# Allowed momentum values for a plane wave

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1. Jun 22, 2016

### alec_grunn

• Member advised to use the homework template for posts in the homework sections of PF.
Hi all,
This is from a past exam paper: At t=0 the state of a particle is described by the wavefunction

Code (Text):
$$\Psi (x,0) =A(iexp(ikx)+2exp(-ikx))$$
This is between positive and negative infinity - not in a potential well.

What values of momentum are allowed, and with what probability in each case?

Relevant Equations:
$\hat p = -i \hbar \frac{\partial}{\partial x}$

My attempted solution:
Since there's only one k value present I was thinking there is one momentum value: $p = \hbar k$ with 100% chance of measuring this. And the fact that the uncertainty in position is infinite means that its momentum is sharp. But if this is the case, then why isn't it an eigenfunction of momentum?

Cheers

2. Jun 22, 2016

### blue_leaf77

Momentum is a vector operator - the sign matters.

3. Jun 22, 2016

### alec_grunn

Ok so that means 20% chance of +p and 80% chance of -p?

4. Jun 22, 2016

Yes.