A faculty colleague of mine posed a statistical physics question for me, since I'm teaching that subject to undergraduates this semester, and I want to throw it out there to my readership. I'll give some context, explain the question, and then explain why it's actually rather subtle. If someone has a good answer or a reference to a good (that is, rigorous) answer, I'd appreciate it.
In statistical physics one of the key underlying ideas is the following: For every macroscopic state (e.g., a pressure of 1 atmosphere and a temperature of around 300 K for the air in your room), there are many microstates (in this example, there are many possible arrangements of positions and momenta of oxygen and nitrogen molecules in the room that all look macroscopically about the same). The macroscopic states that we observe are those that have the most possible microstates associated with them. There is nothing physically forbidden about having all of the air in your room just in the upper 1 m of space; it's just that there are vastly more microstates where the air is roughly evenly distributed, so that's what we end up seeing.
Crucial to actually calculating anything using this idea, we need to be able to count microstates. For pointlike particles, that means that we want to count up how many possible positions and momenta they can have. Classically this is awkward because position and momentum are continuous variables - there are an infinite number of possible positions and momenta even for one particle. Quantum mechanically, the uncertainty principle constrains things more, since we can never know the position and momentum precisely at the same time. So, the standard way of dealing with this is to divide up phase space (position x momentum) into "cells" of size hd, where h is Planck's constant and d is the dimensionality. For 3d, we use h3. Planck's constant comes into it via the uncertainty principle. Here's an example of a typical explanation.
Here's the problem: why h3, when we learn in quantum mechanics that the uncertainty relation is, in 1d, (delta p)(delta x) >= hbar/2 (which is h/4 pi, for the nonexperts), not h ? Now, for many results in classical and quantum statistical mechanics, the precise number used here is irrelevant. However, that's not always the case. For example, when one calculates the temperature at which Bose condensation takes place, the precise number used here actually matters. Since h3 really does work for 3d, there must be some reason why it's right, rather than hbar3 or some related quantity. I'm sure that there must be a nice geometrical argument, or some clever 3d quantum insight, but I'm having trouble getting this to work. If anyone can enlighten me, I'd appreciate it!
UPDATE: Thanks to those commenting on this. I'm afraid that I wasn't as clear as I'd wanted to be in the above; let me try to refine my question. I know that one can start from particle-in-a-box quantum mechanics, or assume periodic boundary conditions, and count up the allowed plane-wave modes within a volume. This is equivalent to Igor(the first response post)'s discussion of applying the old-time Bohr-Sommerfeld quantization condition (that periodic orbits have actions quantized by h). My question is, really, why does h show up here, when we know that the minimal uncertainty product is actually hbar/2. Or, put another way, should all of the stat mech books that argue that the h3 comes from uncertainty be reworded instead to say that it comes from Bohr-Sommerfeld quantization?