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Wave function on slope potential

  1. Jun 24, 2010 #1
    What are solutions to

    [tex]
    \psi''(x) = (a_0 + a_1 x)\psi(x)
    [/tex]
    ?

    First idea I've had was that I could try some kind of perturbation with respect to the [itex]a_1[/itex] variable, so that

    [tex]
    \psi(x) = A_1e^{\sqrt{a_0}x} + A_2e^{-\sqrt{a_0}x} + \psi_1(x)
    [/tex]

    would be an attempt. But I couldn't find anything useful for [itex]\psi_1[/itex] when [itex]a_1\neq 0[/itex], so it got stuck there.

    Second idea was to use Fourier transform to transform the problem into this form

    [tex]
    \phi'(x) = (b_0 + b_1x^2)\phi(x)
    [/tex]

    Unfortunately the solutions

    [tex]
    \phi(x) = Be^{b_0x + \frac{1}{3}b_1x^3}
    [/tex]

    don't have converging Fourier transforms, so that doesn't help either.
     
  2. jcsd
  3. Jun 24, 2010 #2

    phyzguy

    User Avatar
    Science Advisor

    The solution is called an Airy function, and it is a much-studied solution to Schrodinger's equation in the semi-classical approximation. In addition to the link below, you might try searching on "semiclassical approximation" or "WKB method".

    http://en.wikipedia.org/wiki/Airy_Function
     
  4. Jun 24, 2010 #3
    I see and appreciate.
     
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