The Ideal Gas Behavior is a Macroscopic Manifestation of Quantum Mechanics
This article first consists in the quantum mechanical study of an adiabatically compressed particle, residing in an infinitely high potential well, which we will consider, as the basis of an ideal gas. Thus we prove in both non- relativistic and relativistic cases, that, all the compression energy is transformed into extra kinetic energy of the particle. The result may be intuitive, but does not seem trivial. Thus it shows a full accordance between Quantum Mechanics, Newton's Second Law of Motion, the Law of Energy Conservation, Special Theory of Relativity, and furthermore Classical Thermodynamics and Kinetic Theory of Gases. In effect, on the same basis, one can quantum mechanically derive, an essential relationship associated with an ideal gas, that is Pressure x (Volume)5/3=Constant, with regards to any set of adiabatic transformations, the gas at hand, would undergo. More important, by doing so, one as well, specifically derives the value of the Constant making the RHS of this relationship, which has been otherwise left obscure since the time Quantum Mechanics came into the scene. Our finding means that, the ideal gas behavior, is just a macroscopic manifestation of Quantum Mechanics. In other words i) it can be derived, by all means, based on the quantum mechanics of a single particle, ii) along that line, it excludes any interaction of the constituents, the gas, embodies. The latter is something known, for it is classically assumed, based on the Kinetic Theory of Gases. However we arrived at it, via the quantum mechanical approach we have deployed. Thus we end up with a precise definition of an ideal gas: It is a gas, where the behavior can be predicted based on the quantum mechanics of a single particle imprisoned in a box. This is how the classical assumption as to, it should be free of any interaction of its constituents, whatsoever, comes to find a deeper root.