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UTILIZATION OF THE DENSITY OPERATOR FOR THE DETERMINATION OF THE EIGENENERGIES AND EIGENFUNCTIONS OF A CANONICAL GROUP

Pieter Smet, Jacques Tillieu

Published May 1, 1962• 0 citations
OpenAlex

Abstract

Two methods are indicated for obtaining eigenenergies and corresponding projectors of a closed quantum system from the density operator describing this system in thermodynamic equilibrium and considered as a solution of Bloch's equation. The first method uses the limit of null temperatures; the second one, the properties of Laplace's transform. The harmonic oscillator is treated as example. (auth)

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DETERMINATION OF THE EIGEN-ENERGIES AND FUNCTIONS FROM THE MAXWELL- BOLTZMANN DENSITY OPERATOR. II. CASE OF PERTURBED SYSTEMS

Pieter Smet, Jacques Tillieu

Previously described methods for the determination of eigen-energies and corresponding projectors from the Maxwell-Boltzmann density operator are adapted to a perturbed system. As an example, the anharmonic oscillator problem is examined. (auth)

•Nov 1, 1962•0 citations

The quantum mechanical density operator and its time evolution

Abraham Nitzan

Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.

•Jun 14, 2024•0 citations

STATISTICAL QUANTUM MECHANICS. DENSITY OPERATORS AND THERMODYNAMIC VALUES

Jacques Yvon

The simple and double density operators, as functions of temperature and activity, are computed for a system in equilibrium. The scattering operator allows, in some cases, separating the thermodnamic variables. In particular, these variables do not appear in the expression of the double density operator, as a function of the simple density operator; which explains the fact that this expression is still valid out of equilibrium. (auth)

•Jul 1, 1960•0 citations

Oscillator phase states, thermal equilibrium and group representations

Yakir Aharonov, E. C. Lerner, Huajun Huang +1 more

Eigenstates of the annihilation type operator U = C + iS, where C and S are the ``cosine'' and ``sine'' operators for harmonic oscillator phase, are shown to be closely related to thermal equilibrium states of the oscillator and to provide a new interpretation of the thermal equilibrium density operator. The problem of creating such states is considered and a general theorem is established leading to the construction of interaction Hamiltonians which transform the eigenstates of U among themselves and, in particular, create them from the oscillator ground state. These Hamiltonians lead to representations of the Lie algebras of O(2,1) and O(3). It is suggested that the mathematical technique used, in which generalized U-type operators provide the link between a group and its representations, has its own intrinsic interest for the study of Lie groups.

Journal of Mathematical Physics•Jun 1, 1973•50 citations

ON QUANTUM MECHANICAL SYSTEMS WITH STOCHASTIC HAMILTONIAN OPERATOR

Hans Primas

The theory of quantum mechanical systems with a stochastic Hamiltonian which are of importance in the theory of dissipative systems and in experimental investigations of the response of physical systems by means of electronic devices is discussed. A new formal development of quantum mechanical density matrices is given that is valid even for strong stochastic perturbations. If the stochastic part of the Hamilionian has a Gaussian distribution and an almost constant spectral density the given solution reduces to an expansion in terms of Hermite functionals which are othonormal respective to the Wiener measure (Cameron-Martin development). This expansion is operationally meaningful and characterized by good convergence and simple properties. As an example of the application of the theory a new foundation of Bloch's relaxation theory is sketched. (auth)

•Jan 1, 1961•0 citations

Classical-quantum correspondence for multilevel systems

Robert Gilmore, C. M. Bowden, Lorenzo M. Narducci

The properties of $r$-mode harmonic-oscillator coherent states are reviewed. In particular, the $\mathcal{D}$-algebra differential-operator realization of the creation and annihilation operators on the coherent states and their diagonal projectors is constructed. A homomorphism between the algebra describing $r$-field modes and the algebra describing $r$-level systems is exhibited explicitly. This homomorphism allows the projection of the multimode calculus onto the multilevel calculus. In particular, multimode coherent states and projectors can be used as generating functions for multilevel coherent states and projectors. In addition, the multilevel $\mathcal{D}$ algebra is constructed directly from the multimode $\mathcal{D}$ algebra under this homomorphism. For illustrative purposes, the $\mathcal{D}$ algebra for the diagonal coherent-state projectors for two-level atomic systems is presented explicitly in terms of a parametrization in the Bloch angles $\ensuremath{\theta}$ and $\ensuremath{\phi}$. Two classes of applications are treated: (a) the mapping of atomic-density-operator equations of motion into phase-space equations of motion for the quasiprobability weighting function $P$; (b) the construction of equations of motion for the diagonal elements $Q$ of the density operator in the coherent states. It is shown that the solution to either equation with the appropriate initial condition gives complete statistical information for the atomic system. It is shown explicitly that the functions $P$ and $Q$ are related by a convolution integral.

Physical Review A•Sep 1, 1975•150 citations
Two methods are indicated for obtaining eigenenergies and corresponding projectors of a closed quantum system from the density operator describing this system in thermodynamic equilibrium and considered as a solution of Bloch's equation. The first method uses the limit of null temperatures; the second one, the properties of Laplace's transform. The harmonic oscillator is treated as example. (auth)
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
Eigenstates of the annihilation type operator U = C + iS, where C and S are the ``cosine'' and ``sine'' operators for harmonic oscillator phase, are shown to be closely related to thermal equilibrium states of the oscillator and to provide a new interpretation of the thermal equilibrium density operator. The problem of creating such states is considered and a general theorem is established leading to the construction of interaction Hamiltonians which transform the eigenstates of U among themselves and, in particular, create them from the oscillator ground state. These Hamiltonians lead to representations of the Lie algebras of O(2,1) and O(3). It is suggested that the mathematical technique used, in which generalized U-type operators provide the link between a group and its representations, has its own intrinsic interest for the study of Lie groups.
The theory of quantum mechanical systems with a stochastic Hamiltonian which are of importance in the theory of dissipative systems and in experimental investigations of the response of physical systems by means of electronic devices is discussed. A new formal development of quantum mechanical density matrices is given that is valid even for strong stochastic perturbations. If the stochastic part of the Hamilionian has a Gaussian distribution and an almost constant spectral density the given solution reduces to an expansion in terms of Hermite functionals which are othonormal respective to the Wiener measure (Cameron-Martin development). This expansion is operationally meaningful and characterized by good convergence and simple properties. As an example of the application of the theory a new foundation of Bloch's relaxation theory is sketched. (auth)
The properties of rrr-mode harmonic-oscillator coherent states are reviewed. In particular, the D\mathcal{D}D-algebra differential-operator realization of the creation and annihilation operators on the coherent states and their diagonal projectors is constructed. A homomorphism between the algebra describing rrr-field modes and the algebra describing rrr-level systems is exhibited explicitly. This homomorphism allows the projection of the multimode calculus onto the multilevel calculus. In particular, multimode coherent states and projectors can be used as generating functions for multilevel coherent states and projectors. In addition, the multilevel D\mathcal{D}D algebra is constructed directly from the multimode D\mathcal{D}D algebra under this homomorphism. For illustrative purposes, the D\mathcal{D}D algebra for the diagonal coherent-state projectors for two-level atomic systems is presented explicitly in terms of a parametrization in the Bloch angles $\ensuremath{\theta}$ and $\ensuremath{\phi}$. Two classes of applications are treated: (a) the mapping of atomic-density-operator equations of motion into phase-space equations of motion for the quasiprobability weighting function PPP; (b) the construction of equations of motion for the diagonal elements QQQ of the density operator in the coherent states. It is shown that the solution to either equation with the appropriate initial condition gives complete statistical information for the atomic system. It is shown explicitly that the functions PPP and QQQ are related by a convolution integral.