1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prob of part being in Lowest Energy Level after Potential Change

  1. Apr 13, 2015 #1
    First off sorry for the badly worded title.

    1. The problem statement, all variables and given/known data
    Beginning of Question:

    Consider a single quantum particle of mass M trapped in the infinite square well potential, V(x), given by

    V(x)= 0 if 0 < x < L
    infinity otherwise

    The wave function for a particle in the n-th energy level is: Ψn(x) = √(2/L) sin(nπx/L)

    a.) I found the expected position and momentum of a particle in the n-th energy level.

    b.) I calculated the expectation value for the energy of a particle in the n-th energy level using the hamiltonian.

    Bit of Question I'm stuck on:

    c.)
    Suppose that the particle initially starts in the lowest energy level and the potential is instantaneously changed to:

    V(x) = 0 if 0 < x < L/2
    infinity otherwise

    Find the probability that the particle ends up in the lowest energy level of the new potential.

    2. Relevant equations


    3. The attempt at a solution.

    I'm not exactly sure what to do here. I assume it must be something along the lines of:
    1st - Finding the lowest allowed energy level
    2nd - Finding the probability that the particle would be in this state.

    I was thinking that I might be able to use a method along these lines:
    Renormalise the wave equation first to account for the change in potential?
    Then repeat what I did in part b.) to find the expectation value of the energy of the particle in the n-th energy level?
    I would surely then be able to find the lowest expected value for the energy?
    And then I would be able to find the probability that a particle is in that state?

    Or have I got totally the wrong idea here? It seems as though I'm ignoring the fact that the potential changed instantaneously.



     
  2. jcsd
  3. Apr 13, 2015 #2

    BvU

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    2. relevant equations. Nothing ? :rolleyes:

    a) What did you find ?
    b) What did you find ? Isn't that what you put in in the first place ?
    c) This is very strange (not your fault): your renormalization idea set me thinking about how to bring about this potential change -- without affecting the particle wave function. After all, the particle has to be somewhere - so what happens to the wave function in the [L/2, L] section ? Usually this kind of exercise expands the size of the box, so you can extend ##\Psi## with zero.

    3. Your attempt isn't really an attempt: you are musing, considering, ...

    The basic idea is to assume some wave function (in this case the lowest energy steady state wavefunction associated with the [0,L] box; the wavefunction you might mention under 2. relevant equations) and claim that that wavefunction stays the same during the instantaneous change in potential. Then expand the old ##\Psi## in terms of the eigenfunctions (steady state wave functions) associated with the [0,L/2] box -- see under 2: relevant equations :wink: )

    I see all kinds of problems on the way, so we might need a real expert. I know one...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Prob of part being in Lowest Energy Level after Potential Change
Loading...