Ideal Gas Law, Quantum Thermodynamics and Pure Bound States
Francesco R. Ruggeri
In (1) the ideal gas result P V (to the power gamma) = constant ((1)) where gamma=cp/cv=5/3 is applied to a pure quantum bound state for a particle in a box with infinite potential walls and it is seen that ((1)) is a constant just as in the case of an ideal gas. This, however, depends on the result PV=nRT. {An ideal gas, however, follows from a Maxwell-Boltzmann distribution which is linked to the equipartition theorem. The value cp/cv=5/3 reflects this. }Furthermore, T and V are distinct physical variables for an ideal gas. For a particle in a well with infinite potential walls, E=constant/LL. The energy depends only on L whereas in an ideal gas it only depends on T i.e. E=3/2 T. In this note we try to analyze why the ideal gas relationship ((1)) holds for a particle in a box with infinite walls. We also try to link the relationship E proportional to 1/LL to scaling features which ensure that S=Sp(n)+Sx does not depend on L allowing for an adiabatic transformation to keep entropy constant. (This holds for both a particle in a box and a harmonic oscillator.) We use the thermodynamical law dE=TdS-PdV and find that PV=nRT only holds in the one dimensional gas i.e. PL= constant T. In such a case we are able to justify ((1)), but not in the 3-D situation.
•Feb 11, 2022•0 citations