Abstract
Using results of number theory we develop an approximate statistical model of energy levels of particles in a three-dimensional infinite potential well depending on whether there is exactly one particle or more than one particles in the well. The model is used to perform a statistical inference about the number of particles in the well. The estimation procedure is developed within the Bayesian framework.
Highlights
An idealized statistical model of a potential well is considered in this article
We use number theory to describe and distinguish energetic spectrum of a single particle enclosed within a cubic potential well from energetic spectrum of total energy of a system of several particles
We introduce two important theorems which we use later to determine the asymptotic density of spectrum
Summary
An idealized statistical model of a potential well is considered in this article. Its link with some of known results of number theory and quantum physics is described. Theoretical model of a finite potential square well has found its application in theory of quantum well lasers [8]. We use number theory to describe and distinguish energetic spectrum of a single particle enclosed within a cubic potential well from energetic spectrum of total energy of a system of several particles. 2ml belongs to the set of natural numbers. It means that the rescaled energy on the left–hand side of Eq (2) of a single particle belongs to the set SE = {a2 + b2 + c2 : a, b, c ∈ N}. We call this set an energetic spectrum. If K ⊆ N is a finite set, there exists k ∈ N such that for every sufficiently large n ∈ N it holds A(n) ≤ (A ∪ K)(n) ≤ A(n) + k and A(n) ≥ (A − K)(n) ≥ A(n) − k
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