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Sudden Perturbation of Potential Well

  1. Apr 18, 2014 #1
    1. The problem statement, all variables and given/known data

    ettkz6.png

    Part (a): Particle originally sits in well V(x) = 0 for 0 < x < a, V = ∞ elsewhere. The well suddenly doubles in length to 2a. What's the probability of the particle staying in its ground state?

    Part (b): What is the duration of time that the change occur, for the particle to most likely remain in ground state?

    2. Relevant equations



    3. The attempt at a solution

    Part (a)

    For a well of length a, eigenfunction is:

    [tex]u = \sqrt{\frac{2}{a}} sin \left(\frac{\pi x}{a}\right) [/tex]

    For a well of length 2a, eigenfunction is:

    [tex]v = \frac{1}{\sqrt a} sin \left(\frac{\pi x}{2a}\right) [/tex]

    For probability amplitude, overlap these 2:

    [tex] \frac{1}{2} \frac{\sqrt 2}{a} \int_0^{2a} 2 sin (\frac{\pi x}{a}) sin (\frac{\pi x}{2a}) dx [/tex]

    [tex]\frac{1}{2} \frac{\sqrt 2}{a} \int_0^{2a} cos (\frac{3\pi x}{2a}) - cos (\frac{\pi x}{2a}) dx[/tex]

    which goes to zero.

    Part (b)

    Am I supposed to use time evolution: ## |\psi_t> = U^{-i\frac{E}{\hbar}t}## ? Doesn't seem to help..
     
  2. jcsd
  3. Apr 18, 2014 #2
    What is the initial wavefunction in the region [itex]a<x<2a[/itex]?
     
  4. Apr 18, 2014 #3
    It is 0 before the change in the potential, since it is a forbidden region.
     
  5. Apr 18, 2014 #4
    Right. Do you see how that affects your overlap integral? It might help to explicitly sketch your before and after wavefunctions.
     
  6. Apr 19, 2014 #5
    So for a < x < 2a: the overlap is zero. While for 0 < x < a, the overlap is usual - so the limits are from 0 to a, and not from 0 to 2a.

    [tex]\frac{1}{2} \frac{\sqrt 2}{a} \int_0^{a} cos (\frac{3\pi x}{2a}) - cos (\frac{\pi x}{2a}) dx [/tex]

    [tex] = - \frac{8}{3\pi \sqrt 2} [/tex]

    So the probability = ##\frac{32}{9\pi^2}##, which is less than 1, reassuringly.


    But for part (b), I haven't got a clue.
     
  7. Apr 21, 2014 #6
    Do I need to use adiabatic approximation for part (b)?
     
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