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Zeitschrift für Naturforschung A

Coherent States and the Magnetic Operators

A. D. Jannussis, Loucas C. Papaloucas

Published May 1, 1973• 5 citations
Publisher

Abstract

In the present paper a method of calculation of the density matrix in the Schrödinger picture is given, in terms of the known creation and annihiliation operators a+ and a. New magnetic operators are given with the help of the coherent states, which depend on two free parameters μ and ν. For the case μ=ν= (e H )/ (2 ħ c), these operators lead to the well-known magnetic operators, as they are given in the current litterature.

DOI
10.1515/zna-1973-0528

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In the present paper a method of calculation of the density matrix in the Schrödinger picture is given, in terms of the known creation and annihiliation operators a+ and a. New magnetic operators are given with the help of the coherent states, which depend on two free parameters μ and ν. For the case μ=ν= (e H )/ (2 ħ c), these operators lead to the well-known magnetic operators, as they are given in the current litterature.
In this paper we study the generalized coherent states (g.c.s.) in n -dimensional space. Their annihilation operators consist of linear combinations of the annihilation operators a l , and their coherent states are the product of the simple coherent states. All the known annihilation operators which have been studied as yet, and also the magnetic ones, are special cases of these operators. The new operators A k + A m form the Lie algebra U ( n ) of the Lie group U ( n ). Finally, instead of the operators a l and a l + we can use the new operators, which depend on a complex parameter.
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.
We provide a brief overview of approaches for calculating the density of states of quantum systems and random matrix Hamiltonians using the tools of free probability theory. For a given Hamiltonian of a quantum system or a generic random matrix Hamiltonian, which can be written as a sum of two non-commutating operators, one can obtain an expression for the density of states of the Hamiltonian from the known density of states of the two component operators by assuming that these operators are mutually free and by using the free additive convolution. In many examples of interacting quantum systems and random matrix models, this procedure is known to provide a reasonably accurate approximation to the exact numerical density of states. We review some of the examples that are known in the literature where this procedure works very well, and also discuss some of the limitations of this method in situations where the free probability approximation fails to provide a sufficiently accurate description of the exact density of states. Subsequently, we describe a perturbation scheme that can be developed from the subordination formulas for the Cauchy transform of the density of states and use it to obtain approximate analytical expressions for the density of states in various models, such as the Rosenzweig-Porter random matrix ensemble and the Anderson model with on-site disorder.
Eigenfunctions and eigenvalues of the free magnetic Schrödinger operator, describing a spinless particle confined to an infinite layer of fixed width, are discussed in detail. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space. Four different classes of temporally stable coherent states associated with the operator are presented. The first two classes are derived as coherent states with one degree of freedom and the last two classes are derived with two degrees of freedom. The dynamical algebra of each class is found. Statistical quantities associated to each class of coherent states are calculated explicitly.
The solutions of the Schrodinger (Schrodinger – Pauli) equations and the Dirac equation in the 2+1 space for the two-dimensional motion of a charged particle in a uniform and constant magnetic field, as well as the construction of coherent states based on the solutions obtained are considered. A specific feature of this problem is the presence of a discrete spectrum with infinite multiplicity of degeneracy. Two pairs of creation and annihilation operators allow one to construct coherent states which are a special superposition of the states of two oscillators. The influence of the multiplicity of the spectrum on the average values of coordinates and momenta due to the appearance of two pairs of creation and annihilation operators is shown. The constructed coherent states depending on two complex numbers lead to two independent Poisson distributions for two types of vibration quanta. The introduced operators allow one to represent the solutions of the Dirac equation in the 2+1 space for the discrete spectrum problem under consideration in a simple form.
In the noncommutative phase space, a new mapping is proposed to express the noncommutative coordinate and momentum operators in terms of the ordinary coordinate and momentum operators under the case of large noncommutativity parameters (μν>1). Using this mapping matrix, the deformed boson operators can be expressed in terms of the ordinary boson operators. Thus, the normal ordering expansion form of vacuum projection operator is obtained. As an application, the completeness relation of the two-mode deformed coherent states is verified by using the vacuum projection operator.