arXiv:2603.20703

Quantum corrections and multioccupancy in a semi-classical gas

Loris Ferrari

quant-phSubmitted Mar 21, 2026• Announced Mar 24, 2026

Abstract

The quantum corrections to the behavior of a semi-classical gas can be expressed as a power series of the ratio $eta$ between the cube of De Broglie's thermal wavelength and the specific volume. The connection between $eta$ and the multioccupancy of quantum states is the aim of the present work. By means of a chemical/physical approach it is possible to associate $eta$ to the concrete realization of multioccupancy in momentum space, through the formation of what we call "pseudo-molecules", i.e. multiplets of particles sharing momenta whose differences are negligeable in the continuous spctrum approximation. The pseudo-molecules result as casual consequences of multiple scattering processes among non interacting (scattering apart) particles.

Comments: 10 pages

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arXiv:2603.20703

Quantum corrections and multioccupancy in a semi-classical gas

Loris Ferrari

quant-phSubmitted Mar 21, 2026• Announced Mar 24, 2026

arXiv HTML PDF

Abstract

The quantum corrections to the behavior of a semi-classical gas can be expressed as a power series of the ratio

eta

between the cube of De Broglie's thermal wavelength and the specific volume. The connection between

eta

and the multioccupancy of quantum states is the aim of the present work. By means of a chemical/physical approach it is possible to associate

eta

to the concrete realization of multioccupancy in momentum space, through the formation of what we call "pseudo-molecules", i.e. multiplets of particles sharing momenta whose differences are negligeable in the continuous spctrum approximation. The pseudo-molecules result as casual consequences of multiple scattering processes among non interacting (scattering apart) particles.

Comments: 10 pages

Related papers

Quantum corrections and multioccupancy in a semi-classical gas

L. Ferrari

Jan 1, 20260 citations

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Jakob Petersen, Eli Pollak, Salvador Miret-Artés

Quantum threshold reflection is a well-known quantum phenomenon which prescribes that at threshold, except for special circumstances, a quantum particle scattering from any potential, even if attractive at long range, will be reflected with unit probability. In the past, this property had been associated with the so-called badlands region of the potential, where the semiclassical description of the scattering fails due to a rapid spatial variation of the de Broglie wavelength. This badlands region occurs far from the strong interaction region of the potential and has therefore been used to ``explain'' the quantum reflection phenomenon. In this paper we show that the badlands region of the interaction potential is immaterial. The extremely long wavelength of the scattered particle at threshold is much longer than the spatial extension of the badlands region, which therefore does not affect the scattering. For this purpose, we review and generalize the proof for the existence of quantum threshold reflection to stress that it is only a consequence of continuity and boundary conditions. The nonlocal character of the scattering implies that the whole interaction potential is involved in the phenomenon. We then provide a detailed numerical study of the threshold scattering of a particle by a Morse potential and an Eckart potential, especially in the time domain. We compare exact quantum computations with incoherent results obtained from a classical Wigner approximation. This study shows that close to threshold the time-dependent amplitude of the scattered particle is negligible in the badlands region and is the same whether the potential has a reflecting wall as in the Morse potential or a steplike structure as in the Eckart smooth step potential. The mean flight time of the particle is not shortened due to a local reflection from the badlands region or due to the lower density of the wave function at short distances. This study should serve to definitely rule out the badlands region as a qualitative guide to the properties of quantum threshold reflection.Quantum threshold reflection is a well-known quantum phenomenon which prescribes that at threshold, except for special circumstances, a quantum particle scattering from any potential, even if attractive at long range, will be reflected with unit probability. In the past, this property had been associated with the so-called badlands region of the potential, where the semiclassical description of the scattering fails due to a rapid spatial variation of the de Broglie wavelength. This badlands region occurs far from the strong interaction region of the potential and has therefore been used to ``explain'' the quantum reflection phenomenon. In this paper we show that the badlands region of the interaction potential is immaterial. The extremely long wavelength of the scattered particle at threshold is much longer than the spatial extension of the badlands region, which therefore does not affect the scattering. For this purpose, we review and generalize the proof for the existence of quantum threshold reflection to stress that it is only a consequence of continuity and boundary conditions. The nonlocal character of the scattering implies that the whole interaction potential is involved in the phenomenon. We then provide a detailed numerical study of the threshold scattering of a particle by a Morse potential and an Eckart potential, especially in the time domain. We compare exact quantum computations with incoherent results obtained from a classical Wigner approximation. This study shows that close to threshold the time-dependent amplitude of the scattered particle is negligible in the badlands region and is the same whether the potential has a reflecting wall as in the Morse potential or a steplike structure as in the Eckart smooth step potential. The mean flight time of the particle is not shortened due to a local reflection from the badlands region or due to the lower density of the wave function at short distances. This study should serve to definitely rule out the badlands region as a qualitative guide to the properties of quantum threshold reflection.

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Semiclassical Trajectory-Coherent Approximation in Quantum Mechanics I. High-Order Corrections to Multidimensional Time-Dependent Equations of Schrödinger Type

Vladislav G. Bagrov, Владимир Владимирович Белов, Andrey Yu. Trifonov

Annals of PhysicsMar 1, 199679 citations

The quantum corrections to the behavior of a semi-classical gas can be expressed as a power series of the ratio

eta

between the cube of De Broglie's thermal wavelength and the specific volume. The connection between

eta

and the multioccupancy of quantum states is the aim of the present work. By means of a chemical/physical approach it is possible to associate

eta

We analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum

L=2

, assuming short range interactions. By solving the Schrödinger equation asymptotically, we derive two expansions of the wave function when three bosons are far apart or a pair of bosons and the third boson are far apart. The scattering hypervolume

D

is defined for this collision. Unlike the scattering hypervolume defined by one of us in 2008, whose dimension is length to the fourth power, the dimension of

D

studied in the present paper is length to the eighth power. We then derive the expression of

D

when the interaction potentials are weak, using the Born's expansion. We also calculate the energy shift of such three bosons with three different momenta

\hbar \mathbf{k_{1}}

\hbar\mathbf{k_{2}}

and

\hbar\mathbf{k_{3}}

in a large periodic box. The obtained energy shift depends on

D^{(0)}/Ω^{2}

and

D/Ω^{2}

, where

D^{(0)}

is the three-body scattering hypervolume defined for the three-body

L=0

collision and

Ω

is the volume of the periodic box. We also calculate the contribution of

D

to the three-body T-matrix element for low-energy collisions. We then calculate the shift of the energy and the three-body recombination rate due to

D^{(0)}

and

D

in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from

D

is proportional to

T^2

if the temperature

T

is much larger than the quantum degeneracy temperature but still much lower than the temperature scale at which the thermal de Broglie wave length becomes comparable to the physical range of interaction.

s

-wave scattering, the correlated parts of the observables are calculated and a sign change of the latent heat is reported at a universal ratio of scattering length to the thermal de Broglie wavelength. This is interpreted as a change from correlational heating to cooling.