Question

Use a classical analogy to explain why it makes sense that a quantum mechanical particle-in-a-box has an average momentum of zero even though it has a non-zero kinetic energy? [particle, box, momentum, positive, negative, zero, kinetic energy, non- zero...]

Answer 1

Suppose the particle is confined to move to and fro inside a box from

The average momentum of the particle is mass times its average velocity

Velocity is a vector. The particle will go towards the negative x direction as often as it goes to the positive x direction.

When it moves to the negative x direction, its velocity is negative and when it moves in the positive x direction, its velocity is positive.

The average velocity is zero because the positive and negative velocities cancel out.

Hence the average momentum is also zero

The kinetic energy, on the other hand, is non zero because it is proportional to velocity squared and we know that square of any real number(positive or negative) is always positive.