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Physical Review A

Nonpermuting zero-damping and infinite-system limits in an exactly solvable model of a particle interacting with a field

H. Dekker

Published Mar 1, 1986• 5 citations

Abstract

The quantum mechanics of an oscillator interacting with a field in the limit of zero damping (\ensuremath{\lambda}+0) and infinite field extension (\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}) depends on the order in which the limits are taken. If the limit \ensuremath{\lambda}\ensuremath{\rightarrow}0 is taken after the limit \ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}, one obtains Gaussian wave functions for all stationary states (m=0,1,2,. . .) of the particle. These Gaussians do have the correct standard values for the variances 〈${\ensuremath{\zeta}}^{2}$${〉}_{n}$ but otherwise obviously differ from the standard harmonic-oscillator result. The latter arises only if the zero-damping limit is taken before the infinite-system limit.

DOI: 10.1103/physreva.33.2140

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The quantum mechanics of an oscillator interacting with a field in the limit of zero damping (\ensuremath{\lambda}+0) and infinite field extension (\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}) depends on the order in which the limits are taken. If the limit \ensuremath{\lambda}\ensuremath{\rightarrow}0 is taken after the limit \ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}, one obtains Gaussian wave functions for all stationary states (m=0,1,2,. . .) of the particle. These Gaussians do have the correct standard values for the variances 〈

{\ensuremath{\zeta}}^{2}$${〉}_{n}

but otherwise obviously differ from the standard harmonic-oscillator result. The latter arises only if the zero-damping limit is taken before the infinite-system limit.

A solvable model of a generic trapped bosonic mixture, N 1 bosons of mass m 1 and N 2 bosons of mass m 2 trapped in an harmonic potential of frequency ω and interacting by harmonic inter-particle interactions of strengths λ 1 , λ 2 , and λ 12 , is discussed. It has recently been shown for the ground state [J. Phys. A 50 , 295002 (2017)] that in the infinite-particle limit, when the interaction parameters λ 1 ( N 1 − 1), λ 2 ( N 2 − 1), λ 12 N 1 , λ 12 N 2 are held fixed, each of the species is 100% condensed and its density per particle as well as the total energy per particle are given by the solution of the coupled Gross-Pitaevskii equations of the mixture. In the present work we investigate properties of the trapped generic mixture at the infinite-particle limit, and find differences between the many-body and mean-field descriptions of the mixture, despite each species being 100%. We compute analytically and analyze, both for the mixture and for each species, the center-of-mass position and momentum variances, their uncertainty product, the angular-momentum variance, as well as the overlap of the exact and Gross-Pitaevskii wavefunctions of the mixture. The results obtained in this work can be considered as a step forward in characterizing how important are many-body effects in a fully condensed trapped bosonic mixture at the infinite-particle limit.

The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈

{\mathit{q}}^{2}

(t)〉, 〈

{\mathit{p}}^{2}

(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.