Eigen Value Equation: Dirac Particles and Dirac Oscillators

Eigen Value Equation Dirac Particles and Dirac OscillatorsThe thermodynamic entities with the GUP for Dirac particlesand Dirac oscillatorsXin-feng Diao,Chao-yun Long,Guang-yu sun,Yuan-sheng wang,Hongling liuAbstractIn this paper we placevass the Eigen regard as comparison for Dirac particles and Dirac Oscillators, considering the spin and Generalized Uncertainty dominion. Then we compute the thermodynamic entities for them with the Generalized Uncertainty Principle corrected. We find that an electron of circumstances m and spin 1/2 in relativistic quantum mechanism confined in a box with the length L which the eigenvalues were cerebrate to the length of box and the correction terms of the Helmholtz free susceptibility prevail to increase the thermodynamic quantities.PACS number 03.65.-w, 11.10.NxKey words GUP Dirac particles Dirac Oscillators thermodynamic entities. demonstrationVarious candidates of quantum gravity such as string theory and shocking hole physics concern the existence of a minimum mensural 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 discernible length 3 (1)where is the GUP parameter and is a positive constant which depends on the hope value of the momentum street girl. On the other hand, Braun Majumder has discussed the harmonic oscillators pursual Maxwell-Boltzmann statistics by solving the Schrdinger compare4. However for the high animation 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 office and other thermodynamic quantities for the relativistic particles side by side(p) 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.+ telecommunicate emailprotected.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 inner(a) the box, where potential energe while and outside. The Dirac equation can be given as 3where and are the Dirac matrices with the following representation , 4Using the Jacobi identity element 5. And we can solve the equation with the method in the paper 6. By defining, 5considering the boundary conditions, we can get the efficacy Eigen value6Wrong calculationAnd we utilized the GUP corrected energy spectrum to report 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 7So the partition function is ill-treat toowith 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 8where N is total number of Dirac particles. We arrange that the correction terms hightail it to increase the thermodynamic quantities.Then, we consider the Dirac oscillator and get the exact antecedent below 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 termUsing the operator relation (16)the equation becomes (17)And we can get (18)With the method of 2, the energy of the e quation (18) will be obtained. (19)If we set , the result becomes , which was well agree with non-relativistic quantum mechanics. And hence 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 rst term as the partition function of Dirac Oscillators, the equation can be rewritten aswith. This equation guides us to write the GUP modify 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.SummaryIn this paper, we analyze 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 oscil lator and get the exact solution under a harmonic term, although the GUP corrected Hamiltonian of the harmonic oscillator has investigated7-10. We consider the dissimilar 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 jump of Guizhou Province Science and Technology OfficeNo. 20132255 and Guizhou Normal College project 12YB005 .References1 K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B 234 (1990) 276.2 M. Maggiore, Phys. Lett. B 304 (1993) 65.3 A. Kempf, G. Mangano, R.B. Mann, Phys. Rev. D 52 (1995) 1108.4 Barun Majumder , Sourav Sen. Physics garner B 717 (2012) 2912945 H. Hassanabadi a, S. Zarrinkamar b, A.A. Rajabia. Physics Letters B 718 (2013) 111111136 A. Kempf, J. Phys. A 30 (1997) 2093.7 Pouria Pedram. Phy sics Letters B 710 (2012) 4784858 P. Pedram, Phys. Rev. D 85 (2012) 024016, arXiv1112.2327.9 K. Nozari, T. Azizi, Gen. Rel. Grav 38 (2006) 735742K. Nozari, H. Mehdipour, Chaos Solitons Fractals 32 (2007) 1637K. Nozari and A.S. 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