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Particle in 1-D Potential

  1. Sep 22, 2010 #1
    1. The problem statement, all variables and given/known data

    The quaestion asks to determine the ground an first excited state of the wavefuntion of a particle in a 1-D potential given by [tex]V(x)=B\left | x \right |[/tex].

    2. Relevant equations

    The Time Independent Schrodinger Equation (TISE):

    [tex]-\frac{\hbar}{2m}\frac{d^{2}\Psi }{dx^{2}}+V\Psi=E\Psi[/tex]

    3. The attempt at a solution

    I substituted the potential into the TISE and with some rearraging of terms I get the following differential equation.

    [tex] {\Psi}''+\frac{2m}{\hbar^{2}}(E-B\left | x \right |)\Psi=0[/tex]

    This is where I'm stuck. I don't know how to solve this equation because of the potential is dependent on [tex]x[/tex]. Any suggestions would be greatly appreciated.
     
  2. jcsd
  3. Sep 22, 2010 #2
    The general solutions will be Airy functions (or Bessel functions of order 1/3).
     
  4. Sep 22, 2010 #3
    Ok, but I don't see how I would get bessel's equation. I'm guessing I would need to multiply the equation first by [tex]x^{2}[/tex].
     
  5. Sep 23, 2010 #4
    Its much easier to get Airy's equation. Check how you can get that ;)
     
  6. Sep 23, 2010 #5
    It turns out for this question I don't actually need to solve for the wavefunction. I just need to determine the curvature of it. Than from there I can get a rough sketch of the excited states, which is all the question asks for. I should have posted the full question, although I know what I need to do now. Thanks for your input Thaakisfox.
     
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