Is This Wave Function an Eigenfunction of Energy and Momentum Operators?

[Craptola]

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 $\Psi (x,t)= \textrm{exp}(-i(kx+\omega t))$ 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:
$$\hat{E}\Psi =k\Psi$$
Where K is the eigenvalue
Energy operator:
$$\hat{E}=i\hbar\frac{\partial }{\partial t}$$

The Attempt at a Solution

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

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.

 

[BruceW]

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 :)

 

[tiny-tim]

Hi Craptola!

hmmm …

$$\hat{E}\Psi =i\hbar\frac{\partial }{\partial t}[\textrm{exp}(-i(kx+\omega t))]$$
$$=-i\hbar i\omega \Psi =\hbar\omega \Psi$$

stop there? ​

 

[Craptola]

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.

 

[tiny-tim]

… it seems that E could be anything depending on other variables.

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