Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

An operator method of evaluating the Bloch density matrix for an oscillator in a constant magnetic field

Katsumi Yonei

Published Jul 7, 1989• 8 citations
Publisher

Abstract

A new method is proposed in evaluating the Bloch density matrix for a three-dimensional charged harmonic oscillator placed in a constant magnetic field. The method first reduces the operator exp (- beta H) to a product of several factors of a simple nature by using the commutation relations among the Hamiltonian components and by performing some transformations on the Hamiltonian, and then calculates the matrix element. The method is especially efficient when the system has a rotational symmetry around the axis parallel to the magnetic field, but it is also useful in other cases. A generalised problem in which a uniform electric field coexists is also discussed. To make the whole discussion consistent, the density matrix for a one-dimensional harmonic oscillator, which plays an essential role in the whole problem, is recalculated in the spirit of this method, and it is shown that the present method derives the well known form of the Bloch density matrix for this system in a quite elementary way, without reference to any advanced knowledge of eigenfunctions.

DOI
10.1088/0305-4470/22/13/034

Related papers

The coherent-state method of evaluating the density matrix for an oscillator in a constant magnetic field

Gong Ren-shan

The coherent-state method is proposed for evaluating the Bloch density matrix for a charged isotropic oscillator placed in a constant magnetic field. The method first reduces the Bloch equation, satisfied by the density matrix e - beta H , to a partial differential equation by means of the coherent-state representation, and then calculates the density matrix. A generalised problem in which a uniform electric field coexists is also discussed.

Journal of Physics A Mathematical and General•Jun 7, 1990•1 citation

Evaluation of the Bloch density matrix for a charged oscillator in a magnetic field by canonical transformations

Mohammed A. Z. Habeeb

The problem of a charged particle moving in an anisotropic harmonic oscillator potential in the presence of a constant external magnetic field is reduced to that of a particle in an anisotropic oscillator by using canonical transformations. The Bloch density matrix for the original problem is then exactly evaluated by transforming the well known result for the anisotropic oscillator with the aid of the integral kernel corresponding to the canonical transformation under consideration.

Journal of Physics A Mathematical and General•Nov 11, 1987•6 citations

Use of the coherent states in evaluating the canonical density matrix for a harmonic oscillator

Katsumi Yonei

An alternative method for evaluating the canonical density matrix (x mod exp(- beta H) mod x') for a one-dimensional harmonic oscillator is proposed. The method uses the coherent states, i.e. the eigenstates of the annihilation operator a, as the basis of the expansion of exp(- beta H)) mod x') in place of the Hermite functions which are used in the conventional treatment. It is shown that the use of the coherent states makes it much easier to evaluate the density matrix than the other methods. As a direct extension of the method, the density matrix for a three-dimensional harmonic oscillator placed in a uniform magnetic field is recalculated in the creation-annihilation operator formalism. Some other uses of the coherent states in operator problems, such as the Wigner distribution function, are also discussed. Two alternative derivations of the Gaussian integral formula are also given, which plays an important role in the present problem as well as in various other problems in physics.

Journal of Physics A Mathematical and General•May 21, 1990•2 citations

Path integral evaluation of the Bloch density matrix for an oscillator in an electric and magnetic field

M. L. Glasser

The Bloch density matrix for an electron in a three-dimensional oscillator potential in the presence of parallel electric and magnetic fields is presented in cylindrical coordinates.

Journal of Physics A Mathematical and General•Feb 21, 1987•5 citations

Exact results for a charged harmonically trapped quantum gas at arbitrary temperature and magnetic field strength

Patrick Shea, Brandon P. van Zyl

An analytical expression for the first-order density matrix of a charged, two-dimensional, harmonically confined quantum gas, in the presence of a constant magnetic field is derived. Our approach is nonpertabative, and is therefore exact for any temperature and magnetic field strength. We also present a useful factorization of the Bloch density matrix in the form of a simple product with a clean separation of the zero-field and field-dependent parts. This factorization provides an alternative way of analytically investigating the effects of the magnetic field on the system, and also permits the extension of our analysis to other dimensions, and/or anisotropic confinement.

arXiv:cond-mat/0610308•Physical Review B•Nov 29, 2006•9 citations

System of fermions confined in a harmonic potential and subject to a magnetic field or a rotational motion

Houda Naïdja, K. Bencheikh, Johann Bartel +1 more

Making use of the Bloch density matrix technique, we derive exact analytical expressions for the density profile in Fourier space, for the current density and the so-called integrated current for fermionic systems confined by a two-dimensional harmonic oscillator, in the presence of a magnetic field or in a rotating trap of arbitrary strength. We present numerical, illustrative examples with or without magnetic field (with or without rotation).

Physical Review A•May 27, 2011•6 citations
A new method is proposed in evaluating the Bloch density matrix for a three-dimensional charged harmonic oscillator placed in a constant magnetic field. The method first reduces the operator exp (- beta H) to a product of several factors of a simple nature by using the commutation relations among the Hamiltonian components and by performing some transformations on the Hamiltonian, and then calculates the matrix element. The method is especially efficient when the system has a rotational symmetry around the axis parallel to the magnetic field, but it is also useful in other cases. A generalised problem in which a uniform electric field coexists is also discussed. To make the whole discussion consistent, the density matrix for a one-dimensional harmonic oscillator, which plays an essential role in the whole problem, is recalculated in the spirit of this method, and it is shown that the present method derives the well known form of the Bloch density matrix for this system in a quite elementary way, without reference to any advanced knowledge of eigenfunctions.
An alternative method for evaluating the canonical density matrix (x mod exp(- beta H) mod x') for a one-dimensional harmonic oscillator is proposed. The method uses the coherent states, i.e. the eigenstates of the annihilation operator a, as the basis of the expansion of exp(- beta H)) mod x') in place of the Hermite functions which are used in the conventional treatment. It is shown that the use of the coherent states makes it much easier to evaluate the density matrix than the other methods. As a direct extension of the method, the density matrix for a three-dimensional harmonic oscillator placed in a uniform magnetic field is recalculated in the creation-annihilation operator formalism. Some other uses of the coherent states in operator problems, such as the Wigner distribution function, are also discussed. Two alternative derivations of the Gaussian integral formula are also given, which plays an important role in the present problem as well as in various other problems in physics.
An analytical expression for the first-order density matrix of a charged, two-dimensional, harmonically confined quantum gas, in the presence of a constant magnetic field is derived. Our approach is nonpertabative, and is therefore exact for any temperature and magnetic field strength. We also present a useful factorization of the Bloch density matrix in the form of a simple product with a clean separation of the zero-field and field-dependent parts. This factorization provides an alternative way of analytically investigating the effects of the magnetic field on the system, and also permits the extension of our analysis to other dimensions, and/or anisotropic confinement.