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Homework Help: Quick QM Question

  1. Sep 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Given an initial ave function [tex]\Psi(x)[/tex] at t=0, and a complete set of energy eigenfunctions [tex]\varphi_{n}(x)[/tex] with corresponding eigenenergies E_n for a particle, and no other information, in therms of the given find:


    a. Find the state of the particle at a later time

    Solution: [tex]\Psi(x,t) = T(t)*\varphi_{n}(x) = e^{-iE_{n}t/\hbar}\varphi_{n}(x)[/tex]

    b. what is the probability that a measurement of energy will yield a particular value E_n?
    [tex]<E> = \sum_{n}C_{n}P(C_n)[/tex]
    Where P is the probability but I don't know where to go from here.

    c. Find an expression for the average energy of the particle in terms of the enrgy eigenvalues:
    [tex]<E>=1/n \sum_{i=1}^{n}E_{n}[/tex]

    All seems right besides b? Any hints on b?
     
  2. jcsd
  3. Sep 29, 2010 #2
    One important point you're missing is that [tex] \psi [/tex] can be any linear combination of the energy eigenfunctions [tex]\phi_n[/tex]. Try the problem armed with the knowledge that:

    [tex]
    \Psi(x,t) = \Sigma_n \phi_n(x,t)
    [/tex]

    I would be glad to answer any further questions, but this seemed to be the major stumbling block.
     
    Last edited: Sep 29, 2010
  4. Sep 29, 2010 #3
    oh, duh...

    a.
    [tex]\Psi(x) = \sum b_{n}\varphi_{n}(x)[/tex]

    which implies

    [tex]\Psi(x,t) = \sum b_{n}e^{-iE_{n} t/ \hbar} \varphi_{n}[/tex]

    Which means in part b the probability of yielding E_n is simply b_n, correct?

    [tex]b_{n} = <\varphi_{n}|\Psi(x)>[/tex]
     
  5. Sep 29, 2010 #4
    hrm, LaTeX is either messing up for me or I missed something in my formatting but hopefully you can see what I meant for Psi in my second post
     
  6. Sep 29, 2010 #5
    Note that
    [tex]
    b_{n} = <\varphi_{n}|\Psi(x)>
    [/tex]
    Can be a complex number. Does a complex probably mean anything? Hint hint...
     
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