The ground state of the infinite square wel

You should take these kinds of questions to the homework section of the forum.

 

Write down the ground state in x representation and try the momentum operator on it. Do you get back a multiple of the ground state?

 

I think that Dick has adequately covered the mathematical aspect of it. Now let's think about some common sense physics. Imagine a classical ball bouncing elastically between two rigid walls (let's neglect gravity here). So kinetic energy is conserved at each collision, right? But the momentum is obviously not conserved, as the ball spends half its time going left and the other half going right. (Recall that momentum has direction).

Caveat: Normally, thinking of quantum particles as classical particles is ill advised, but this is one of those situations in which the classical result carries over to the quantum scale.