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arXiv:2207.11970

Non-relativistic Fermi particle in one-dimensional pseudoscalar $δ$-function potential

Fuad M. Saradzhev

Published Jul 25, 2022• 0 citations
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Abstract

It is shown that a non-relativistic Fermi particle with a non-zero rest energy moving in a pseudoscalar $δ$-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.

DOI
10.48550/arxiv.2207.11970

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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.
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.
The solution of the one-dimensional Dirac equation is examined for the symmetrical double delta-function potential. A formula for the relativistic transmission coefficient is derived from the solution. The maximum condition as well as the minimum condition for the transmission coefficient is determined. The transmission coefficient is shown to be equal to unity in its maximum. In its minimum, it is smaller than the transmission coefficient for the single delta-function potential of the twofold strength. Further, some differences between the relativistic and non-relativistic transmission coefficient are presented. It is found that the wave numbers, at which the maxima and minima occur according to the non-relativistic treatment, are shifted by the relativistic treatment. Unlike the non-relativistic transmission coefficient, the relativistic transmission coefficient does not approach unity in the range of high energies. It oscillates between unity and a constant. The value of this constant depends on the strength of the delta-interaction. Thus, the relativistic corrections to the transmission coefficient for the symmetrical double delta-function potential are found to represent significant shifts in its characteristics.
We have observed two-particle bound states of atoms confined in a one-dimensional matter waveguide. These bound states exist irrespective of the sign of the scattering length, contrary to the situation in free space. Using radio-frequency spectroscopy we have measured the binding energy of these dimers as a function of the scattering length and confinement and find good agreement with theory. The strongly interacting one-dimensional Fermi gas which we create in an optical lattice represents a realization of a tunable Luttinger liquid.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
Abstract Introduced eighty years ago, the Fermi pseudopotential has been a powerful concept in multiple fields of physics. It replaces the detailed shape of a potential by a delta-function operator multiplied by a parameter giving the strength of the potential. For Cartesian dimensions d > 1, a regularization operator is necessary to remove singularities in the wave function. In this study, we develop a Fermi pseudopotential generalized to d dimensions (including non-integer) and to non-zero wavenumber, k . Our new fractional calculus approach has the advantage of circumventing singularities that occur in the wave function at integer values of d while being valid arbitrarily close to integer d . In the limit of integer dimension, we show that our generalized pseudopotential is equivalent to previously derived s -wave pseudopotentials. Our pseudopotential generalizes the operator to non-integer dimension, includes energy ( k ) dependence, and simplifies the dimension-dependent coupling constant expression derived from a Green’s function approach. We apply this pseudopotential to the problem of two cold atoms ( k → 0) in a harmonic trap and extend the energy expression to arbitrary dimension.