The non-relativistic and relativistic quantum theory with temperature

In this paper, we have proposed the principle of quantum thermodynamics, including energy principle and microcosmic entropy principle, and given the quantum thermodynamics of non-relativistic and relativistic quantum theory, i.e., the temperature-dependent schrodinger equation, Dirac equation and photon equation. We given the solution for wave function and energy level with temperature. Taking the hydrogen atom as an example, we given the temperature correction to hydrogen atom energy level and wave function.


Introduction
The study of the physical properties of materials, focusing on its thermodynamic properties, is of great interest in condensed matter physics, solid-state physics, and materials science [1,2]. The classical thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. The behaviour of quantum many-body systems driven out-of-equilibrium is one of the grand challenges of modern physics. The problem is particularly challenging when treating the dynamics of realistic finite-temperature systems, rather than systems evolving from the zero-temperature ground state. Recent development in the field of quantum thermodynamics taking place under nonequilibrium state [3][4][5][6][7]. The quantum thermodynamics which has grown rapidly over the last decade. It is fuelled by recent equilibration experiments [8] in cold atomic and other physical systems, the introduction of new numerical methods [9], and the discovery of fundamental theoretical relationships in non-equilibrium statistical physics and quantum information theory [10][11][12][13]. In this paper, we have proposed the principle of quantum thermodynamics, including energy principle and microcosmic entropy principle, and given the quantum thermodynamics of non-relativistic and relativistic quantum theory, i.e., the temperature-dependent schrodinger equation, Dirac equation and photon equation. We given the solution for wave function and energy level with temperature. Taking the hydrogen atom as an example, we given the temperature correction to hydrogen atom energy level and wave function, which should be tested by experiment. 2. The principle of quantum thermodynamics 1. Energy Principle: At temperature T , the classical total energy E of microcosmic particle is Where p 2 2m is particle kinetic energy, V (r) is particle potential energy, T S is called particle thermal potential energy, T is the temperature of particle in the external environment and S is particle microcosmic entropy.
At the i − th energy level, the classical microcosmic entropy of particle is where k B is the Boltzmann constant, n i is the particle numbers of each state in the i − th energy level, and the dimension of T S is the energy dimension. The Eq. (1) is the classical total energy of particle, it should become operator form in quantum theory, it isÊ =T +V (r) + TŜ.
2. Microcosmic Entropy Principle: The microcosmic entropy operator of particle iŝ The Eq. (1) can be explained. In thermodynamics, for the infinitely small processes, the entropy is defined as For the finite processes, there is At temperature T , when a particle has the microcosmic entropy S, it should has the thermal potential energy Q = T S. The total energy of particle should be the sum of kinetic energy, potential energy and thermal potential energy, the Eq. (1) is obtained.
The Eq. (4) can be explained. We can prove the following operator relation: With Eqs. (7)-(9), the operator T (−i) ∂ ∂T is not hermitian operator, but the operator 1 T ] is hermitian operator. The microcosmic entropy operator can be written as: The Eq. (4) has been obtained.

The Schroding equation including temperature
With the canonical quantization, (3), we can obtain the Schroding equation including temperature By separating variables substituting Eq. (12) into (11), we obtain the Eq. (14) can be written as: where E n = E 1n + E 2n , they are the undefined constants. The general solution of Eq. (11) is For the Eq. (16), we can obtain the solution and the solution of Eq. (19) is Where A is the normalization constant, and T 0 is the temperature constant. From the above results, we can give the temperature correction of hydrogen energy level and eigenfunction function, they are and From Eqs. (21) and (22), we can find the temperature has effect on the hydrogen atom energy level and wave function, but it has not effect on hydrogen atom spectrum. By the accurate measurement of hydrogen atom ionization energy, we can determine the temperature constant T 0 .
where µ is the reduced mass of hydrogen atom, e s = e(4πε 0 ) − 1 2 , E exp is the experimental value of the hydrogen atom ionization energy, the temperature constant T 0 can be determined by the following formula For the free particle of momentum p, the plane wave solutions and total energy are When T 0 → 0, it becomes the quantum theory without temperature.

The Dirac equation including temperature
In section 3, we have studied the quantum theory with temperature for the low energy non-relativistic particle. In the following, we should consider the high energy relativistic case. As is known to all, Dirac equation describes the particle of spin 1 2 , such as electron, by factorizing Einstein's dispersion relation, such that the field equation becomes the first order in time derivative [16]. Namely, Dirac factorized the relativistic dispersion relation employing four by four matrices, which is expressed as thus we get By canonical quantization Eq. (28), i.e., E → i ∂ ∂t , p → −i ∇, we can obtain the Dirac spinor wave equation where α and β are Dirac matrices, and ψ( r, t) is spinor wave function, they are and the Eq. (29) is the Dirac equation without temperature, its plane wave solution is Where the energy E = ± m 2 c 4 + c 2 p 2 , the u(p) matrix is When the particle of spin 1 2 at temperature T , we should consider the thermal potential energy Q = T S, the Eq. (27) should be written as it is similar to Eq. (29), we obtain the Dirac equation with temperature, it is substituting Eq. (10) into (33), we obtain by separating variables, we obtain the plane wave solution and energy of Eq. (36), they are Where

The photon quantum theory with and without temperature
With Dirac's factorization approach, we can obtain the spinor wave equation of free photon. For a photon, its mass m 0 = 0, Eq. (28) becomes By canonical quantization Eq. (39), we obtain the spinor wave equation of photon where H = −ic α · is Hamiltonian operator, and ψ is the spinor wave function of photon. For the proper Lorentz group L p , the irreducibility representations of spin s = 1 photon are D 10 , D 01 and D 6 four, respectively. We choose the spinor wave function of photon as the basis vector of three dimension irreducibility representation, i.e.
and the α matrices are denoted by The Eq. (40) is the spinor wave equation of photon without temperature, its plane wave solution and energy are where the u(p) matrix is The detailed theory can see the Appendix A of Ref. [15].
When the photon is in the external environment of temperature T , we should consider the thermal potential energy Q = T S, the Eq. (39) should be written as By canonical quantization Eq. (46), we obtain its spinor wave equation the Eq. (47) is the spinor wave equation of photon with temperature, its plane wave solution and energy are

Conclusions
In this paper, we have proposed the Principle of quantum dynamics, i.e., energy Principle and microcosmic Entropy Principle. With the canonical quantization, we obtained the temperature-dependent schrodinger equation, i.e., finite-temperature quantum thermodynamics equation. By separating variables, we given the eigen solution of quantum thermodynamics equation. Taking the hydrogen atom as an example, we given the correction of temperature to hydrogen atom energy level and wave function. By the accurate measurement of hydrogen atom ionization energy, we can determine the temperature constant T 0 . On that basis, we can further study multi-body quantum thermodynamics and finite-temperature quantum field theory.

Acknowledgment
This work was supported by the Scientific and Technological Development Foundation of Jilin Province (no.20190101031JC).