Particle in a Markov Cube by the Non-Classical Information Entropy Space

1. Introduction

The particle-in-the-box model it is commonly used as an approximation for more complicated quantum systems in which a particle is trapped in a narrow region of low electric potential between two high-potential barriers. These quantum well systems are important in optoelectronics [1,2,3,4,5,6,7,8,9]; they are used in devices such as the quantum well laser, the quantum well-infrared photodetector, and the quantum-confined Stark effect modulator [1,2,3,4,5,6,7,8,9]. The particle-in the-box model is also used to model a lattice in the Kronig-Penney model and for a finite metal with the free electron approximation [1,2,3,4,5,6,7,8,9]. Recently, researchers in China (C.Geng, X.Di, X.Tang, H.Han, 2023: [10]) developed a way to improve the quality of Cherenkov images using a flexible, non-toxic sheet of carbon quantum dots (cQDs) attached to the patient [11].

A Quantum Dot (QD) is a crystal of semiconductor material whose diameter is on the order of several nanometers. At this size, the free charge carriers experience “quantum confinement” in all three spatial dimensions [5]. QD particles are present inside the semiconductors used in microelectronics.These particles may contain an electron and a “hole” (absence of an electron); therefore, they are real-world “particle in a box” systems. Indeed, the electrons will never leave the interior of the QD particle of the semiconductor [5]. We note that, in QDs, the effects of particle size on the energy levels of the system can be easily investigated.

Santamato (1980 [12]) formulated de Broglie-Bohm quantum mechanics by relating the quantum potential to fundamental geometrical properties in Weyl’s geometry [13,14]. Based on the Weyl picture, J.T. Wheeler (1990 [15]) proposed a new picture of quantum theory in the context of the standard interpretation. Based on Weyl space, R. Wood and G. Papini (1995[4]16) developed a modified Weyl-Dirac theory that joins the particle aspects of matter and Weyl symmetry breaking. Novello, Salim and Falciano (2011[24]17) showed that there is a close connection between the de Broglie-Bohm interpretation of quantum mechanics and the Weyl integrable space. They demonstrated that Bohm’s quantum potential Q can be identified with the curvature scalar of the Weyl integrable space.

More recently, G.Resconi, I.Licata, and D.Fiscaletti (2013 [18,19]) showed that Bohm’s quantum potential Q can be derived from the minimum condition of Fisher information connected to the entropy S of a quantum system. In the non-classical entropic picture, the curvature of space indicates the Weyl-like gauge potential B_μ connected with the Fisher metric. Since Bohm’s quantum potential Q is a consequence of the Schrödinger Equation (SE) and vice versa, one can give a thermodynamic justification of the quantum mechanics modeled by the SE [18,19].

The Markov numbers are the positive integers that appear in the solutions of the equation: $m_1^2+m_2^2+m_3^2=3m_1{m}_2{m}_3$, where $\left(m_1,m_2,m_3\right)\in {\mathbb{N}}^3$ (Markov1879 [20]). These numbers are a classical subject in number theory, and have important ramifications in hyperbolic geometry, algebraic geometry, and combinatorics [21,22,23,24]. D.Zagier (1982 [25]) showed that the number M(x) of Markov numbers below the real number x is:

$$M(x)=c{(\log x)}^2+O(\log x{(\mathrm{loglogx})}^2)$$

(1)

where 𝑐 = 0.180717104711507 … is an explicit constant [25].

In the present article, we are investigating the exotic possibility that the quantum numbers (n_x,n_y,n_z) satisfy the Markov equation for a particle in the cube system, which we call the Markov cube. Based on Resconi,Licata, and Fiscaletti (2013 [18,19]), we find that the proposed particle follows from a non-classical information entropic space by the quantum potential Q. We also study the classical correspondents for this particle.

2. Particle in a Markov cube

The energy levels of the particle in the box system are given by:

$$E_{n_x,n_y,n_z}=\frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)$$

(2)

where

n_x = 1, 2, 3, . . . ∞, n_y = 1, 2, 3, . . . ∞, n_z = 1, 2, 3, . . . ∞

(3)

are the triplet of quantum numbers $\left(n_x,n_y,n_z\right)\in {\mathbb{N}}^3$. In the case of cubic symmetry$\left(L=L_x=L_y=L_z\right)$, Equation (2) becomes:

$${(E_{n_x,n_y,n_z})}_{Cube}=\frac{h^2}{8m{L}^2}\left(n_x^2+n_y^2+n_z^2\right)$$

(4)

The value of energy levels with the corresponding combinations and sum of the squares of the quantum numbers

$$n^2=n_x^2+n_y^2+n_z^2$$

(5)

is shown in Table 1.

The energy levels of the particle in a Markov-Cube system is given by:

$${(E_{m_x,m_y,m_z})}_{Markov-Cube}=\frac{h^2}{8m{L}^2}(m_x^2+m_y^2+m_z^2)=\frac{3h^2}{8m{L}^2}(m_x{m}_y{m}_z)$$

(6)

where

$$m_x^2+m_y^2+m_z^2=3m_x{m}_y{m}_z$$

(7)

is the Markov equation satisfied by the triplet of quantum numbers $\left(m_x,m_y,m_z\right)\in {\mathbb{N}}^3$ . The first few solutions of Equation (7) are the following quantum numbers:

$$\left(m_x,m_y,m_z\right)=\{\left(1,1,1\right),\left(1,1,2\right),\left(1,2,5\right),\left(1,5,13\right),\left(2,5,29\right)\}. \left[26\right]$$

(8)

The first two of these solutions {(1, 1, 1), (1, 1, 2)} can generate all other solutions: since Equation (7) is a quadratic in each of the variables, one integer solution leads to a second, and all solutions other than the first two singular ones have distinct values of $m_x$,$m_y$, and$m_z$, and share two of their three values with three other solutions (Guy 1994, p. 166 [27]). The Markov quantum numbers are then given by 1, 2, 5, 13, 29, 34 ... (OEIS A002559) [28]. The value of energy levels for a particle in a Markov Cube (see Equation (6)), with the corresponding combinations and sum of the squares of the quantum numbers are shown in Table 2.

Following Ref [26,29], we introduce the Quantum Vieta Jumping (QVJ) involutions as follows: If (m_x,m_y,m_z) is a Markov triple of quantum numbers, then, by the QVJ proposition, we have (m_x,m_y,3m_xm_y − m_z). Applying this quantum involution twice returns the same triple we started with. Joining each normalised Markov triple to the first, second, or third normalised triples we obtain from QVJ involution gives a graph starting from the ground state (1,1,1) of the quantum system (Figure 1). This graph is connected; in other words, every Markov triple can be connected to the (1,1,1) ground state of the quantum system by a sequence of QVJ involutions . For instance, starting with the (1,5,13) quantum state, we get its three neighbours, the (5,13,194), (1,13,34) and (1,2,5) quantum states, in the Markov quantum tree (Figure 1).

From Table 1 and Table 2, we observe that the Markov cube system and the ordinary cube system have the (1,1,1) ground state and the first excitation (2,1,1) in common, while all the other excitations are strongly divergent: for example, the difference between the nine excitation levels of the Markov and ordinary cube systems is in the order of O(MeV). This suggests that, at the O(eV) energy scale, in the Markov cube system, the particle is trapped in the first two energy levels; as we know, however, in the ordinary cube system, the particle can be at higher excitation energy levels. This strong divergence between the excitation energy levels may be an experimental signature of the proposed Markov cube system.

3. The Markov particle by non-classical information entropy space

Following G. Resconi, I.Licata, and D.Fiscaletti (2013 [18,19]), in the de Broglie-Bohm interpretation of quantum mechanics, the quantum potential emerges as an information medium determined by the vector of the superposition of entropies:

$$\begin{array}{l}{S}_1=k_B\log W_1\left({\theta }_1,{\theta }_2,\dots ,{\theta }_p\right)\\ S_2=k_B\log W_2\left({\theta }_1,{\theta }_2,\dots ,{\theta }_p\right)\\ \dots \\ S_n=k_B\log W_n\left({\theta }_1,{\theta }_2,\dots ,{\theta }_p\right)\end{array} [18,19]$$

(9)

where W is the number of the microstates for the same parameters, θ (temperature, pressure, etc.), and $k_B=1.38\times {10}^{-23}J/K$is the Boltzmann constant. In this picture, the quantum effects are equivalent to the geometry described by the following equation:

$$\frac{\partial }{\partial x^k}+\frac{{\partial }^2{S}_j}{\partial x^k\partial x^p}\frac{\partial x^i}{\partial S_j}=\frac{\partial }{\partial x^k}+k_B\frac{\partial W_j}{\partial x_h}=\frac{\partial }{\partial x^k}+B_{j,h}$$

(10)

where

$$B_{j,h}=\frac{\partial S_j}{\partial x_h}=k_B\frac{\partial \log W_j}{\partial x_h}=\frac{k_B}{W_j}{\partial }_h{W}_j$$

(11)

is a Weyl-like gauge potential [30,31]. A deformation of the moments p for the non-local change of this geometry is represented by the following expression:

$$A={\displaystyle \int \rho \left[\frac{\partial A}{\partial t}+\frac{1}{2m}\left(p_i+B_{k,i}\right)\left(p_j+B_{k,j}\right)+V\right]}dt{d}^{n}x$$

(12)

which gives

$$A={\displaystyle \int \rho \left[\frac{\partial A}{\partial t}+\frac{1}{2m}\left(p_i{p}_j+B_{k,i}{B}_{k,j}\right)+V\right]}dt{d}^{n}x$$

(13)

namely

$$A={\displaystyle \int \rho \left[\frac{\partial A}{\partial t}+\frac{1}{2m}\left(p_i{p}_j+\frac{\partial \log W_k}{\partial x_i}\frac{\partial \log W_k}{\partial x_j}\right)+V\right]}dt{d}^{n}x$$

(14)

that is,

$$A={\displaystyle \int \rho \left[\frac{\partial A}{\partial t}+\frac{1}{2m}{p}_i{p}_j+V+\frac{1}{2m}\left(\frac{\partial \log W_k}{\partial x_i}\frac{\partial \log W_k}{\partial x_j}\right)\right]}dt{d}^{n}x$$

(15)

The quantum action assumes the minimum value when $\delta A=0$, namely:

$$\delta {\displaystyle \int \rho \left[\frac{\partial A}{\partial t}+\frac{1}{2m}{p}_i{p}_j+V\right]}dt{d}^{n}x+\delta {\displaystyle \int \frac{\rho }{2m}\left(\frac{\partial \log W_k}{\partial x_i}\frac{\partial \log W_k}{\partial x_j}\right)}dt{d}^{n}x=0$$

(16)

and thus

$$\begin{array}{l}\frac{\partial A}{\partial t}+\frac{1}{2m}{p}_i{p}_j+V(x,y,z)+\frac{1}{2m}\left(\frac{1}{W_k^2}\frac{\partial W_k}{\partial x_i}\frac{\partial W_k}{\partial x_j}-\frac{2}{W_k}\frac{{\partial }^2{W}_k}{\partial x_i\partial x_j}\right)\\ =\frac{\partial S_k}{\partial t}+\frac{1}{2m}{p}_i{p}_j+V(x,y,z)+Q\end{array}$$

(17)

where

$$Q=\frac{1}{2m}\left(\frac{1}{W_k^2}\frac{\partial W_k}{\partial x_i}\frac{\partial W_k}{\partial x_j}-\frac{2}{W_k}\frac{{\partial }^2{W}_k}{\partial x_i\partial x_j}\right) [18]$$

(18)

Q is the Bohm quantum potential that is a consequence for the extreme condition of Fisher information [18]. On the basis of Ref [18], we interpret Q as an information channel determined by the functions W_k that define the number of microstates of the physical system under consideration. The latter depend on the parameters θ of the distribution probability. Now, by the definition of the quantum potential as an informational entity that pertains to the geometry (18), the quantum Hamilton-Jacobi equation is given by:

$$\frac{|\nabla S{|}^2}{2m}+V(x,y,z)+Q=-\frac{\partial S}{\partial t}$$

(19)

Equation (19) provides a new way to read the energy conservation law in quantum mechanics in the non-classical entropic picture, where the curvature of space indicates the Weyl-like gauge potential B_μ connected with the Fisher metric [18,19]. For the particle in the cube system, it is:

$$V\left(x,y,z\right)=\left\{\begin{array}{l}0,0<x<L,0<y<L,0<z<L\\ \infty ,elsewhere\end{array}\right\}$$

(20)

Physically, we set $V\left(x,y,z\right)=\infty$, so there is no chance that the particles penetrate the cube. This means that the wave function $\psi \left(x,y,z\right)$ must be exactly zero in the region where $V\left(x,y,z\right)=\infty$, making it impossible for the particle to occur beyond the cube.

Following D.Bohm (1951 [32,33]), and using Equation (18), we evaluate the quantum potential Q for the particle of mass m inside the cube, where $0<x<L$,$0<y<L$,$0<z<L$, as follows:

$$\begin{array}{l}{Q}_{n_x,n_y,n_z}=\frac{1}{2m}\left({\left(\frac{\nabla W}{W}\right)}^2-2\frac{{\nabla }^{2}W}{W}\right)=-\frac{{\hslash }^2}{2m}\frac{{\nabla }^2\psi (x,y,z)}{\psi (x,y,z)}\\ =\frac{1}{2m}\left[{\left(\frac{\pi n_x\hslash }{L}\right)}^2+{\left(\frac{\pi n_y\hslash }{L}\right)}^2+{\left(\frac{\pi n_z\hslash }{L}\right)}^2\right]=\frac{{\pi }^2{\hslash }^2}{2m{L}^2}\left(n_x^2+n_y^2+n_z^2\right)\end{array}$$

(21)

where

$$\psi (x,y,z)=\sin \left(\frac{n_x\pi }{L}x\right)\sin \left(\frac{n_y\pi }{L}y\right)\sin \left(\frac{n_z\pi }{L}z\right) [33]$$

(22)

and the triplet of quantum numbers $\left(n_x,n_y,n_z\right)\in {\mathbb{N}}^3$. Now, following D.Zagier (1982 [25]), for a particle in the cube system, the number M(W_j) of the quantum Markov triplet (m_x,m_y,m_z) below the number of microstates W_j of the physical system is:

$$M(W_j)=c{({\mathrm{logW}}_j)}^2+O({\mathrm{logW}}_j{({\mathrm{loglogW}}_j)}^2)$$

(23)

where 𝑐 = 0.180717104711507 … is an explicit constant [25,26]. Using Equation (9), the number M(W_j) of quantum the Markov triplet (m_x,m_y,m_z) below the number of microstates W_j of the physical system can be written in terms of the quantum entropy S_j:

$$M(W_j)\approx c{S}_{(Markov)}^2(W_j)$$

(24)

The Weyl-like gauge potential $B_{j,h}$ (Equation (11)) can be written in terms of the number M(W_j) of the quantum Markov triplet (m_x,m_y,m_z) below the number of microstates W_j of the physical system, as follows:

$$B_{j,h}^{(Markov)}=\frac{\partial S_j^{(Markov)}}{\partial x_h}=\frac{k_B}{2\sqrt{cM(W_j)}}{\partial }_{h}M(W_j)$$

(25)

By the Equation (11), the following condition holds:

$$B_{j,h}=B_{j,h}^{(Markov)}$$

(26)

namely,

$$\frac{k_B}{W_j}{\partial }_h{W}_j=\frac{k_B{\partial }_{h}M(W_j)}{2\sqrt{cM(W_j)}}$$

(27)

Thus

$${\partial }_h{W}_j=\frac{W_j}{2\sqrt{cM(W_j)}}{\partial }_{h}M(W_j)$$

(28)

Using Equation (28), the quantum potential Q given by Equation (21) can be written in terms of the number M(W_j) of the quantum Markov triplet (m_x,m_y,m_z) below the number of microstates W_j of the physical system, as follows:

$$\begin{array}{l}{Q}_{m_x,m_y,m_z}=\frac{1}{2m}\left(\frac{{\left(\nabla M\right)}^2}{4cM}-\frac{{\nabla }^{2}M}{\sqrt{cM}}\right)=-\frac{{\hslash }^2}{2m}\frac{{\nabla }^2{\psi }^{(Markov)}(x,y,z)}{{\psi }^{(Markov)}(x,y,z)}\\ =\frac{1}{2m}\left[{\left(\frac{\pi m_x\hslash }{L}\right)}^2+{\left(\frac{\pi m_y\hslash }{L}\right)}^2+{\left(\frac{\pi m_z\hslash }{L}\right)}^2\right]=\frac{{\pi }^2{\hslash }^2}{2m{L}^2}\left(m_x^2+m_y^2+m_z^2\right)=\frac{3{\pi }^2{\hslash }^2}{2m{L}^2}{m}_x{m}_y{m}_z\end{array}$$

(29)

where

$${\psi }^{(Markov)}(x,y,z)=\sin \left(\frac{m_x\pi }{L}x\right)\sin \left(\frac{m_y\pi }{L}y\right)\sin \left(\frac{m_z\pi }{L}z\right)$$

(30)

and the Markov triplet of the quantum numbers $\left(m_x,m_y,m_z\right)\in {\mathbb{N}}^3$ satisfies the Markov Equation (7).

4. The quantum–classical correspondence

The classical limit of quantum mechanics remains an unresolved problem [34]. Planck’s h → 0 limit [35] and Bohr’s n →∞ limit [36] are the oldest proposals for the formulation of the classical limit of quantum theory. For a particle of mass m in the cube system, from Equation (4) we note that the energy between adjacent levels is not constant [37]:

$$E_{n+1}-E_n=2n+1$$

(31)

This leads to

$$\frac{E_{n+1}-E_n}{E_n}=\frac{{\left(n+1\right)}^2-n^2}{n^2}=\frac{2n+1}{n^2}$$

(32)

In the Bohr’s limit n →∞,

$$\underset{n\to \infty }{\lim }\frac{E_{n+1}-E_n}{E_n}=\underset{n\to \infty }{\lim }\frac{2n+1}{n^2}=0$$

(33)

The adjacent energy levels become so close as to be practically indistinguishable.For a particle of mass m in the proposed Markov cube system, the situation is very different than that in the cube system due to the QVJ involutions between adjacent levels. Let the QVJ involution acting on the Markov quantum triplet$\left(m_x,m_y,m_z\right)\in {\mathbb{N}}^3$ be as follows:

$$QVJ:\left(m_x,m_y,m_z\right)\to \left({m^{\prime }}_x,{m^{\prime }}_y,{m^{\prime }}_z\right)=\left(m_x,m_y,3m_x{m}_y-m_z\right)$$

(34)

where

$${m^{\prime }}_x=m_x,{m^{\prime }}_y=m_y{m^{\prime }}_z=3m_x{m}_y-m_z$$

(35)

We calculate:

$$\begin{array}{l}{m^{\prime }}^2-m^2={m^{\prime }}_x^2+{m^{\prime }}_y^2+{m^{\prime }}_z^2-\left(m_x^2+m_y^2+m_z^2\right)=(m_x^2-{m^{\prime }}_x^2)+(m_y^2-{m^{\prime }}_y^2)+(m_z^2-{m^{\prime }}_z^2)\\ =m_z^2-{\left(3m_x{m}_y-m_z\right)}^2=-9m_x^2{m}_y^2+6m_x{m}_y{m}_z\end{array}$$

(36)

Therefore, we have

$$\begin{array}{l}{\left(\frac{E_{m^{\prime }}-E_m}{E_m}\right)}_z={\left(\frac{{m^{\prime }}^2-m^2}{m^2}\right)}_z=\frac{-9m_x^2{m}_y^2+6m_x{m}_y{m}_z}{m_x^2+m_y^2+m_z^2}\\ =\frac{-9m_x^2{m}_y^2+6m_x{m}_y{m}_z}{3m_x{m}_y{m}_z}=-\frac{3m_x{m}_y}{m_z}+2\end{array}$$

(37)

similarly:

$${\left(\frac{E_{m^{\prime }}-E_m}{E_m}\right)}_y={\left(\frac{{m^{\prime }}^2-m^2}{m^2}\right)}_y=-\frac{3m_x{m}_z}{m_y}+2$$

(38)

$${\left(\frac{E_{m^{\prime }}-E_m}{E_m}\right)}_x={\left(\frac{{m^{\prime }}^2-m^2}{m^2}\right)}_x=-\frac{3m_y{m}_z}{m_x}+2$$

(39)

In the Bohr’s limit n →∞,

$$\underset{m_z\to \infty }{\lim }{\left(\frac{E_{m^{\prime }}-E_m}{E_m}\right)}_z=\underset{m_y\to \infty }{\lim }{\left(\frac{E_{m^{\prime }}-E_m}{E_m}\right)}_y=\underset{m_x\to \infty }{\lim }{\left(\frac{E_{m^{\prime }}-E_m}{E_m}\right)}_x=2$$

(40)

Thus, for a Markov quantum state in the proposed Markov cube system, a classical limit n →∞ does not exist.

5. Conclusions

G.Resconi,, I.Licata,, and D.Fiscaletti (2013 [18,19]) derived Bohm potential Q from the minimum condition of Fisher information connected to the entropy S of a quantum system. In the non-classical entropic picture, the curvature of space indicates the Weyl-like gauge potential B_μ connected with the Fisher metric. Based on this interpretation of the Bohm potential Q, we show that the Markov cube system follows from a non-classical information entropic space by the quantum potential. We find that a classical limit n →∞ does not exist for a Markov quantum state due to the quantum Vieta jumping involutions between adjacent levels.