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Fortschritte der Physik

Quantum Fictitious Forces

Iwo Białynicki‐Birula, M. A. Cirone, Joachim Dahl, T. H. Seligman, F. Straub, Wolfgang P. Schleich

Published May 1, 2002• 10 citations

Abstract

We present Heisenberg's equation of motion for the radial variable of a free non-relativistic particle in D dimensions. The resulting radial force consists of three contributions: (i) the quantum fictitious force which is either attractive or repulsive depending on the number of dimensions, (ii) a singular quantum force located at the origin, and (iii) the centrifugal force associated with non-vanishing angular momentum. Moreover, we use Heisenberg's uncertainty relation to introduce a lower bound for the kinetic energy of an ensemble of neutral particles. This bound is quadratic in the number of atoms and can be traced back to the repulsive quantum fictitious potential. All three forces arise for a free particle: “Force without force”.

DOI: 10.1002/1521-3978(200205)50:5/7<599::aid-prop599>3.0.co;2-g

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We present Heisenberg's equation of motion for the radial variable of a free non-relativistic particle in D dimensions. The resulting radial force consists of three contributions: (i) the quantum fictitious force which is either attractive or repulsive depending on the number of dimensions, (ii) a singular quantum force located at the origin, and (iii) the centrifugal force associated with non-vanishing angular momentum. Moreover, we use Heisenberg's uncertainty relation to introduce a lower bound for the kinetic energy of an ensemble of neutral particles. This bound is quadratic in the number of atoms and can be traced back to the repulsive quantum fictitious potential. All three forces arise for a free particle: “Force without force”.

In a two-dimensional world, a free quantum particle of vanishing angular momentum experiences an attractive force. This force originates from a modification of the classical centrifugal force due to the wave nature of the particle. For positive energies the quantum anticentrifugal force manifests itself in a bunching of the nodes of the energy wave functions towards the origin. For negative energies this force is sufficient to create a bound state in a two-dimensional \ensuremath{\delta}-function potential. In a counterintuitive way, the attractive force pushes the particle away from the location of the \ensuremath{\delta}-function potential. As a consequence, the particle is localized in a band-shaped domain around the origin.

This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and momentum observables are taken with respect to one-particle states and their time-dependence is discussed. A potential is added to the free-particle Hamiltonians and q-analogs of the Ehrenfest theorem are derived from the Heisenberg equations of motion. The conservation of probability is proved.

We regard the non-relativistic Schrödinger equation as an ensemble mean representation of the stochastic motion of a single particle in a vacuum, subject to an undefined stochastic quantum force. The local mean of the quantum force is found to be proportional to the third spatial derivative of the probability density function, while its associated pressure is proportional to the second spatial derivative. The latter arises from the single particle diluted gas pressure, and this observation allows to interpret the quantum Bohm potential as the energy required to put a particle in a bath of fluctuating vacuum at constant entropy and volume. The stochastic force expectation value is zero and is uncorrelated with the particle location, thus does not perform work on average. Nonetheless, it is anti-correlated with volume and this anti-correlation leads to an uncertainty relation. We analyze the dynamic Gaussian solution to the Schrödinger equation as a simple example for exploring the mean properties of this quantum force. We conclude with a few possible interpretations as to the origins of quantum stochasticity.