Can a Particle in a Bounded System Have Momentum in One Direction?

[Syrus]

Homework Statement

Consider a particle in a one-dimensional box whose (time-independent) state is given by
e^ikx. Of course, this corresponds to the particle moving with a momentum of magnitude hk in the positive (let's call it x) direction.

I know momentum in the quantum sense cannot be regarded as momentum in the classical sense. I am struggling to conceive how a particle in such a bounded system can always have momentum in one direction- how can it renew itself when it reaches the right boundary? I am thinking it has something to do with (complex) traveling waves, but i can't put my finger on it. I also looked in a few books on quantum mechanics and was able to find the following lines, which I believe applies to my inquiry somehow:

"In a nutshell: the formal problem with applying the classical definition of momentum to an ensemble of microscopic particles arises because it is impossible to measure the position of such particles without altering their state. This vicissitude is inevitable, because prior to measurement, the particles do not have a position or a momentum."

 

[vela]

The problem is the picture you have in your head of a little ball bouncing back and forth between the walls isn't really a good one when it comes to quantum mechanics. You're trying to understand quantum mechanics in terms of classical concepts but it should be the other way around. Quantum mechanics is the more general theory.