Study on the Phenomena of Potential Well of Infinite Depth on the View Point of Natural Statistical Physics

In this paper, we study the phenomena of potential well of infinite depth on the view point of natural statistical physics. The mathematical model of this physical system is the system of micro-particles moving periodically with constant velocity in the interval D = [−a, a], (a > 0).
Thereby we obtain the structure of this physical system at the stationary state.
Thus we clarify that this physical system is the composed state of the proper physical systems at the stationary state and the ratio of their composition is given by the sequence

{|an|2}n=0 ∪ {|bn|2}∞n=1,

where an, (n ≥ 0) and bn, (n ≥ 1) are the Fourier type coefficients of the initial state ψ ∈ L2.
Then we obtain the energy expectation of the total physical system
　　　 　
Ē =∞Σn=0 n2π2ℏ2/2ma2 (|an|2 + |bn|2),
　　　 　
where m denotes the mass of one micro-particle.