Highlighting relations between Wave-particle duality, Uncertainty principle, Phase space and Microstates

Quantum and Relativistic corrections to Maxwell-Boltzmann ideal gas model from a Quantum Phase Space approach

“…This formulation has been introduced and developed firstly in the references [1,[33][34][35].This approach deals with a formulation of quantum mechanics but the phase space which is considered is a classical one. Another approach has been considered and developed through the works [22][23][24][25] in which the concept of "quantum phase space" is introduced. For a monodimensional case, this quantum phase space was defined as the set {(〈𝑥〉, 〈𝑝〉)} of the possible mean values 〈𝑥〉 and 〈𝑝〉 of the coordinate operator𝒙and momentum operator𝒑corresponding to quantum states denoted |〈𝑧〉⟩ which are themselves eigenstates of the operator 𝒛defined by the relation [22][23][24][25] 𝒛 = 𝒑 − 2𝑖 ℏ ℬ𝒙 (7) in which ℬ is the statistical variance of the momentum operator corresponding to the state |〈𝑧〉⟩ itself.…”

“…It is possible to define a phase space representation of quantum mechanics using the set {|〈𝑧〉⟩} of the eigenstates |〈𝑧〉⟩of the operator 𝒛 [24][25].…”

“…in which 𝜁 𝑄 is the one particle partition function which is given by the relation (25).From the relations ( 25) and ( 30), one can deduces…”

“…In this work, we consider the study on new kind of corrections for the Maxwell-Boltzmann ideal gas model which can be considered as related with the quantum nature of phase space. The approach uses results from the references [22][23][24][25].…”

“…• 𝜆 𝑡ℎ = ℎ (2𝜋𝑚𝑘 𝐵 𝑇) 1/2 is the thermal de Broglie wavelength The integration which is considered in this expression is performed over the phase space domain {(𝑥 , 𝑝 )} which corresponds to the set of the possible classical microstates of the particle. The Planck constant ℎ appears here because it is related to the fact that the phase space hypervolume of a microstate is finite (not equal to zero) [25][26]. For an ideal gas composed of 𝑁 particles, the canonical classical partition function 𝒵 𝐶 is approximately given by the relation (particles are considered as indiscernible)…”