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Communications in Theoretical Physics

A New Method of the Functional Integral Constructed Directly from Hamiltonian

Zhong-Can Zhang, Chong-Sheng Li, An-Sheng Wei

Published Dec 1, 1993• 1 citation
Publisher

Abstract

We establish a new method of constructing the functional integral for the matrix elements of the time evolution operator of quantum system directly from the Hamiltonian. In this first paper, we present the general principles and methods, and realize the construction of the functional integral for a two-dimensional Hamiltonian. In the second paper, the method is extended to arbitrary -dimensional state space.

DOI
10.1088/0253-6102/20/4/461

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We establish a new method of constructing the functional integral for the matrix elements of the time evolution operator of quantum system directly from the Hamiltonian. In this first paper, we present the general principles and methods, and realize the construction of the functional integral for a two-dimensional Hamiltonian. In the second paper, the method is extended to arbitrary -dimensional state space.
Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.
This paper is devoted to establishing the relation between the holomorphic form of the functional integral and its Feynman form, as well as presenting a new way to evaluate the Feynman transformation matrix element by using the coherent state. The Feynman kernel for the harmonic oscillator with two time-dependent sources is thus derived. Besides these, by the use of the normal product form of the Wigner operator derived in Ref.[1] and the Weyl correspondence, the general holomorphic form of the functional integral and its evolution to the Feynman kernel are discussed.
A simple systematic way of obtaining analytically solvable Hamiltonians for quantum two-level systems is presented. In this method, a time-dependent Hamiltonian and the resulting unitary evolution operator are connected through an arbitrary function of time, furnishing us with new analytically solvable cases. The method is surprisingly simple, direct, and transparent and is applicable to a wide class of two-level Hamiltonians with no involved constraint on the input function. A few examples illustrate how the method leads to simple solvable Hamiltonians and dynamics.
A free nonrelativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Nonlocal boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.