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The non-relativistic and relativistic quantum theory with temperature

Published Apr 5, 2022• 0 citations
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Abstract

In this paper, we have proposed the non-Hermitian theory of quantum thermodynamics, and given the quantum thermodynamics equations, including non-relativistic and relativistic quantum theory with temperature, i.e., the temperature-dependent schrodinger equation, Dirac equation and photon equation. We give the solution of wave function and energy level with temperature, and introduced the temperature constant T0, which should be determined by the atom or molecule spectrum experiments. PACS: 03.65.-w, 05.70.Ce, 05.30.Rt.

DOI
10.33140/pcii.05.02.05

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In this paper, we have proposed the non-Hermitian theory of quantum thermodynamics, and given the quantum thermodynamics equations, including non-relativistic and relativistic quantum theory with temperature, i.e., the temperature-dependent schrodinger equation, Dirac equation and photon equation. We give the solution of wave function and energy level with temperature, and introduced the temperature constant T0, which should be determined by the atom or molecule spectrum experiments. PACS: 03.65.-w, 05.70.Ce, 05.30.Rt.
Abstract 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.
This chapter introduces the concepts of the relativistic theory to basic problems of quantum physics, in particular, the extension of the Schrodinger equation to a relativistic form. Starting from the fundamental laws of relativity, the chapter evaluates the Klein–Gordon equation and investigates if the equation can be considered a generalization of the non-relativistic Schrodinger equation to the case of the relativistic energy. The chapter shows that the wave function, which is a solution to the Klein–Gordon equation, cannot be connected with the probability wave function. For this reason, the Dirac equation is used to solve the problems faced by the Klein–Gordon equation. The Dirac equation also includes the spin. The advantage of the Dirac equation over the Schrodinger and Klein–Gordon equations is that it naturally includes the spin. In the Schrodinger and Klein–Gordon equations, the spin is not present and has to be added to the wave function manually whenever the wave function determines a system with the spin.
The major problem in quantum physics is to find the wave function of a given physical system and to understand (predict) how the wave function of the system evolves in time, or how it changes under external influences. In 1926, Erwin Schrodinger predicted that the wave function of a given physical system might be completely determined if the total energy of the system was known. Since the equation represents a real physical system, it must satisfy the following conditions: the equation must be linear and coefficients appearing in this equation should only depend on the parameters characteristic of the particle. This chapter focuses on to the non-relativistic case only. It considers the case of a free particle moving in one dimension. The chapter shows that the Schrodinger equation can be obtained from the energy of the free particle. It shows that the Schrodinger equation guarantees the conservation of normalization of the wave function.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
We have developed a theoretical formalism to introduce temperature as a parameter into the framework of non-relativistic quantum mechanics using the laws of classical thermodynamics and the canonical ensemble scheme of statistical mechanics. A self-consistent Hamiltonian has then been constructed for a given quantum many-body system which includes the effect of temperature in the form of correction terms added to the corresponding zero-temperature Hamiltonian of the system. Investigating some quantum mechanical systems with exact zero-temperature solutions including the particle-in-a-box model, the free particle, and the harmonic oscillator within our finite-temperature approach up to the first order of self-consistency has led to temperature-dependent Hamiltonians describing these systems above absolute zero without encountering any physically unacceptable brand of behavior for their wave functions and energy spectra. Results firmly support the view that a quantum mechanical system at a finite temperature behaves as if it is in a zero-temperature excited state.