Why Helium Doesn't Freeze When Cooled: Zero-Point Energy

Let's say I've got, uh, an atom in a liquid, and it has some freedom. It can move around, but, you know, it's very closely packed in with all of its neighboring atoms.

Let's say that it occupies some spherical volume with a radius R. Well, we can say that that R is the uncertainty in its position.

So the uncertainty in the momentum is gonna look like H-bar, Planck's constant, over R. But if it's got an uncertainty in momentum, it means it must be jiggling around inside that sphere.

We know if it's jiggling, we can say it's got an energy.

Well, we know what its momentum is. It's Planck's constant over, over R. So how do we get the energy? Well, it's easy.

We say it's the momentum squared over twice the mass.

Now, helium, being such a light atom, that mass term is very small.

So that means that the energy that it has, because it's confined, is very, very large.

Now, this is what we call the zero-point energy.

This is the energy it cannot get rid of.

It's nothing to do with thermal energy. It's not because there's a temperature in the liquid.

In fact, when you're in the liquid state, that zero-point energy accounts for more than 98% of the total energy in the fluid.

Wait, what? It is, it is enormous. That is crazy. I had no clue.

This zero-point energy is too large for the atoms ever to lock down in place, so helium never freezes just by cooling it down.

So when you're thinking of this kind of ladder of states, this kind of BEC picture, like, you know, everything's macroscopically occupying this bottom state, we think, we-- the, the intuition is to think of it as like a ladder,

where each rung is nice and evenly spaced.

But the way to think of it in terms of this zero-point energy is that that first rung is 10 stories off the floor.