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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 [  t  ,   P  ]= 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 Δ t Δ 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 ǀΔ F ǁΔ(1/ T )ǀ≥ 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 ǀΔ F ǀ→0 and ǀΔ T ǀ→0 are negligible at low temperatures  T →0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties ǀΔ F ǀ→∞ and ǀΔ T ǀ→∞ grow unlimited at high temperatures  T →∞, 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  σ =ε/ T = ak . The dimensionless entropy number a connects the particle energy ε= akT  with the temperature  T . The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrödinger equation   A   s  φ=aφ . Here   A   s =−( Λ  2 /4 π )∇ 2  means the dimensionless entropy operator describing the entropy number a and thus the particle entropy  σ = ak .  Λ  is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies  σ  the internal energy  E , the Helmholtz free energy  F , the entropy  S , the chemical potential  μ , 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 / Λ  3  of a single free particle within the volume  V . Here  Λ  3  is a small volume element taking into account the wave–particle dualism of a single free particle of mass  m  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.

Relation between Quantum Thermodynamics and Classical Thermodynamics