Suppression of Quantum-Mechanical Collapse in Bosonic Gases with Intrinsic Repulsion: A Brief Review

Boris Malomed

doi:10.3390/condmat3020015

Abstract

It is known that attractive potential ~− 1 / r 2 gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schrödinger equation. This article summarizes theoretical analysis, chiefly published in several original papers, which demonstrates suppression of the collapse caused by this potential, and the creation of the otherwise missing ground state in a 3D gas of bosonic dipoles pulled by the same potential to the central charge, with repulsive contact interactions between them, represented by the cubic term in the respective Gross–Pitaevskii equation (GPE). In two dimensions (2D), quintic self-repulsion is necessary for the suppression of the collapse; alternatively, this may be provided by the effective quartic repulsion produced by the Lee–Huang–Yang correction to the GPE. 3D states carrying angular momentum are constructed in the model with the symmetry reduced from spherical to cylindrical by an external polarizing field. Interplay of the collapse suppression and miscibility–immiscibility transition is considered in a binary condensate. The consideration of the 3D setting in the form of the many-body quantum system, with the help of the Monte Carlo method, demonstrates that, although the quantum collapse cannot be fully suppressed, the self-trapped states predicted by the GPE exist in the many-body setting as metastable modes protected against the collapse by a tall potential barrier.

Highlights

One of standard exercises given to students taking a course in quantum mechanics is solving the three-dimensional (3D) Schrödinger equation with an isotropic attractive potential [1], U(r) = − U0 2r2 U0 > (1)This exercise offers a unique example of critical phenomena in the nonrelativistic quantum theory.the corresponding classical (Newton’s) equation of motion for the particle’s coordinates, r = {x, y, z}, d2r dt2 − ∂U ∂r ≡
This article aims to produce a review of results reported in works [5,6,7,21] that offer a solution to the known problem of the quantum collapse, alias “fall onto the center” [1], in nonrelativistic quantum mechanics
The quantum collapse occurs in the three-dimensional Schrödinger equation with 3D isotropic attractive potential −U0/(2r2)

Summary

Introduction

One of standard exercises given to students taking a course in quantum mechanics is solving the three-dimensional (3D) Schrödinger equation with an isotropic attractive potential [1], U(r) = − U0 2r2 U0 > (1). This exercise offers a unique example of critical phenomena in the nonrelativistic quantum theory. In the framework of the mean-field theory, Vint(r) is characterized solely by the s-wave scattering length [8], while the many-body system should be introduced with a particular form of the interaction potential. Two basic forms of the interaction potential chosen for the analysis are specified below, see Equations (88) and (89)

Objectives

Results

Conclusion