Gray and Verosky have recently studied the reformulation of the N-state matrix representation of the time-dependent Schrödinger equation as an N-degrees of freedom classical Hamiltonian system. This opens the possibility of using in quantum dynamics numerical integrators originally devised for classical mechanics. When the Hamiltonian matrix is time-dependent, Gray and Verosky suggest the use of a Magnus approximation before reducing the quantum system to its classical format. We show that Magnus approximations are not necessary and suggest an alternative technique. With the new technique it is possible to obtain simple integrators of arbitrarily high orders of accuracy that can be applied to all matrix Schrödinger problems with a, possibly time-dependent, real Hamiltonian matrix. The connection between the new approach and high-order split-operator methods is studied.Gray and Verosky have recently studied the reformulation of the N-state matrix representation of the time-dependent Schrödinger equation as an N-degrees of freedom classical Hamiltonian system. This opens the possibility of using in quantum dynamics numerical integrators originally devised for classical mechanics. When the Hamiltonian matrix is time-dependent, Gray and Verosky suggest the use of a Magnus approximation before reducing the quantum system to its classical format. We show that Magnus approximations are not necessary and suggest an alternative technique. With the new technique it is possible to obtain simple integrators of arbitrarily high orders of accuracy that can be applied to all matrix Schrödinger problems with a, possibly time-dependent, real Hamiltonian matrix. The connection between the new approach and high-order split-operator methods is studied.

The Journal of Chemical PhysicsFeb 8, 199661 citations

Solution of time-dependent problems in quantum mechanics

Kálmán Varga, Joseph A. Driscoll

Cambridge University Press eBooksApr 14, 20110 citations

Aspects of Time-Dependent Quantum Mechanics

Princeton University Press eBooksJun 5, 20180 citations

New U -matrix theory in quantum mechanics

C. C. Lam, P. C. W. Fung

We have analyzed Dyson's $U$ -matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the $U$ matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our $U$ matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our $U$ matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our ${U}^{(s)}$ matrix, which is the $U$ matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.We have analyzed Dyson's $U$-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the $U$ matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our $U$ matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our $U$ matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our ${U}^{(s)}$ matrix, which is the $U$ matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.

Physical Review AApr 1, 198314 citations

Problems & Solutions in Nonrelativistic Quantum Mechanics

ANTON Z. CAPRI

The Breakdown of Classical Mechanics Review of Classical Mechanics Elementary Systems One-Dimensional Problems More One-Dimensional Problems Mathematical Foundations Physical Interpretation Distributions and Fourier Transforms Algebraic Methods Central Force Problems Transformation Theory Non-degenerate Perturbation Theory Degenerate Perturbation Theory Further Approximation Methods Time-Dependent Perturbation Theory Particle in a Uniform Magnetic Field Angular Momentum, Etc. Scattering-Time Dependent Scattering-Time Independent Systems of Identical Particles Quantum Statistical Mechanics.The Breakdown of Classical Mechanics Review of Classical Mechanics Elementary Systems One-Dimensional Problems More One-Dimensional Problems Mathematical Foundations Physical Interpretation Distributions and Fourier Transforms Algebraic Methods Central Force Problems Transformation Theory Non-degenerate Perturbation Theory Degenerate Perturbation Theory Further Approximation Methods Time-Dependent Perturbation Theory Particle in a Uniform Magnetic Field Angular Momentum, Etc. Scattering-Time Dependent Scattering-Time Independent Systems of Identical Particles Quantum Statistical Mechanics.

Dec 18, 20021 citation

Exact solutions for time-dependent phenomena in quantum mechanics

M. Kleber

Physics ReportsJan 1, 1994155 citations

We have analyzed Dyson's

U

-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the

U

matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our

U

matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our

U

matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our

{U}^{(s)}

matrix, which is the

U

matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.