Entropy and probability I read "The NEW world of Mr. Tompkins" and I'm not sure with one of the Gamow's equation. When he calculated the probability of entropy, he used this reasoning:
"How likely is a situation that all the atoms in this dining room will get under this table?
In this room is about $10^{27}$ molecules. Space under this table is maybe 1% of all space in the room. So the probability is $(1/100)*10^{27} = 10^{25}.$
I think that is wrong, because all the atoms would have the same probability on all the space in a room - but that is not what quantum mechanics says. There is some probability waves and some places, where atom could be rather than elsewhere.
Btw: What the probability actually means here?
 A: You will almost never encounter a calculation that is intended to account for every detail of a phenomenon with perfect accuracy. That isn't possible, and in fact many times adding more detail to a calculation only takes away from the insight it grants. Why make a complicated calculation when a simple one tells you everything you want to know?
Gamow is using the simple approximation that


*

*a given molecule is equally-likely to be found anywhere in the room

*the molecules are all independent of each other


The first assumption is not exactly true. The biggest correction is that, due to gravity, molecules are a bit more likely to be at the bottom of the room than at the top of the room. However, the scale height of the atmosphere is $10^4 \mathrm{m}$, so the few-meter height difference from floor to ceiling only makes a fraction of a percent difference.
The second assumption is the ideal gas approximation. We could make the calculation more accurate by finding the time-average of the potential energy of a nitrogen molecule in air of a given density and comparing it to the time-average kinetic energy. As long as the potential energy is much smaller than the kinetic energy, the ideal gas approximation is good. At 300K, air is very nearly ideal, even at 100 atmospheres. (Its compressibility factor at 300K is 0.9999 at 1 atm and 0.993 at 100atm.) There are no large corrections to be made to Gamow's calculation.
Quantum mechanics does not say much about where any particular atom will be in a room, or favor one location over another. The wavelength of an atom is extremely-small compared to a room. The uncertainty in the location of the atom is dominated by classical ignorance, i.e. entropy, not by quantum uncertainty. However, quantum mechanics is important for understanding an ideal gas, as it sets the scale of volumes in phase space necessary for calculating the entropy of the gas. See the Wikipedia article on the Gibbs Paradox.
