# Heisenberg's Uncertainty Principle Applied to Absolute Zero

But absolute zero ($$0$$ K or $$-273.15$$ $$^\circ$$C) is an impossible goal to reach. Practically, the work needed to remove heat from a gas increases the colder you get, and an infinite amount of work would be needed to cool something to absolute zero. In quantum terms, you can blame Heisenberg's uncertainty principle, which says that the more precisely we know about its position, and vice versa. If you know your atoms are inside your experiment, there must be some uncertainty in their momentum keeping them above absolute zero -- unless your experiment is the size of the whole universe.

-New Scientist/Instant Expert, The Quantum World p. 76

I'm having a hard time relating the concept of absolute zero to the quantum world. What is exactly meant by, "unless your experiment is the size of the whole universe?"

My guess is that because we know "the atoms are inside our experiment," we can't know their speed, but how is this solved by expanding the size of our experiment to include the whole universe?

I think they mean than, for a very large experiment size and hence possible spatial uncertainty, the uncertainty on the momentum (and hence on the zero-point energy) goes to zero.

If $$\Delta x \Delta p \sim \hbar$$, then $$\Delta p \sim \frac{\hbar}{\Delta x}$$.

With $$\Delta x \rightarrow \infty$$, $$\Delta p \rightarrow 0$$ and so that the kinetic energy $$E \propto p^2$$.

• I might be overthinking, but doesn't the concept of "the whole universe" imply something different, rather than merely "very large?" Specification of it seems weird. Commented Aug 6, 2021 at 18:52
• No I think it relates to the sentence just before that, " If you know your atoms are inside your experiment". What they are saying is: if you really have no clue where they are, they you have to take the whole universe as your box. But it's the same idea, very large $\Delta x$. Commented Aug 8, 2021 at 0:24

When you cool down atoms their momentum decrease, and thus the uncertainty in their momentum also has to decrease. By the Heisenberg's uncertainty principle, $$\Delta x \Delta p \geq \frac{\hbar}{2}$$, this means that the uncertainty in the position has to increase. Atoms grow bigger as they are cooled down.

This is what happens in a Bose-Einstein condensate of atoms: the gas temperature is so low, and atoms get so large, so that their atomic wavefunctions start to overlap, and begin oscillate in-phase. Thus the condensate behaves as if it was a huge single atom, allowing for applications like atom-lasers (see here).

At zero temperature, thus zero momentum, the size of an atom would be infinite, if that was even possible to achieve.