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Asymptotic form of a wave function

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data
    For one of my physics assignments I need to do some numeric integration (and I know its physics but my question doesn't pertain to that).

    [itex]\hat{H}\Phi = E\Phi[/itex]
    [itex]\frac{\delta^{2}\Phi}{\delta x^{2}}(x) = \frac{-2m}{h^{2}}(E - V(x))\Phi(x) = -Z(x)\Phi(x)[/itex]

    We have the taylor expansion:
    [itex]\Phi(x + 2d) = \Phi(x) + 2d\frac{\delta \Phi}{\delta x} + 2d^{2}\frac{\delta^{2}\Phi}{\delta x^{2}}[/itex]

    Using definition of the derivative:
    [itex]\frac{\delta \Phi(x)}{\delta x} = lim_{d\rightarrow 0} \frac{\Phi(x + d) - \Phi(x - d)}{2d}[/itex]

    we can obtain:
    [itex]\Phi(x + 2d) = \Phi(x + d) + \Phi(x) - \Phi(x - d) - 2d^{2}Z(x)\Phi(x)[/itex].

    For the potential V(x) = Ax with A = -1, m = 1, h = 1, and E = 0, find the asymptotic form of the wave function for x >> 0. Use this form to find the three initial points.

    This is where I run into problems. Maybe I'm missing something very obvious but I don't know how to find this initial asymptotic form. Any hints?
  2. jcsd
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