Momentum values in quantum mechanics

[shakaflaka]

Homework Statement

In an infinite potential hole, the ground state is described by the wavefunction ψ=A*sin($\frac{πx}{a}$). Does the ground state have a definite momentum? If not, then what are the values of momentum in ground state?

Homework Equations

Wavefunction ψ=A*sin($\frac{πx}{a}$)
Momentum operator $\hat{p}$=-iħ$\frac{d}{dx}$

The Attempt at a Solution

Momentum has a definite value when $\hat{p}$ψ=pψ.
In this case $\frac{dψ}{dx}$=$\frac{π}{a}$A*cos($\frac{πx}{a}$)
and $\hat{p}$ψ=-$\frac{iħπA}{a}$*cos($\frac{πx}{a}$)
So this is the point where I am stuck, what does this result show me?

 

[Modulated]

You're almost there! You're correct that the momentum has a definite value when $\hat{p}\psi = p\psi$ for some constant $p$. Is that the case here? If not, you will have to conclude that this state has no definite momentum.

As for the second half, we know that the allowable momentum values are the eigenfunctions of the momentum operator. Do you know what these are? Can you express the given wavefunction in terms of these eigenfunctions?

HTH

 

[shakaflaka]

The first part of your reply was just what I was expecting, so thank you for that. I have given all the information concerning the problem, so I don't know if I can express it in terms of eigenfunctions, in case I haven't missed anything. Can you help me formulate an answer, what are the values of momentum in ground state? Sorry if you didn't understand something, I am finding it hard to "talk physics" in English.