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The Journal of Chemical Physics

On the Exponential Form of Time-Displacement Operators in Quantum Mechanics

Philip Pechukas, John C. Light

Published May 15, 1966• 549 citations
Publisher

Abstract

We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.

DOI
10.1063/1.1726550

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In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.

arXiv:1409.0236•Aug 31, 2014•1 citation
We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.
The Magnus expansion offers a powerful geometric method to express a time-ordered exponential operator solution, typical of time-dependent linear differential equations, as an ordinary operatorial exponential. This representation possesses advantageous theoretical and qualitative features, such as preserving the underlying Lie algebraic structure of the system while remaining an exact solution to the original differential equation. For any finite-dimensional Hermitian Hamiltonian governing a quantum system, the standard Magnus expansion guarantees a manifestly unitary representation at every finite order of truncation. However, this foundational property is no longer preserved if the underlying Hamiltonian is non-Hermitian. In this work, we derive a generalized, unitarized expansion framework that systematically restores and maintains the property of geometric unitarity for all bounded finite-dimensional non-Hermitian Hamiltonians.
The authors introduce a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time-dependent Hermitian operator. Rigorous upper bounds are derived for two different 'time-ordered' approximation schemes and for errors resulting from approximating a time-ordered exponential by an ordinary exponential operator. The properties and the usefulness of the product formula algorithms are examined by applying them to the problem of Zener tunnelling. The most efficient algorithm is employed to solve the time-dependent Schrodinger equation for a wavepacket incident on a time-modulated rectangular barrier.
In this paper we introduce an alternative approach to studying the evolution of a quantum harmonic oscillator subject to an arbitrary time dependent force. With the purpose of finding the evolution operator, certain unitary transformations are applied successively to Schrödinger's equation reducing it to its simplest form. Therefore, instead of solving the original Schrödinger's partial differential equation in time and space the problem is replaced by a system of ordinary differential equations. From the obtained evolution operator we workout the propagator. Even though we illustrate the use of unitary transformations on the solution of a forced harmonic oscillator, the method presented here might be used to solve more complex systems. The present work addresses many aspects regarding unitary transformations and the dynamics of a forced quantum harmonic oscillator that should be useful for students and tutors of the quantum mechanics courses at the senior undergraduate and graduate level.