Quantum thermodynamics describes dynamic processes by means of the operators of entropy production P and time t. P and t do not commute. It exists the non-vanishing t-P commutator [1, P1= ik. Here the Boltzmann constant k has the physical meaning of a quantum of entropy. The t-P commutator immediately leads us to the t-P uncertainty relation Delta t Delta P >= k/2. Hence the observables t and P are not sharply defined simultaneously. Similar uncertainty relations can also be expected for other pairs of conjugate variables with products of the physical meaning of an entropy. The free energy F and the reciprocal temperature (1/T) are the respective conjugate variables of an isolated system of many particles, which leads us to the F-(1/T) uncertainty relation vertical bar Delta F vertical bar vertical bar Delta(1/T)vertical bar >= k/2. It can be traced back to the t-P uncertainty relation mentioned above. In this way the Helmholtz free energy F and the temperatur T are introduced into quantum thermodynamics. The uncertainties vertical bar Delta F vertical bar -> 0 and vertical bar Delta T vertical bar -> 0 are negligible at low temperatures T -> 0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties vertical bar Delta F vertical bar -> infinity and vertical bar Delta T vertical bar infinity grow unlimited at high temperatures T -> infinity, and classical thermodynamics loses its sense. In the limit of one particle the uncertainties cannot be neglected even at low temperatures. However a detailed discussion shows that the free energy f of a single particle vanishes within the whole range of temperatures T. This defines the particle entropy sigma= epsilon/T =ak. The dimensionless entropy number a connects the particle energy epsilon = akT with the temperature T. The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrodinger equation A(s)phi= a phi. Here A(s)= -(Lambda(2)/4 pi)del(2) means the dimensionless entropy operator describing the entropy number a and thus the particle entropy sigma = ak. Lambda is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies sigma the internal energy E, the Helmholtz free energy F, the entropy S, the chemical potential mu and the equation of state of an ideal gas of N monatomic free particles in full agreement with classical thermodynamics. We also calculate the partition function q = V/Lambda(3) of a single free particle within the volume V. Here Lambda(3) is a small volume element taking into account the wave-particle dualism of a single free particle of mass to at temperature T. Extension to a system of N free particles leads us to a simple geometrical model and to the conclusion that an ideal gas of independent particles becomes instable below a critical temperature T(C). T(C) corresponds to the critical temperature T(BE) of Bose-Einstein condensation.
