Entropy of State of Quantum System and Dynamic of This System

Malkhaz Mumladze

Gori University, Gori, Georgia.

DOI: 10.4236/oalib.1109052                

Abstract

This work defined the concept of entropy for states of a quantum system and built a stochastic model of a dynamic closed quantum system. This paper also is considered new approaches to the concepts of statistics and spin of elementary particles.

Keywords

State, Quantum System, Entropy, Random Value, Random Process

1. Introduction

In the algebraic approach, a quantum system is represented by the pair $\left(U,\Im \right)$ where the U is some $C^{\ast }$ -algebra observables, the $\Im$ is some set of states: positive linear functionals $f:U\to C$ with unit norm [1]. In work [2], for any state f we define probability measure on the spectrum $P_U$ of any $C^{\ast }$ -algebra $U$, which is the set of all pure states on this algebra. The set $P_U\subset {\displaystyle \underset{u}{\prod }{\sigma }_u}$ where ${\displaystyle \underset{u}{\prod }{\sigma }_u}$ is Tikhonov product of spectrums ${\sigma }_u$ of all Hermite operators in U.

On any ${\sigma }_u\subset \left[{\alpha }_u,{\beta }_u\right]$ we have Lebesgue measure $l_u$. On ${\displaystyle \underset{u}{\prod }{\sigma }_u}$ we also have product measure $L=\underset{u}{\otimes }{l}_u$ [3]. This measure induces on $P_U$ measure $L_{P_U}$. In work [4] we prove that there exists density distribution function ${\varphi }_f:P_U\to R$ for measure ${\mu }_f$, such that $0\le {\varphi }_f\left(p\right)\le 1$, $f={\displaystyle \underset{P_U}{\int }p{\varphi }_f\left(p\right)\text{d}L}$, ${\displaystyle \underset{P_U}{\int }{\varphi }_f\left(p\right)\text{d}L}=1$. Consider the function $F_f\left(p,t\right)$ which the module squar ${\phi }_f^2={\varphi }_f$.

Every elementary particle is dinamical system in the unperturbed state. They are stationary, it follows that function or functional which represents the particle must depend on the time. The function which depends on time and has module ${\phi }_f$ is function $F_f\left(p,t\right)={\phi }_f\left(p\right)\left(\cos \beta \left(p,t\right)+i\sin \beta \left(p,t\right)\right)={\phi }_f{\text{e}}^{i\beta \left(p,t\right)}$, this complex value function uniquely defines state f, the stationarity of $F_f$ gives that $\frac{\text{d}{\beta }^{\prime }\left(p,t\right)}{\text{d}t}=const$ and $\beta \left(p,t\right)=\omega \left(p\right)t+\theta \left(p\right)$. Hence $F_f\left(p,t\right)={\phi }_f\left(p\right){\text{e}}^{i\left(\omega \left(p\right)t+\theta \left(p\right)\right)}$. This function is like the solution of stationary Sch-rodinger equation. Let’s call it the wave function. $\omega t=\frac{Et}{\hslash }$, $k\left(p\right)=-\frac{\theta \left(p\right)}{d\left(p_0,p\right)}$, where the E energy of particle, d any metric on the space $P_U$ It follows $F_f\left(p,t\right)={\phi }_f{\text{e}}^{i\left(\frac{E}{\hslash }t-k\left(p\right)\left(d\left(p_0,p\right)\right)\right)}$. Identical particles in the same state have equal energies, so their wave functions may differ only in phase $\alpha$, $F_f\left(p,t\right)={\phi }_f{\text{e}}^{i\left(\frac{E}{\hslash }t-k\left(p\right)\left(d\left(p_0,p\right)+\alpha \right)\right)}$ We, known $\omega \left(p\right)$ is called frequency and $k\left(p\right)=-\frac{\theta \left(p\right)}{d\left(p_0,p\right)}$ is called wave number.

2. Concept of Entropy of State of Quantum State and It’s Dynamic

Let f is the state of quantum system it has some support $Supp{\mu }_f$, $\Gamma =\left\{h\right\}$ is a family of homeomorphisms $h:P_U\to P_U$ which save minimal open pseudo- convex cover [5] ${\left\{O_i\right\}}_{i=1,2,\cdots ,k}$ of support $Supp{\mu }_f$, and save distribution function ${\phi }_f\circ h={\phi }_f$ of measure ${\mu }_f$ defined by state f. Let $k_f$ is the number of orbit of action of homeomorphism $\left\{h\right\}$ on the cover ${\left\{O_i\right\}}_{i=1,2,\cdots ,k}$ and $n_f$ number of elements in family ${\left\{O_i\right\}}_{i=1,2,\cdots ,k}$. Let’s call the number $H_f=\frac{n_f}{k_f}$ the entropy of state f.

Let sequence of states $f_1,f_2,\cdots ,f_n,\cdots$ represents dynamic $h:P_U\to P_U$ of closed quantum system. As known, the entropy at the evolution closed system decreases, so $H_{f_1}\le H_{f_2}\le \cdots \le H_{f_n}\le \cdots$

Let $t_1<t_2<\cdots <t_n<\cdots$ ascending sequence of points in time i.e. entropy value ${\Im }_{t_n},n=1,2,\cdots$ the set of states of a quantum system whose entropy of is less than $t_n$, proceeding from the consideration that the entropy of the state of a quantum system should be quantized that the sets ${\Im }_{t_n},n=1,2,\cdots$ will be finite Define on the sets ${\Im }_{t_n}$ probability measures [6] ${\mu }_{t_n}$ as the measure ${\mu }_{t_n}$ of one point subset $\left\{f_{\beta }\right\}$ be number:

${\mu }_{t_n}\left(\left\{f_{\beta }\right\}\right)=\frac{H_{f_{\beta }}}{{\displaystyle \underset{\beta =1}{\overset{l_n}{\sum }}{H}_{f_{\beta }}}}$,

where $f_{\beta }\in {\Im }_{t_n}$ and $l_n$ number of elements in the set ${\Im }_{t_n}$. Measure for subset ${\left\{f_i\right\}}_{i=1,2,\cdots ,k\le l}\subset {\Im }_{t_n}$ is number:

${\mu }_{t_n}\left({\left\{f_i\right\}}_{i=1,2,\cdots ,k\le l_n}\right)={\displaystyle \underset{\beta =1}{\overset{k}{\sum }}{\mu }_{t_n}\left(\left\{f_{\beta }\right\}\right)}$.

This probability measure defines on the set ${\Im }_{t_n}$ random value ${\Psi }_{t_n}$ with distribution law [3] [5]:

${\Psi }_{t_n}=\begin{array}{cccc}\left\{f_1\right\},& \left\{f_2\right\},& \cdots ,& \left\{f_{l_n}\right\}\\ {\mu }_{t_n}\left(\left\{f_1\right\}\right),& {\mu }_{t_n}\left(\left\{f_2\right\}\right),& \cdots ,& {\mu }_{t_n}\left(\left\{f_{l_n}\right\}\right)\end{array}$

Let $E\left({\Psi }_{t_n}\right)={\displaystyle \underset{\beta =1}{\overset{l_n}{\sum }}{f}_{\beta }}{\mu }_{t_n}\left(f_{\beta }\right)$ is the mathematical expectation of random value ${\Psi }_{t_n}$ [6] [7]. The stochastic dynamics of a closed quantum dynamic system which is represented by states on any $C^{\ast }$ -algebra observables A may described by a sequence of random values:

${\Psi }_{t_1},{\Psi }_{t_2},\cdots ,{\Psi }_{t_n},\cdots ;n=1,2,3,\cdots$

The most probability realization of this random process [6] [7] will be a sequence of states:

$E\left({\Psi }_{t_1}\right),E\left({\Psi }_{t_2}\right),\cdots ,E\left({\Psi }_{t_n}\right),\cdots$

3. New Approaches to the Concepts of Statistics and Spin of Elementary Particles

The family ${\left\{O_i\right\}}_{i=1,2,\cdots ,k}$ is renumbered whit natural numbers, Let homeomorphism $h_{1j_{1}2j_2\cdots k{j}_k}:P_U\to P_U$ which save the cover ${\left\{O_i\right\}}_{i=1,2,\cdots ,k}$ of $Supp{\mu }_f$ such that $h_{1j_{1}2j_2\cdots k{j}_k}\left(O_i\right)=O_{j_i}$, $h_{1j_{1}2j_2\cdots k{j}_k}\left(O_{j_i}\right)=O_i$, $i=1,2,\cdots ,k$. We call such a map mirror map.

Let the family $\Phi =\left\{F_f^{\delta }\left(p,t\right)={\phi }_f{\text{e}}^{i\left(\frac{E}{\hslash }t-k\left(p\right)\left(d\left(p_0,p\right)\right)\right)}\right\}$ of all wave function of all particles which are in state f. They have same module, and they differ only in phase $k\left(p\right)\left(d\left(p_0,p\right)\right)$.

Theorem 1. If $F_f^{\delta }\circ h_{1j_{1}2j_2\cdots k{j}_k}=-F_f^{\delta }$, where $F_f^{\delta }\in \Phi =\left\{F_f^{\delta }\left(p,t\right)\right\}$ is the family of wave functions of identical particles which are in one state such, mirror map $h_{1j_{1}2j_2\cdots k{j}_k}:P_U\to P_U$ is only one.

Proof: let $F_f^{\delta }\circ h_{1j_{1}2j_2\cdots k{j}_k}=-F_f^{\delta }$, and ${h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}:P_U\to P_U$ other mirror map, for which also $F_f^{\delta }\circ {h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}=-F_f^{\delta }$. For composition $h_{1j_{1}2j_2\cdots k{j}_k}\circ {h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}$ mast be

$F_f^{\delta }\circ h_{1j_{1}2j_2\cdots k{j}_k}\circ {h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}=F_f^{\delta }\circ {h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}\circ h_{1j_{1}2j_2\cdots k{j}_k}=F_f^{\delta }$.

It means that $h_{1j_{1}2j_2\cdots k{j}_k}\circ {h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}={h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}\circ h_{1j_{1}2j_2\cdots k{j}_k}=I{d}_{Supp{\mu }_f}$. Thus ${h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}=h_{1j_{1}2j_2\cdots k{j}_k}^{-1}$, for mirror maps $h_{1j_{1}2j_2\cdots k{j}_k}=h_{1j_{1}2j_2\cdots k{j}_k}^{-1}$, It follows that $h_{1j_{1}2j_2\cdots k{j}_k}={h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}$. The theorem is proved.

Definition 1. We say that elementary particle is subject to the Bose-Einstein statistics if it’s wave function is symmetric with respect to mirror maps $h_{1j_{1}2j_2\cdots k{j}_k}:P_U\to P_U$, ( $F_f^{\delta }\circ h_{1j_{1}2j_2\cdots k{j}_k}=F_f^{\delta }$ ) and we say that elementary particle Is subject to the Fermi-Dirac statistic if it’s wave function is anti-symmetric with respect mirror maps:

$h_{1j_{1}2j_2\cdots k{j}_k}:P_U\to P_U,\left(F_f^{\delta }\circ h_{1j_{1}2j_2\cdots k{j}_k}=-F_f^{\delta }\right)$.

Theorem 2. In given quantum state may be located in only one Fermi-Dirac elementary particle.

Proof: let $F_f^{\delta }\in \Phi$, $F_f^{{\delta }^{\prime }}\in \Phi$, and $F_f^{{\delta }^{\prime }}\left(O_i\right)=F_f^{\delta }\left(h\left(O_i\right)\right)$ two wave function which located in one state, where $h:P_U\to P_U$ homeomorphism which saves the cover $O_i\in {\left\{O_i\right\}}_{i=1,2,\cdots ,k}$ and the module of the wave function. Every such homeomorphism performs a rotation at some angle area of value of function $F_f^{\delta }$ on complex plain around zero point. Obviously in case of mirror map this angle is $\pi$ radian. Let for $h:P_U\to P_U$ angle of rotation is $\alpha$, and $F_f^{\delta }\ne F_f^{{\delta }^{\prime }}$. If $h_{1j_{1}2j_2\cdots k{j}_k}:P_U\to P_U$ corresponds to $F_f^{\delta }$ and ${h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}:P_U\to P_U$ corresponds to $F_f^{{\delta }^{\prime }}$, we have:

$h\circ F_f^{\delta }\circ {h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}=-h\circ F_f^{\delta }$,

it follows:

$F_f^{\delta }\circ {h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}=-F_f^{\delta }$,

hence

${h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}=h_{1j_{1}2j_2\cdots k{j}_k}$.

From the theorem 1, follows

${h^{\prime }}_{1j_{1}2j_2\cdots k{j}_k}\ne h_{1j_{1}2j_2\cdots k{j}_k}$.

We took the opposite, our assumption $F_f^{\delta }\ne F_f^{{\delta }^{\prime }}$ was not correct. Hence $F_f^{\delta }=F_f^{{\delta }^{\prime }}$.

The theorem is proved.

From Theorem 2, follows that in case Fermi-Dirac particles the angle of rotation which corresponds to cover saving homeomorphism $h:P_U\to P_U$ may be $n\pi$, for mirror maps is $\pi \left(2k+1\right)$ and $\pi 2k$, $k=0,1,2,\cdots$ another for rotation maps.

For Bose-Einstein particles the mirror map $h_{1j_{1}2j_2\cdots k{j}_k}$ for which $h_{1j_{1}2j_2\cdots k{j}_k}\circ F_f^{\delta }=F_f^{\delta }$.

For Bose-Einstein particles, we do not have a theorem analogous to theorem 2. Therefore in one state may be any number of elementary particles.

Let new we have representation $T:G\to \text{turns}\left(E^2,E^2\right)$ of group G of mirror maps in tarns of complex flatness, which is considered a two-dimensional Euclidian vector space. For only one wave function $F_f^{\delta }$ of Fermi-Dirac particle which is in given state f, we have only one mirror map which transfers $F_f^{\delta }$ in $-F_f^{\delta }$. Let this mirror map is $h_{1j_{1}2j_2\cdots k{j}_k}$ and $T\left(h_{1j_{1}2j_2\cdots k{j}_k}\right)=h$ where h is tarn at angle $\left(2n+1\right)\pi ,n\in Z$. In state In state f is particle with wave function $F_f^{\delta }$ or particle with wave function $-F_f^{\delta }$, if assume that on each particle comes the half of the rotation angle $\frac{\left(2n+1\right)\pi }{2}=\left(n+\frac{1}{2}\right)\pi ,n\in Z$. Call the number $\left(n+\frac{1}{2}\right)\hslash ,n\in Z$ a spin of Fermi-Dirac particle.

For Bose-Einstein particles for each mirror map and for wave function $F_f^{\delta }$ of this particle we have $F_f^{\delta }\circ h_{1j_{1}2j_2\cdots k{j}_k}=F_f^{\delta }$ it means that for representation $T:G\to \text{turns}\left(E^2,E^2\right)$ $T\left(h_{1j_{1}2j_2\cdots k{j}_k}\right)=h=const=n\pi ,n\in Z$ in this case on each particle comes the total rotation angle $n\pi ,n\in Z$. Call the number $n\hslash ,n\in Z$ a spin of Bose-Einstein particle.

4. Conclusions

1) In the paper is defined entropy of states of some quantum system.

2) In the paper is constructed discrete random process describing the evolution of the states of a closed dynamical system.

3) In the article, we are in a new way to prove the fact that a particle subject to Fermi-Dirac statistics can only be one in a given state, while particles that are subject to Bose-Einstein statistics can be of any number in a given state.

4) In the article, we are a new way to introduce the notion of spin for elementary particles.

Conflicts of Interest

The author declares no conflicts of interest.

References

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