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The quantum thermodynamics of the ideal gas is treated by means of the particle entropies  σ = ka . The non-dimensional entropy numbers  a  are calculated by the extended, temperature-dependent Schrödinger equation applying periodic boundary conditions. The Boltzmann constant  k  represents the atomic entropy unit. The quantized particle entropies  a  are used to calculate the Boltzmann distribution  f =exp( α − a ), the density of entropy levels  D ( a ), the dimensionless chemical potential  α = μ / kT ,  etc . – The thermodynamics of the ideal gas is treated on the basis of entropy quanta by means of the density matrix formalism. In this way we obtain the total particle entropy  A , the thermodynamic entropy  S  (equation of Sackur-Tetrode), the internal and the Helmholtz free energy  E  and  F , respectively,  etc . Moreover, it can also be shown on the basis of entropy quanta that the entropy of a closed system in thermal equilibrium has a maximum value,  S = S  max . Against this, the entropy  S  ′  of an arbitrary non-equilibrium state is always smaller,  S ′ &lt;  S  max  (second law of thermodynamics). – Application of ordinary density matrix theory to a closed system of  N  particles leads to the result that the entropy  S  does not change in time, d S /d t =0, regardless whether we consider an equilibrium state,  S = S  max , or a non-equilibrium state,  S = S ′. Consequently,  S  cannot irreversibly change from  S ′ to  S  max . However, any irreversible process is accompanied by a positive entropy production  P =d S /d t &gt;0. In order to overcome this obvious contradiction, we discuss the entropy evolution in time by means of the commutator equation   G  =i[  P  ,  t  ], which is deduced from very general assumptions. Here   P   and   t   are the operators of the entropy production  P  and the time  t . Accordingly,   P   and   t   do not commute in general, and hence  P  and  t  are not sharply defined simultaneously. Instead we have uncertainties Δ P  and Δ t , which are expressed by the uncertainty relation of the  N -particle system, Δ P Δ t ≥(1/2)|〈  G  〉|=( γ /2) k . This  P  −  t  uncertainty relation easily allows a discussion of the evolution of the entropy  S  in time  t  from  S ′ to  S  max . Now the irreversible steps are correctly described by the entropy production  P =d S /d t  &gt; 0, and the thermal equilibrium by  P =0, ∆ P =0, and thus the lifetime ∆ t =∞.

Particle Entropies and Entropy Quanta: IV. The Ideal Gas, the Second Law of Thermodynamics, and the P – t Uncertainty Relation