The following passage has been extracted from the book "The Feynman Lectures on Physics-Vol l":
The mean kinetic energy is a property only of the "temperature." Being a property of the "temperature," and not of the gas, we can use it as a definition of the temperature. The mean kinetic energy of a molecule is thus some function of the temperature. But who is tell us what scale to use to use for the temperature? We may arbitrarily define the scale of the temperature so that the mean energy is linearly proportional to the temperature. The best way to do it would be to call the mean energy itself "the temperature." That would be the simplest possible function. Unfortunately, the scale of temperature has been chosen differently, so instead of calling it temperature directly we use a constant conversion factor between the energy of a molecule and a degree of absolute temperature called a degree kelvin.
The constant of proportionality is $k=1.38\times10^{-23}$ joule for every degree. So if T is a absolute temperature, our definition says that the mean kinetic energy is $(3/2) kt$ (The $3/2$ is put in as a matter of convenience, so as to get rid of it somewhere else.)
From the above passage, at absolute zero, by definition, mean kinetic energy of a molecule should be zero-"completely frozen." There is a giant principle which stands against the view of atoms getting completely frozen; the following passage from the same book introduces the principle:
As we decrease the temperature, the vibration decreases
and decreases until, at absolute zero, there is a minimum amount of vibration
that the atoms can have, but not zero…..Remember that when a crystal is cooled to absolute zero, we said that the atoms do not stop moving, they still
jiggle. Why? If they stopped moving, we would know where they were and that
they had zero motion, and that is against the uncertainty principle. We cannot
know where they are and how fast they are moving, so they must be continually
wiggling in there!
Aren't the above two passages in contradiction with each other? Didn't we mess up with temperature?
Best Answer
If carefully interpreted and converted to mathematics, the passages are not contradicting one another.
The uncertainty principle guarantees that there is some zero-point energy that can't be eliminated (the second passage) – it is the energy of the ground state of the physical object (atom or a macroscopic piece of a material).
On the other hand, the kinetic energy that increases with $T$ and reaches $E_k=0$ for $T=0$ is the excess energy above the energy of the ground state: if we want $E=0$ for $T=0$, the additive shift $\Delta E$ of the energy has to be chosen appropriately to guarantee this condition. The right additive shift is really the "subtraction of the ground state energy", i.e. of the minimum energy eigenvalue that the physical system may have.
If a physical system is frozen to $T=0$, it means that this physical system is "certain" (100%) to be found in the ground state i.e. the energy eigenstate with the minimum allowed value of $E$. In most contexts, this ground state is pretty much unique. It contains "some motion", by the uncertainty principle, but "the amount of motion above the minimum level allowed by the uncertainty principle" is zero.
In the first passage, Feynman really talks about the overall kinetic energy of the atom as it moves through space. This is indeed $3kT/2$ and strictly goes to zero for $T=0$. No subtraction is needed here. The minimum kinetic energy of the "overall motion of the object through space" is really $E_k=0$ and the corresponding momentum is $\vec p =0$. This doesn't violate the uncertainty principle because, indeed, $\Delta x = \infty$ or $\Delta x \to \infty$. The position of an atom frozen to $T=0$ (which can only be approached in the real world) is absolutely undetermined. The subtle issues of the zero-point energy only arise for the "internal energy" i.e. the relative motion of parts of a bound state (e.g. the motion of electrons around the nuclei). Only for the bound state, we really know that $\Delta x$ cannot be infinite. The constituents' being "bound" means that the distance between them is finite and bounded from above.