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Homework Help: Question about QM eigenvalues.

  1. Oct 7, 2013 #1
    I've been wrestling with this question for a while and can't seem to find anything in my notes that will help me.

    1. The problem statement, all variables and given/known data
    Determine whether the wave function [itex]\Psi (x,t)= \textrm{exp}(-i(kx+\omega t))[/itex] is an eigenfunction of the operators for total energy and x component of momentum, and if it is, calculate the eigenvalues.

    2. Relevant equations
    Condition for an eigenfunction:
    [tex]\hat{E}\Psi =k\Psi [/tex]
    Where K is the eigenvalue
    Energy operator:
    [tex]\hat{E}=i\hbar\frac{\partial }{\partial t}[/tex]

    3. The attempt at a solution
    Determining that psi is an eigenfunction is easy enough.
    [tex]\hat{E}\Psi =i\hbar\frac{\partial }{\partial t}[\textrm{exp}(-i(kx+\omega t))][/tex]
    [tex]=-i\hbar i\omega \Psi =\hbar\omega \Psi =\frac{h}{2\pi }2\pi f\Psi =hf\Psi =E\Psi [/tex]

    I can't figure out how to calculate the value of E from this information alone. I imagine the same method works for momentum when I figure out what it is.
  2. jcsd
  3. Oct 7, 2013 #2


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    hmm? You have just calculated the value of E. Now, yeah it is pretty much a similar method to find out if it is also an eigenstate of momentum, once you remember what the operator looks like :)
  4. Oct 7, 2013 #3


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    Hi Craptola! :smile:

    hmmm …
    stop there? :wink:
  5. Oct 7, 2013 #4
    I assumed that the question wanted me to calculate an actual number for the eigenvalue, which is what confused me as it seems that E could be anything depending on other variables. This is all stuff we covered fairly recently so I wasn't sure if there was some kinda law that wasn't in the lecture notes which limited the possible values of E.
  6. Oct 7, 2013 #5


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    Ah, but k and ω aren't variables, they're your given constants! :wink:
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