zero-point energy textbook question

[AnnaNovoselov1G]

The textbook says
"a particle in a container cannot have zero energy. Because the lowest allowed value of n is 1 (corresponding to a wave of one-half wavelength fitting into the box), the lowest energy is E 1 = h^2 / 8mL^2. This lowest possible energy is called the zero-point energy. The existence of a zero-point energy means that, according to quantum mechanics, a particle can never be perfectly still when it is confined between two walls: it must always possess an energy—in this case, a kinetic energy of at least E 1 = h^2 / 8mL^2."

Could someone please clarify why a particle cannot have zero energy inside the box? What happens if the particle is not confined to the box? I understand that the particle has a finite limit on momentum if it is confined to a region, but why exactly does it need to have a nonzero value for momentum?

[Zaid Bustami 1B]

Hi Anna!

So if you think about Heisenberg uncertainty, you know that uncertainty in position is inversely proportional to uncertainty in momentum. If an electron were to stand still, its uncertainty in velocity would be so low that the uncertainty in position would be larger than the size of the box it's confined within. That is, as the uncertainty in the velocity (and therefore momentum) of an electron approaches 0, the uncertainty in position approaches infinity. So in a confined space, the electron cannot stand still.
Due to the electron's wave-like properties, the electron's De Broglie wavelengths are quantized (if you remember the standing wave representation for an electron in an atom) because the electron can only occupy discrete energy states (n=1, 2, etc.). An e- cannot occupy an n=0 state, so according to the textbook's derivation for the energy of an electron in a confined 1D space, the energy of a particle inside a box cannot be zero. Assuming that the potential energy of the particle is zero inside the box, that energy is present as kinetic energy, or movement.
Momentum is nonzero because the e- always has a velocity and a rest mass (it's not pure energy).
I'm not entirely sure about what happens when a particle is not confined to a box, but I think it would help to think about what happens when you increase the size of the box. So, the answer could com from what happens when you increase the size to infinity.
This is stuff that I'm not too familiar with, so if something I said isn't right or doesn't make sense feel free to correct me. Also, this material is from 1C in the textbook, so it's not part of 14A's curriculum but it's still pretty interesting.