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Solution to the Schrödinger equation for a non rigid step

  1. Oct 15, 2014 #1
    I've been having troubles resolving the Schödinger's time independent one-dimensional equation when you have a particle that goes from a zone with a constant potential to a zone with another constant potential, yet the potential is a continuos function of the form:

    $$
    V(x)=\left\{
    \begin{array}{lcl}
    0&\text{if}&x<0\\
    \displaystyle\frac{V_{0}}{d}x&\text{if}&0<x<d\\
    V_{0}&\text{if}&d<x
    \end{array}\right.
    $$

    My main problem is around the solution in the second region of the potential, the non-constant region, in which looks like:
    $$E\psi(x)=\frac{\hbar^{2}}{2m}d_{x}^{2}\psi(x)+\frac{V_{0}}{d}x\,\psi(x)$$
    If tried solving the differential equation by lowering it's order, yet I have not managed to do so. Is there another way of attacking the problem? Or how may I resolve the diff. equation?
     
  2. jcsd
  3. Oct 16, 2014 #2

    Simon Bridge

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  4. Oct 16, 2014 #3
  5. Oct 16, 2014 #4

    Simon Bridge

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    No worries - it's not something you were going to guess.
    Note: this sort of thing happens a lot.
     
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