Physical Review

The Continuous Spectrum Approximation in Quantum Statistics

Peter T. Landsberg

Published Apr 15, 1954• 1 citation

Abstract

A class $O$ of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class $O$. A "limit $L$" is defined as the limiting case which arises of the number of particles $N$ and the volume $V$ of the system are both allowed to tend to infinity so as to keep the ratio $\frac{N}{V}$ a finite and nonzero constant. It is shown that if one wishes to calculate the mean value of an extensive variable, as averaged over a canonical or over a grand canonical ensemble, for a system of class $O$ in the limit $L$, one can apply the continuous spectrum approximation directly to the particle quantum states (excepting, possibly, the states which belong to the lowest energy level) without using a limiting process.

DOI: 10.1103/physrev.94.469

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Physical Review

The Continuous Spectrum Approximation in Quantum Statistics

Peter T. Landsberg

Published Apr 15, 1954• 1 citation

Publisher

Abstract

A class

O

of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class

O

. A "limit

L

" is defined as the limiting case which arises of the number of particles

N

and the volume

V

of the system are both allowed to tend to infinity so as to keep the ratio

\frac{N}{V}

a finite and nonzero constant. It is shown that if one wishes to calculate the mean value of an extensive variable, as averaged over a canonical or over a grand canonical ensemble, for a system of class

O

in the limit

L

, one can apply the continuous spectrum approximation directly to the particle quantum states (excepting, possibly, the states which belong to the lowest energy level) without using a limiting process.

DOI: 10.1103/physrev.94.469

Related papers

Quantum Statistics of Closed and Open Systems

P. T. Landsberg

In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit $L$ ," i.e., in the limit in which the number of particles $N$ and the volume $V$ of the system are allowed to tend to infinity so as to keep $\frac{N}{V}$ a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit $L$ the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit $L$," i.e., in the limit in which the number of particles $N$ and the volume $V$ of the system are allowed to tend to infinity so as to keep $\frac{N}{V}$ a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit $L$ the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.

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ABSTRACT This paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of statesN(E) ∝Enfor high quantum numbers, a condition for condensation beingn> 0. These results are obtained on the basis of the following assumptions: (i) For large volumesV(a) all energy levels behave likeV−θ, and (b) there exists a finite integerMsuch that it is justifiable to put for thejth energy levelEj=c V−θand to use the continuous spectrum approximation, wheneverj ≥ Mc θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.ABSTRACT This paper contains a proof that the description of the phenomenon of Bose-Einstein condensation is the same whether (1) an open system is contemplated and treated on the basis of the grand canonical ensemble, or (2) a closed system is contemplated and treated on the basis of the canonical ensemble without recourse to the method of steepest descents, or (3) a closed system is contemplated and treated on the basis of the canonical ensemble using the method of steepest descents. Contrary to what is usually believed, it is shown that the crucial factor governing the incidence of the condensation phenomenon of a system (open or closed) having an infinity of energy levels is the density of statesN(E) ∝Enfor high quantum numbers, a condition for condensation beingn> 0. These results are obtained on the basis of the following assumptions: (i) For large volumesV(a) all energy levels behave likeV−θ, and (b) there exists a finite integerMsuch that it is justifiable to put for thejth energy levelEj=c V−θand to use the continuous spectrum approximation, wheneverj ≥ Mc θ τ are positive constants, (ii) All results are evaluated in the limit in which the volume of the gas is allowed to tend to infinity, keeping the volume density of particles a finite and non-zero constant. The present paper also serves to coordinate much of previously published work, and corrects a current misconception regarding the method of steepest descents.

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The perfect Bose-Einstein gas in the theory of the quantum-mechanical grand canonical ensembles

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A class

O

of systems of noninteracting quantum-mechanical particles is defined, so that it is often justifiable to approximate a given system of particles as a system of class

O

. A "limit

L

" is defined as the limiting case which arises of the number of particles

N

and the volume

V

of the system are both allowed to tend to infinity so as to keep the ratio

\frac{N}{V}

O

in the limit

L

In this note an ideal Bose-Einstein and an ideal Fermi-Dirac gas are considered in the "limit

L

," i.e., in the limit in which the number of particles

N

and the volume

V

of the system are allowed to tend to infinity so as to keep

\frac{N}{V}

a finite and nonzero constant. It is shown that, whatever the system of energy levels, the mean occupation numbers calculated on the basis of the grand canonical ensemble are in the limit

L

the same as those calculated on the basis of the canonical ensemble. The only requirement is that the mean total number of particles for the grand canonical ensemble be equal to the fixed total number of particles for the canonical ensemble.