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Journal of Mathematical Physics

Exact propagators for multidimensional quadratic Hamiltonians—Initial-value approach

Yoel Tikochinsky

Published Jun 1, 1981• 3 citations
Publisher

Abstract

A practical method suitable for both analytic and numerical computation of propagators for multidimensional quadratic Hamiltonians in presented. The traditional two-end-point problem is replaced by a manageable initial-value problem. As a by-product, the classical path and action are also obtained as a solution to an initial-value problem. The method is illustrated analytically as well as numerically by solving for the motion of a particle in a constant magnetic field.

DOI
10.1063/1.525034

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A practical method suitable for both analytic and numerical computation of propagators for multidimensional quadratic Hamiltonians in presented. The traditional two-end-point problem is replaced by a manageable initial-value problem. As a by-product, the classical path and action are also obtained as a solution to an initial-value problem. The method is illustrated analytically as well as numerically by solving for the motion of a particle in a constant magnetic field.
We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to uniform and constant electro-magnetic fields.
In this paper we solve exactly the problem of the spectrum and Feynman propagator of a charged particle submitted to both an anharmonic oscillator in the plane and a constant and homogeneous magnetic field of arbitrary strength aligned with the perpendicular direction to the plane. As we shall see in the beginning of the letter, the Hamiltonian, being a quadratic form, is easily diagonalizable and the Classical Action can be used to construct the exact Feynman Propagator using the Stationary Phase Approximation. The result is useful for the treatment of quasi two dimensional samples in the field of magnetic effects in nano-structures and quantum optics. The presented solution, after minor extensions, can also be used for motion in three dimensions, and in fact it has been used for years in such cases. Also it can be used as a good exercise of a Feynman Path Integral that can be calculated easily with just the help of the Classical Action.
In the eigen-functional theoretical framework, exact/accurate treatment of high-dimensional strongly correlated electron systems may be realized. The two key steps of this method are that under the path integral formulation we first change a D-dimensional quantum many-particle system into an (D+1)-dimensional (time-dependent) effective "single-particle" problem, then by solving the eigen-equation of the propagator operator of the particles, we can obtain the ground state energy functional and action, respectively. Under the eigen-functional theory, the problems of quantum many-particle systems end in to solve the equation of phase fields that are completely determined by the electron interaction. In practice this equation of the phase fields can be numerically solved for large lattice sites. After replacing the real time by the imaginary time, this method can also be applied for finite temperature cases.