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Quantum Mechanics - Infinite Potential Well

  1. Jan 29, 2009 #1
    1. The problem statement, all variables and given/known data

    A particle is trapped in an infinite potential well, with the infinite walls at ±a. At time t=0, the wavefunction of the particle is

    [tex]\psi = \frac{1}{\sqrt{2a}}[/tex]

    between -a and a, and 0 otherwise.

    Find the probability that the Energy of the particle is [tex]\frac{9 \bar{h}^2 \pi^2}{8ma^2}[/tex]

    2. Relevant equations

    [tex]E_n = \frac{n^2\bar{h}^2\pi^2}{8ma}[/tex]

    [tex]\psi = A \cos{\frac{(2r+1) \pi x}{2a}}[/tex] for |x| < a
    [tex]\psi = 0[/tex] otherwise

    3. The attempt at a solution

    I've calculated the above equations, but I'm unsure how to get from them to the probability of the particle having a certain energy. This could be really simple and it's me just having a brain dead moment, but any help would be very much appreciated.
     
  2. jcsd
  3. Jan 29, 2009 #2
    btw there is also a sin solution with an argument (in your notation) 2r(pi)x/a

    personally I prefer the notation n(pi)x/a n even

    however the cos solution you wrote is the one you want with r=1, that 2r+1 thing is just a way of writing n so that n is always odd.

    so just take the projection of psi at t=0 on your cos function and square the answer
     
  4. Jan 29, 2009 #3
    The sin solution isn't valid because this well has even parity (i.e. it's symmetric).

    How do I find A? Presumably I need to use the initial condition of Psi, but I found when doing that that A is x dependent, when it should be a constant.
     
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