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Question about QM eigenvalues.

  • Thread starter Craptola
  • Start date
  • #1
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I've been wrestling with this question for a while and can't seem to find anything in my notes that will help me.

Homework Statement


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.


Homework 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]


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.
 

Answers and Replies

  • #2
BruceW
Homework Helper
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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 :)
 
  • #3
tiny-tim
Science Advisor
Homework Helper
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Hi Craptola! :smile:

hmmm …
[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[/tex]
stop there? :wink:
 
  • #4
14
0
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.
 
  • #5
tiny-tim
Science Advisor
Homework Helper
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… it seems that E could be anything depending on other variables.
Ah, but k and ω aren't variables, they're your given constants! :wink:
 

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