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

Reviews of Modern Physics

Quantum-Mechanical and Semiclassical Forms of the Density Operator Theory of Relaxation

Paul S. Hubbard

Published Apr 1, 1961• 249 citations
Publisher

Abstract

Nuclear magnetic relaxation phenomena are studied by means of a density operator formalism. Both quantum mechanical and semiclassical forms of a density operator relaxation theory are developed using this formalism. Both forms of the theory are applicable when the spin energy depends explicity on time, and both predict correctly the observed density operator. The relaxation of a system of identical spin-1/2 nuclei is calculated as an example. (T.F.H.)

DOI
10.1103/revmodphys.33.249

Related papers

On the Theory of Relaxation Processes

Alfred G. Redfield

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.

IBM Journal of Research and Development•Jan 1, 1957•1,803 citations

Formalism and Interactions in NMR

Hervé Desvaux

In this lecture, a fast overview of the formalism used in Nuclear Magnetic Resonance is provided, with a stress on the density matrix approach. Its equivalence to the classical derivation for particular spin systems is shown. The different interactions which are usually encountered are introduced. Finally, the case of randomly time-dependent interactions inducing relaxation is addressed.

EPJ Web of Conferences•Jan 1, 2012•0 citations

Quantum-Corrected Statistical Method for Many-Particle Systems: The Density Matrix

L. C. R. Alfred

Quantum corrections are obtained for the single-particle density matrix in a semiclassical ensemble where the distribution is unrestricted. A form for the density matrix containing explicitly a function of the wave-mechanical Hamiltonian operator is devised; a formalism is then developed to decompose this operator function into effectively classical and nonclassical parts. The classical part corresponds to the semiclassical density matrix and the quantum corrections are obtained from the nonclassical terms.The quantum-corrected density matrix for a spherically symmetrical and a one-dimensional system have been evaluated. The densities for a linear harmonic oscillator and a Coulomb potential for a Fermi-Dirac distribution have also been examined in some detail.

Physical Review•Mar 1, 1961•21 citations

Density Operators and Matrices

The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalisation techniques.

Royal Society of Chemistry eBooks•Dec 13, 2016•0 citations

Density Operators and Matrices

The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.

Royal Society of Chemistry eBooks•Sep 12, 2025•0 citations

A semiclassical self-consistent-field approach to dissipative dynamics: The spin–boson problem

Gerhard Stock

A semiclassical time-dependent self-consistent-field approach for the description of dissipative quantum phenomena is proposed. The total density operator is approximated by a semiclassical ansatz, which couples the system degrees of freedom to the bath degrees of freedom in a self-consistent manner, and is thus in the spirit of a classical-path description. The capability of the approach is demonstrated by comparing semiclassical calculations for a spin–boson model with an Ohmic bath to exact path-integral calculations. It is shown that the semiclassical model nicely reproduces the complex dissipative behavior of the spin–boson model for a large range of model parameters. The validity and accuracy of the semiclassical approach is discussed in some detail. It is shown that the method is essentially based on the assumption of complete randomization of nuclear phases. In particular, the assumption of phase randomization allows one to perform the trace over the bath variables through quasiclassical sampling of the nuclear initial conditions without invoking any further approximation.

The Journal of Chemical Physics•Jul 22, 1995•113 citations
Nuclear magnetic relaxation phenomena are studied by means of a density operator formalism. Both quantum mechanical and semiclassical forms of a density operator relaxation theory are developed using this formalism. Both forms of the theory are applicable when the spin energy depends explicity on time, and both predict correctly the observed density operator. The relaxation of a system of identical spin-1/2 nuclei is calculated as an example. (T.F.H.)
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.
Quantum corrections are obtained for the single-particle density matrix in a semiclassical ensemble where the distribution is unrestricted. A form for the density matrix containing explicitly a function of the wave-mechanical Hamiltonian operator is devised; a formalism is then developed to decompose this operator function into effectively classical and nonclassical parts. The classical part corresponds to the semiclassical density matrix and the quantum corrections are obtained from the nonclassical terms.The quantum-corrected density matrix for a spherically symmetrical and a one-dimensional system have been evaluated. The densities for a linear harmonic oscillator and a Coulomb potential for a Fermi-Dirac distribution have also been examined in some detail.
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalisation techniques.
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.
A semiclassical time-dependent self-consistent-field approach for the description of dissipative quantum phenomena is proposed. The total density operator is approximated by a semiclassical ansatz, which couples the system degrees of freedom to the bath degrees of freedom in a self-consistent manner, and is thus in the spirit of a classical-path description. The capability of the approach is demonstrated by comparing semiclassical calculations for a spin–boson model with an Ohmic bath to exact path-integral calculations. It is shown that the semiclassical model nicely reproduces the complex dissipative behavior of the spin–boson model for a large range of model parameters. The validity and accuracy of the semiclassical approach is discussed in some detail. It is shown that the method is essentially based on the assumption of complete randomization of nuclear phases. In particular, the assumption of phase randomization allows one to perform the trace over the bath variables through quasiclassical sampling of the nuclear initial conditions without invoking any further approximation.