1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Quantum mechanics-help!

  1. May 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Below is a past paper question from a paper i am working through in preperation for my exams. I am having particular trouble with this question, it doesnt seem as straight forward and formulated as other questions. I am unsure if anything i have done is right so any help would be great.

    2. Relevant equations


    Time independent equation

    Time dependent Schrodinger

    Probabilty Density [tex]=\left|\psi(x,t)\right|^{2}[/tex]

    3. The attempt at a solution





    I tried simply putting the wavefunction into the Time dependent Schrodinger equation and differentiating. This mustnt be the way since there is an m on the LHS.


    Probabilty Density

    [tex]\left|\psi(x,t)\right|^{2}=1/2[\phi^{2}_{0}+\phi^{2}_{1}+\phi^{*}_{0}\phi_{1}exp(-i\omega t)+\phi^{*}_{1}\phi_{0}exp(i\omega t)][/tex]


    [tex]\int^{\infty}_{0}\left|\psi(x,t)\right|^{2}dx=1/2\int^{\infty}_{0}[\phi^{2}_{0}+\phi^{2}_{1}+\phi^{*}_{0}\phi_{1}exp(-i\omega t)+\phi^{*}_{1}\phi_{0}exp(i\omega t)]dx[/tex]

    ^ This doesnt work out too well
  2. jcsd
  3. May 16, 2009 #2
    The purpose of this problem is to show that the given combination of wave functions gives you an oscillating probability density going back and forth at the classical frequency. I don't have time to go through all your equations at the moment but item b, for one, ought to work out the way you've set it up. I think the dimensions are consistent between the LHS and the RHS.
  4. May 16, 2009 #3


    User Avatar
    Science Advisor
    Gold Member

    Your time-dependent Schroedinger equation appears to be missing a hbar on the left, and a hbar squared in front of the partial derivative on the right...
  5. May 16, 2009 #4
    Thanks, Matterwave. Good of you to notice that.

    I wanted to elaborate on my previous answer a bit: the first three parts of the question are just verifying that you know how to correctly manipulate the formulas for these things, and you seem to be OK. The final part, where you actually have to calculate probabilities, is where you get to see some interesting physics. The combination of states chosen here is actually the very simplest manifestation of a classical type of oscillation using a superposition of quantum states. The quantum states are stationary and have the same likelihood of being anywhere for all time. This particular superposition oscillates back and forth at the classical frequency. What they might have asked you to do is calculate the center of mass of the system, or the x expectation: but they actually ask for something a little different, which is the time spent in the Right Hand Side. Perhaps they're doing you a favor by giving you an easier integration...I'm not sure. But you do appear to have set up the integral correctly. The time-independent terms integrate out to 1/2, because the particle spends half the time in the RHS. The cross terms give you the time dependent oscillation. So you should get the right answer if you can execute the integration...good luck.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook