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International Journal of Modern Physics B

QUANTUM VARIATIONAL PRINCIPLE OF IRREVERSIBLE PROCESSES IN A MAGNETIC FIELD

Masumi Hattori, Huzio Nakano

Published Sep 30, 1996• 0 citations
Publisher

Abstract

Quantum variational principle of irreversible processes is generalized to the systems under the uniform magnetic field. For the electron transport in impurity potentials as a typical example, contractions of the density matrix are performed without any approximation in two steps; first the off-diagonal part of the density matrix is eliminated and subsequently the even part with respect to time reversal. In consequence the maximum principle is derived for the odd part of the diagonal elements of the density matrix. This means the contraction of the density matrix brings about irreversibility. The variational principle is shown to be closely related to that for the conventional Boltzmann–Bloch equation and is regarded as its fundamental version. The dependences of the density matrix on the magnetic field and wave number are discussed.

DOI
10.1142/s0217979296001161

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Quantum variational principle of irreversible processes is generalized to the systems under the uniform magnetic field. For the electron transport in impurity potentials as a typical example, contractions of the density matrix are performed without any approximation in two steps; first the off-diagonal part of the density matrix is eliminated and subsequently the even part with respect to time reversal. In consequence the maximum principle is derived for the odd part of the diagonal elements of the density matrix. This means the contraction of the density matrix brings about irreversibility. The variational principle is shown to be closely related to that for the conventional Boltzmann–Bloch equation and is regarded as its fundamental version. The dependences of the density matrix on the magnetic field and wave number are discussed.
The variational principle of irreversible processes, which was previously presented for the von Neumann equation as a stationarity problem and then converted into a maximum problem by contracting the density matrix perturbatively, is reinvestigated w.r.t. the contraction of the density matrix. The present contraction relies on the T-matrix theory of scattering, where no perturbational consideration enters. By taking the electron transport in solids as a typical example, the contraction is performed in two steps: the even component of the density matrix as to time reversal is eliminated first and then the off-diagonal elements in the scheme of diagonalizing the unperturbed Hamiltonian. The maximum problem thus obtained is for the diagonal elements of the odd component of the density matrix. The maximum condition gives the master equation, which is reduced to the Boltzmann-Bloch equation in the scheme of one-body picture. It is noticeable in this equation that the collision term is given in terms of the T-matrix in scattering theory.
A time-dependent variational principle is given for nonequilibrium density matrix of a quantum system. It allows a derivation of dynamical equations for thermodynamical forces which are the Lagrangian parameters of the present theory. If the system is near equilibrium, these equations determine the transport coefficients which satisfy the reciprocity relations. For the understanding of irreversibility, the introduction of the van Hove parameter turns out to be essential, which bases upon a distinct separation of microscopic and macroscopic scales in time.
For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.
The methods developed by Prigogine and his colleagues are applied to the study of quantum mechanical systems. The density matrix is divided into a slowly varying and an oscillatory part, and asymptotic equations are established for the elements of this matrix. In the representation in which the unperturbed hamiltonian is diagonal, these equations take a particularly simple form for weakly coupled systems: the matrix elements lying on diagonals parallel to the principal diagonal undergo separate transformations. A particular case is that of the diagonal elements themselves; for these the well-known Pauli equations are obtained. As an illustrative example the problem of the frictional forces on an oscillator immersed in a thermostat is considered in detail. The frictional forces introduced by Langevin and Lorentz are special cases of this.
A general procedure is given for finding the equation of motion of the density matrix of a system in contact with a thermal bath, as for example a nuclear spin system weakly coupled to a crystal lattice. The thermal bath is treated both classically and quantum mechanically, and the theory is similar to, and a generalization of, conventional theories of time proportional transition probabilities. Relaxation of the system by the thermal bath is expressed by a linear matrix operator, and it is stressed that elements of this operator can be regarded as secular or nonsecular perturbations on the equation of motion and can be treated accordingly. When the motion of the system is slow compared to that of the thermal bath, the equation of motion can be expressed in an operator form which is independent of representation. If the system has a time-dependent Hamiltonian which varies slowly compared to the motion of the thermal bath, the same equation of motion is obeyed and the system is relaxed by the bath toward a Boltzmann distribution with respect to its instantaneous Hamiltonian. If the time variation of the Hamiltonian is more rapid, higher order corrections to the equation of motion must be applied. The theory is applied to spin-lattice relaxation of a coupled nuclear spin system in a metal, for arbitrary externally applied fixed magnetic field.
Forms of canonical (Bloch) and Dirac density matrices for free electrons with and without external applied fields are first considered. The basic tool employed is the Bloch equation with a one-electron Hamiltonian. Such an approach is used to obtain a perturbation theory to all orders for the idempotent Dirac density matrix when a common potential energy V(r) is switched on to originally free electrons. The relation to density functional theory is then considered and the exchange–correlation contribution Vxc(r) to V(r) is expressed in terms of first- and second-order density matrices following Holas and March. These latter density matrices are now for the fully interacting system and, in particular, the first-order density matrix is no longer idempotent, though it must still satisfy generalized Pauli Principle conditions. Reference is also made to a localized Wigner electron in a strong magnetic field.