IIMAS - FENOMEC
UNAM
Suppression of the quantum-mechanical collapse by repulsive interactions
Resumen:
The quantum-mechanical collapse (alias "fall onto the center" of particles attracted by potential -1/r2) is a well-known issue in the elementary quantum theory. It is closely related to the so-called "quantum anomaly", i.e., breaking of the scaling invariance of the respective Hamiltonian by the quantization. We demonstrate that, in a rarefied gas of quantum particles attracted by the above-mentioned potential, the mean-field repulsive nonlinearity induced by collisions between the particles prevents the collapse, and thus puts forward a solution to the quantum-anomaly problem different from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge, and also in the 2D gas of magnetic dipoles attracted by a current filament. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schrödinger equation. The addition of the harmonic trapping potential gives rise to a tristability, in the case when the Schrödinger equation still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it. The analysis is also extended to the 3D anisotropic setting, with the dipoles polarized by an external uniform field.
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