Energetic Spectrum of a Particle in Three-Dimensional Infinite Potential Square Well in Point of View of Number Theory and Bayesian Statistics

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.


Introduction
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].Quantum well structures can be grown by Molecular Beam Epitaxy (MBE) [2] and then used in active regions of lasers [1].Models of finite potential square wells can be approximated by a model of an infinite potential square well.
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.Measured values of total energy are used to estimate whether there is only one or more particles in the potential well.
We are concerned with a well-known model of a three-dimensional infinite potential square well.Infinite depth of the well could be interpreted as the infinite stiffness of walls of this cube.Thus, a particle does not loose anything of its energy by collisions with walls.Potential energy V of a particle is equal to 0 inside the well and V = ∞ outside.In other words, it is impossible for a particle to leave the well.
Solving stationary Schrodinger equation we obtain the energetic spectrum of a particle enclosed within a cubic potential well with an edge of length l.It is given by the formula where m is a mass of the particle, is the reduced Planck constant, and a, b, c ∈ N, referred to as quantum numbers, describe a state of the particle.For details see, e.g.[11].
It follows from the formula Eq. ( 1) that 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 a, b, c ∈ N}.We call this set an energetic spectrum.
From now on we assume that the total energy is expressed in this rescaled form.
We would like to estimate the probability that a randomly chosen natural number from the interval 1, n , n ∈ N, belongs to the set S E .For "large" n we approximate this probability by the number where S E (n) is a number of elements of the set S E which are less than or equal to n.The number d(S E ) is called the asymptotic density of the set S E .

Asymptotic Density of the Spectrum
Asymptotic density of the set A ⊆ N, well-known concept from number theory, is defined as follows.
Definition 1.Let A ⊆ N. A number of elements of the set A ⊆ N, which are less than or equal to n, is denoted by A(n).The number d(A), where is called upper asymptotic density of the set A and the number d(A), where is called lower asymptotic density of the set A.
is called asymptotic density of set A.
We immediately obtain from the definition the following basic properties of asymptotic density, which are further used.Lemma 1.Let A ⊆ N. Then d(A) and d(A) exist and the following statements hold:  Proof.The statements follow from basic properties of limes superior and limes inferior of sequences and from the following facts: • For every natural number n it holds 0 ≤ A(n) ≤ n.
• If K ⊆ N is a finite set, then there exists k ∈ N such that for every sufficiently large n ∈ N it holds and thus: We introduce two important theorems which we use later to determine the asymptotic density of spectrum.
The proof can be found e.g. in [6].
The proof can be found e.g. in [7].
We determine asymptotic density of energetic spectrum of a single particle at first.
Proof.Let us denote First, we determine d(B 2 ).It is known (see [11] and [5] We determine d(B 3 ) now.Theorem 1 says that a natural number n can be expressed as a um of three squares of integers if and only if the number n does not belong to the set We can see that We prove that d(A) = 1 6 .It can be easily verified that sets A j = {4 j (8k − 1) |k ∈ N}, j ∈ N ∪ {0} are pairwise disjoint and Since A = ∪ ∞ j=0 A j and sets A j are pairwise disjoint, it must hold where j n is the least integer such that there exists k ∈ N satisfying 4 jn (8k − 1) ≤ n.Hence and we find out that j n = ln n−ln 7 ln 4 . Thus where 0 ≤ ε j < 1.
Proof.We will prove this theorem by induction with respect to k.First, we will show that d(E 4 ) = 1.

Let us denote
Lagrange four square theorem (Bachet's conjecture) says that B 4 = N.Furthermore, it is known (see [3] and sequence A000534 in the OEIS) that where It is obvious that for a sufficiently large n it holds 0 and, using Eq. ( 14), it follows that Now, let us assume that d(E k ) = 1 and let us denote For a sufficiently large n it holds

Statistical Inference
The three-dimensional infinite potential square well is just a simplified model approximating physical potential wells which exist in real world.Even so, let us assume that such potential wells are available and we want to decide whether there is exactly one particle or there are more particles in a particular well.For that purpose we perform a virtual experiment based on repeated measurements of total energy of the system.We assume certain "ideal" properties of the particles: They are all of the same kind and they do not interact with each other.
We already know that energy E of a single particle belongs to the set S E (Eq. ( 2)) with the asymptotic density 5  6 (Thm.3).If there are k ≥ 2 particles in the well, the total energy of the system is equal to a sum of 3k squares of natural numbers.It means that in this case (Thm.4) the total energy of the particles can be equal to an arbitrary value from N excluding a set of zero asymptotic density.Now, suppose, that we are able to change the energy in a potential well and measure it repeatedly.Moreover, we suppose that the measurements are independent.Under appropriate conditions the asymptotic density of S E can be used to establish a statistical model of observed energies depending on the number of particles in the well.Particularly, if there is exactly one particle in the well, then the energy is surely in S E .If there are more than one particle, then we approximate the probability that the energy is in S E with the asymptotic density d(S E ).The model can be accepted under quite general conditions: Let us denote with p( ) the probability that a system with multiple particles has energy equal to .The probability that the energy is in S E is then is satisfied, e.g., if the energy E is approximately uniformly distributed on a set {1, . . ., N }, where N ∈ N is large enough so that S E (N ) N ≈ d(S E ).Another condition under which Eq. ( 16) is satisfied is that p( ) is concentrated on a discrete interval {M, . . ., N } where M, N ∈ N are large enough so that and points in S E are approximately uniformly distributed on {M, . . ., N }.For example, if M = 1, N = 10 4 and p( ) is the probability mass function of the binomial distributions with parameters (10 4 , 0.5), we get Remark 1.We can verify whether measured energy E belongs to S E algorithmically, see [9].
In what follows we derive a statistical procedure through which the number of particles can be inferred.Bayesian paradigm [10] is used for this purpose as it allows to employ prior knowledge.In problems of this type the prior information can be naturally acquired from similar experiments.

Notational Remarks
In this section random variables are denoted by uppercase Roman or Greek letters and their values by corresponding lowercase letters.However, often the random variables and their values need not be formally distinguished.In such cases lowercase letters are used whereas the exact meaning is clear from the context.All probability density functions -conditional as well as non-conditional -are solely denoted by the symbol f .Random variables to which they are related are distinguished by the arguments.For example, f (x, y) denotes the joint density of the random vector (X, Y ) at the point (x, y), f (x) is the marginal density of X at x , and f (x|y) is the conditional density of X at x given Y = y.By f (x|y = ξ) we denote the conditional density of X at x given Y = ξ.Random vector (X 1 , X 2 , . . ., X t ) and its value (x 1 , x 2 , . . ., x t ) are denoted by X 1:t and x 1:t , respectively.
Throughout this section the distributions of random variables are assumed to have densities with respect to either Lebesgue measure on R or counting measure on N 0 = N ∪ {0}.Distributions of random vectors are represented by densities with respect to a suitable product measure.

Bayesian Approach
Within the Bayesian paradigm the beliefs about the values of unknown quantities are represented with probability measures.In other words, the unknown quantities are taken as random quantities.Let θ be an unknown (vector) parameter.Its probability density function f (θ), referred to as prior probability density, is determined by the statistician so that it reflects the available prior information about θ, i.e., the knowledge which is available before data are observed or processed.The statistical model which describes dependence of a random quantity x on the unknown parameter θ is now understood as a conditional density f (x|θ).
After observing the value of x the conditional density f (θ|x) represents the overall knowledge about the parameter θ.It can be taken as a prior knowledge combined with the information acquired from data.The conditional density f (θ|x) is referred to as posterior density and relates to the model and prior density via Bayes rule where The integral Eq. ( 19) is understood as a Lebesgue integral over the range of θ with respect to the appropriate measure.
Because the denominator in Eq. ( 18) is completely determined by the product in the nominator the relation Eq. ( 18) can be shorten to where ∝ means proportionality up to a constant independent of θ.Bayesian calculations are typically made easier with this convention.

Statistical Model
In what follows it is assumed that individual realizations of a potential well are produced under identical conditions, whereas the numbers of particles in individual wells are independent.Each well is repeatedly inspected by measuring the total energy of particles in the system.Before each measurement the energy of the system is randomly changed so that the energies measured for a particular well can be taken as independent random quantities.Moreover, the energies are assumed to be uniformly distributed over {1, 2, . . ., N } for some large N ∈ N.For simplicity it is also assumed the same number, say n ∈ N, of measurements is done for each well.
Let k i be a number of particles in the i-th potential well and E i,j be a result of j-th energy measurement in c 2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING the i-th well.We define the following random variables: In words, s i = 1 if there is a single particle in the ith well and m i = 1 iff all of the energy measurements of the i-th system can be expressed as a sum of three squares of natural numbers.The variables s i are not observed and constitute a part of unknown parameter vector, while m i are observed and their values represent data.
The statistical model through which the data m i are related to the unobserved states s i is now given by From the assumptions formulated at the beginning of this subsection it follows that the parameters s i can be modelled as conditionally independent Bernoulli random variables given the probability that a well contains exactly one particle.This probability, denoted θ, does not change across individual potential wells and it is assumed to be unknown.Thus, θ is another part of the parameter vector.The joint density of s 1:t , for any t ∈ N, given θ is then where To specify the prior density of the parameter (s 1:t , θ) completely it remains to select a marginal prior distribution of θ.For simplicity, the prior density is selected as a beta density being a conjugate prior for the Bernoulli model, i.e., B (•, •) in Eq. (25) denotes the beta function.If no relevant prior information is available before the first measurement, the parameters can be set to (α, β) = ( 1 2 , 1 2 ), which corresponds to a non-informative Jeffreys prior for the Bernoulli model.For details see, e.g., [10].
The energy levels E ij and thus also data m i depend only on the corresponding parameters s i , i.e., given s i data m i are conditionally independent of all other quantities.Particularly, for all t ∈ N we get The joint density of m 1:t , s 1:t , θ thus can be factorized as follows: The conditional independence structure can be represented through a directed acyclic graph (DAG) D = (V, H) at Fig. 1 with V and H being a set of vertices and a set of edges, respectively.Its vertices are the random quantities θ, s 1 , . . ., s t , m 1 , . . .m t .The joint probability density function Eq. ( 27) is factorized with respect to D in the following sense: where P a(x) denotes a random vector represented by the parents of node x.
From Eq. ( 27) any conditional or marginal density of interest can be derived.Namely, for the posterior density of θ we get Denoting v t = t i=1 m i , the posterior density can be written for θ ∈ [0, 1] and f (θ|m The main concern is in a probability distribution of a number of particles in, say, the (t + 1)-th system given the measurements of the energies in the system and all preceding measurements which serve as a source of prior information about the (t + 1)-th system.The conditional density f (s t+1 |m 1:t , m t+1 ) can be again de- rived from Eq. ( 27) for (t + 1) instead of t.
Particularly, we get and where

Simulations
In this paragraph we give several illustrative examples of posterior densities derived above.In all of the examples the prior density is selected as the Jeffreys prior.We start with posterior densities f (θ|m 1:t ) estimated from simulated data for various numbers of measurements in each well, namely n = 2, 10, 20, and various numbers of produced wells t.The true value of the  parameter θ used for simulation is θ = 0.3.The posterior densities are plotted in Fig. 4. From the graphs it is seen how increasing number of observations, i.e., number of measurements in each well and/or number of wells, provides more information about the unknown parameter θ.With increasing number of observations the posterior densities gradually concentrate about the true value θ = 0.3.
Figure 2 illustrates five examples of posterior densities f (θ|m 1:t ) for different numbers of measurements n in each well.Namely, we assume t = 20, v t = 10, and n = 1, 5, 10, 15, 20.In this case the data are not sampled.Instead, it is assumed that for 10 of the 20 wells all measured energies were in the set S E .Although t and v t are the same in all situations, the acquired information about θ naturally varies with increasing number of measurements n.

Conclusions
The considered model of a potential well is definitely an idealized one.In this case known results of number theory can be used to derive a simple statistical model of the energy levels.In the future it would be interesting to compare the model with models constructed under more realistic assumptions including, e.g., finiteness of the well or specific assumptions about particle energies.Nevertheless, with more realistic models the posterior and predictive densities will be intractable and Monte Carlo algorithms or other approximation techniques will be required.