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Theoretical and Mathematical Physics

Two-Dimensional Motion of a Slow Quantum Particle in the Field of a Central Long-Range Potential

Vasily Veniaminovitch Pupyshev

Published Jun 1, 2019• 4 citations
Publisher

Abstract

We study the two-dimensional motion of a slow quantum particle in the field of a central long-range potential decaying in the limit of long distances r as the power r −β with the exponent β ∈ (1, 2). We find the low-temperature asymptotic behavior for all partial phases and the differential cross section of the particle scattering and derive a rather simple approximation for the weakly bound state energy.

DOI
10.1134/s0040577919060059

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We study the two-dimensional motion of a slow quantum particle in the field of a central long-range potential decaying in the limit of long distances r as the power r −β with the exponent β ∈ (1, 2). We find the low-temperature asymptotic behavior for all partial phases and the differential cross section of the particle scattering and derive a rather simple approximation for the weakly bound state energy.
We study the finite two-dimensional motion of a quantum particle in the field of the potential V(r)=−V0r−βV(r) = - {{V}_{0}}{{r}^{{ - \beta }}}V(r)=−V0​r−β with parameters V0>0{{V}_{0}} > 0V0​>0 and β∈(0,2)\beta \in (0,2)β∈(0,2) . An analysis of an asymptotic equation equivalent to the Bohr–Sommerfeld quantization rule is given. As a result, a simple and explicit approximation for the energies of weakly-bound states of this particle has been obtained.
The weakly interacting Bose gas in two dimensions is considered in the dilute limit n1/2{n}^{1/2}n1/2a\ensuremath{\ll}1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system has two separate divergences: the first, associated with classical phase fluctuations, is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is Tc{T}_{c}Tc​\ensuremath{\approxeq}4\ensuremath{\pi}(${\ensuremath{\Elzxh}}^{2}$/2m)n/[ln ln(1/na2{\mathrm{na}}^{2}na2)], and the condition of validity of the dilute limit is ln ln(1/na2{\mathrm{na}}^{2}na2)\ensuremath{\gg}1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.
The present theory of low-energy scattering by a long-range potential has been examined by omitting the assumptions about the wavefunctions and taking the phase to be a function of the position of the scattered particle. The resulting integro-differential equation for the phase shift has been simplified for a particular case and it is found that the magnitude of the correction terms so obtained is quite large. It is shown that the two-potential formalism can be applied only to light pairs of scattering atoms.
Magnetic potentials have a direct significance to the motion of particles in quantum mechanics. This is known as the Aharonov-Bohm effect. We study this quantum property in the scattering by two magnetic fields at large separation in two dimensions. We derive the asymptotic formula of scattering amplitude when the distance between the centers of two fields goes to infinity. The result depends on the fluxes of fields and on incident and final directions. We here consider only the simple case that at least one of two fluxes is zero.
Quantal systems are predicted to show a changeover from exponential to a slower decay rate at very long times. Asymptotically most models predict power-law decay with integer exponents. However, the postexponential decay of a trapped particle from a potential can occur with a continuous range of power-law exponents. We show that this happens when the outer part of the potential is repulsive and decreases with the inverse square of the distance. We demonstrate a simple relation between the strength of the long-range tail and the power-law exponent. We also give explicit forms for the postexponential decay laws when the outer part of the potential is weakly attractive. These systems are amenable to experimental scrutiny.