Question

Explain, using wave mechanics, why it is
impossible for a particle to have zero kinetic energy
when it is confined to a fixed region in space.

Answer 1

I'm assuming this coming straight from quantum mechanics.

If we confine a particle to a well (I shall not derivate QM here), we argue that the potential energy is 0. Therefore, the total energy is all kinetic energy.

For a infinite potential well, this energy is given by:\[E_{n}=\frac{n^{2}\pi^{2}\hbar^{2}}{2mL^{2}}\]
For the lowest energy possible, n=1:\[E_{1}=\frac{\pi^{2}\hbar^{2}}{2mL^{2}} \neq0\]