Eigen Value Equation: Dirac Particles and Dirac Oscillators

The thermodynamic entities with the GUP for Dirac particles and Dirac oscillators

• Xin-feng Diao, Chao-yun Long, Guang-yu sun, Yuan-sheng wang, Hongling liu

Abstract

In this paper we studied the Eigen value equation for Dirac particles and Dirac Oscillators, considering the spin and Generalized Uncertainty Principle. Then we calculated the thermodynamic entities for them with the Generalized Uncertainty Principle corrected. We find that an electron of mass m and spin 1/2 in relativistic quantum mechanics confined in a box with the length L which the eigenvalues were related to the length of box and the correction terms of the Helmholtz free energy tend to increase the thermodynamic quantities.

PACS number: 03.65.-w, 11.10.Nx

Key words: GUP; Dirac particles ; Dirac Oscillators; thermodynamic entities.

Introduction

Various candidates of quantum gravity such as string theory and black hole physics concern the existence of a minimum measurable length. When energies of particles are much smaller than the scale of Planck mass [1, 2], it gives rise to the so-called Generalized Uncertainty Principle that results in a minimum observable length [3]

(1)

where is the GUP parameter and is a positive constant which depends on the expectation value of the momentum operator. On the other hand, Braun Majumder has discussed the harmonic oscillators following Maxwell-Boltzmann statistics by solving the Schrödinger equation[4]. However for the high energy particles we should consider the relativistic effect, so that it is important to study the effect of the Generalized Uncertainty Principle for Dirac particles and Dirac oscillators. Further more, the canonical partition function and other thermodynamic quantities for the relativistic particles following Maxwell-Boltzmann statistics should be involved. We discused the aspects in this paper.

The Generalized Uncertainty Principle can be obtained from the deformed commutation relation,

(2)

where . The limits andcorrespond to the ordinary quantum mechanics and extreme quantum gravity, respectively.

⁺Email: diaoxinfeng77@126.com.

Now let us consider an electron of mass m and spin 1/2 in relativistic quantum mechanics, confined in a box of length L. The boundaries of the box are located at and. The wave function of the particle satisfies the following GUP corrected Dirac equation inside the box, where potential energe while

and outside. The Dirac equation can be given as

（3）

where and are the Dirac matrices with the following representation

, （4）

Using the Jacobi identity [5]. And we can solve the equation with the method in the paper [6]. By defining

, （5）

considering the boundary conditions, we can get the energy Eigen value

（6）

Wrong calculation

And we utilized the GUP corrected energy spectrum to calculate the canonical partition function and other thermodynamic quantities for the relativistic particles following Maxwell-Boltzmann statistics. So, we first calculated the GUP corrected partition function and it can be expressed as

（7）

So the partition function is wrong too

with the. For the case of indistinguishable particles we use the relation of Helmholtz free energy with partition function . For the Helmholtz free energy and it comes out to be

（8）

where N is total number of Dirac particles. We found that the correction terms tend to increase the thermodynamic quantities.

Then, we consider the Dirac oscillator and get the exact solution under a harmonic term. Firstly, Dirac equation is written as [7]

(9)

where U₀ and V₀ denote scalar and vector interactions, respectively, and the matrices are

， (10)

The spin wave function can be written as

(11)

We expand the equation and get the coupled equations

(12)

Then, (13)

Pluging in to (12), we can obtain

(14)

(15)

Here, we consider the harmonic term

Using the operator relation

(16)

the equation becomes

(17)

，

And we can get

(18)

With the method of [2], the energy of the equation (18) will be obtained

. (19)

If we set , the result becomes , which was well agree with non-relativistic quantum mechanics. And then we calculate the thermodynamic entities with the GUP corrected energy eigenvalue equation for the Dirac Oscillators. The partition function can be evaluated as

. (20)

We do this sum in a perturbative sense to distinguish the first term as the partition function of Dirac Oscillators, the equation can be rewritten as

with. This equation guides us to write the GUP modified Helmholtz free energy as

. (21)

Simply, we write the expressions for the entropy internal energy as

(22)

where N is total number of Dirac oscillators.

Summary

In this paper, we studied an electron of mass m and spin 1/2 in relativistic quantum mechanics, which was confined in a box with the length L, We found that the eigenvalues were related to the length of box and the correction terms of the Helmholtz free energy tend to increase the thermodynamic quantities. We consider the Dirac oscillator and get the exact solution under a harmonic term, although the GUP corrected Hamiltonian of the harmonic oscillator has investigated[7-10]. We consider the different operator relation and get the partition function for the Dirac Oscillator. Moreover, we calculated the thermodynamic entities with the GUP corrected energy Eigen value equation for the Dirac Oscillator. So exploring relations in the basic foundations of the GUP is worth interesting [11].

This work was Supported by the Project of Guizhou Province Science and Technology OfficeNo. [2013]2255 and Guizhou Normal College project: 12YB005 .

References

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