. Thanks for your help!!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook