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arXiv:2201.06688

Rest energy effects for non-relativistic Fermi fields in external potentials

Fuad M. Saradzhev

Published Jan 18, 2022• 0 citations
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Abstract

The generalized Levy-Leblond equation (GLL) is used to study the bound state problem for a non-relativistic Fermi field in external potentials. A condition when the rest energy term can be removed from the GLL equation is determined. For external potentials which do not obey the condition, the bound state spectrum is affected non-trivially by the rest energy of the Fermi field. This is demonstrated for a spherical finite-depth well potential. The existence, number, and energies of bound states depend on the value of the rest energy.

DOI
10.48550/arxiv.2201.06688

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The generalized Levy-Leblond equation (GLL) is used to study the bound state problem for a non-relativistic Fermi field in external potentials. A condition when the rest energy term can be removed from the GLL equation is determined. For external potentials which do not obey the condition, the bound state spectrum is affected non-trivially by the rest energy of the Fermi field. This is demonstrated for a spherical finite-depth well potential. The existence, number, and energies of bound states depend on the value of the rest energy.
Non-relativistic Fermi particle in one-dimensional pseudoscalar δδδ-function potential
It is shown that a non-relativistic Fermi particle with a non-zero rest energy moving in a pseudoscalar δδδ-function potential in one dimension can be confined for both signs of the coupling constant. The binding energies depend on the value of the particle's rest energy, and in the limit of vanishing rest energy only one of the bound states survives. The coefficients of reflection and transmission are determined, and the conditions for complete reflection and transmission are discussed.
Based on the semiclassical(Thomas-Fermi) approximation,the macrocanonical partition function and macrocanonical potential of a relativistic Fermi gas in a general external potential in any dimensional space are studied.The general analytical expressions of the total particle number N,total energy U and heat capacity C have been derived with thermodynamic fundamental equations,where N,U and C are expressed by the modified Bessel function.Moreover,the analytical expressions of the total energy,chemical potential and heat capacity in the high-and low-temperature approximations are given.From these results,how the characteristics of the Fermi gas depend on various factors is discussed.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.
It is shown that $\ensuremath{\int}{|V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})|}^{\frac{3}{2}}{d}^{3}x>(\frac{{\ensuremath{\pi}}^{2}}{4}){(\frac{3{\ensuremath{\hbar}}^{2}}{2m})}^{\frac{3}{2}}$ is a necessary condition for one or more normalizable bound-state solutions to the nonrelativistic Schr\"odinger energy-eigenvalue equation for any potential function $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ such that the integral is finite. Moreover, for potentials such that $\ensuremath{\int}{|V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})|}^{\frac{3\ensuremath{\gamma}}{2}}{d}^{3}x$ is finite for a value or values of $\ensuremath{\gamma}>1$, the magnitude of a negative energy eigenvalue is related to the latter integral by a general inequality derived here.
The exact bound-state wavefunctions and the corresponding energy equation are calculated for a new generalized harmonic oscillator problem describing a spin-1/2 fermion in 3+1-dimensions, involving scalar, vector, and tensor couplings acting simultaneously within a particular plane of motion. For the scalar and vector coupling, singular harmonic oscillator shapes are considered, such that the singular term is needed to allow analytical solutions for the wavefunctions while preserving binding under adequate conditions. For the tensor sector,the Dirac oscillator potential is employed, which adds another independent binding mechanism to the problem. The exact bound-state solutions are computed by specifically tuning coefficients for an appropriate pair of \textit{Ansätze} for the radial functions, which leads to wavefunctions in terms of generalized Laguerre polynomials. Although the energy equation cannot provide a general expression for the energy spectrum, specific constraints on the quantum numbers as simple functions of the external potential parameters can be derived for it, which determines the conditions for bound solutions to exist, and of what type: particle, antiparticle or both. It is shown that the results can be simply mapped to the spherically symmetric analogue problem, and this is used to show that the general result encompass several previous particular cases of spherically symmetric harmonic oscillator problems in the Dirac equation available in the literature.