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Physical Review A

Comparison of the path-integral and the U -matrix theories

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

Published May 1, 1984• 3 citations
Publisher

Abstract

We have previously developed a basic $U$-matrix formalism and have been able to obtain explicit expressions for the $U$ matrix and the wave function of a quantum-mechanical system by solving the operator Schr\"odinger equation directly within the regime of the interaction picture [Phys. Rev. A 27, 1760 (1983)]. In our approach, we introduce the switching function ${S}_{w}$ which describes how the interaction Hamiltonian is introduced to the system. In the present investigation, we list the corresponding $U$ matrix and wave function pertaining to the Schr\"odinger picture. Under the special case where the unperturbed Hamiltonian and the Lagrangian commute, our wave function can be reduced to the form identical to that obtained from the path-integral method. A comparison of the path-integral theory and our $U$-matrix theory is presented. The difference in the wave functions of the two theories lies in the fact that in the $U$-matrix approach the Schr\"odinger equation is treated as an operator equation, whereas the key equation in the path-integral theory is taken as algebraic.

DOI
10.1103/physreva.29.2364

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We have previously developed a basic UUU-matrix formalism and have been able to obtain explicit expressions for the UUU matrix and the wave function of a quantum-mechanical system by solving the operator Schr"odinger equation directly within the regime of the interaction picture [Phys. Rev. A 27, 1760 (1983)]. In our approach, we introduce the switching function Sw{S}_{w}Sw​ which describes how the interaction Hamiltonian is introduced to the system. In the present investigation, we list the corresponding UUU matrix and wave function pertaining to the Schr"odinger picture. Under the special case where the unperturbed Hamiltonian and the Lagrangian commute, our wave function can be reduced to the form identical to that obtained from the path-integral method. A comparison of the path-integral theory and our UUU-matrix theory is presented. The difference in the wave functions of the two theories lies in the fact that in the UUU-matrix approach the Schr"odinger equation is treated as an operator equation, whereas the key equation in the path-integral theory is taken as algebraic.
We have analyzed Dyson's UUU-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the UUU 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 UUU 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 UUU 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){U}^{(s)}U(s) matrix, which is the UUU 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.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to non-commuting operators. Generally, the quantum mechanics based on the Schrodinger equation is preferred for a system of particles. But path integral formulation of quantum mechanics finds applications in more complicated situations, particularly, in quantum field theory. The purpose of solving the Schrodinger equation lies in finding the time evolution operator which is able to generate the time translation of the given system. Richard Feynman noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. Dissipative phenomena with a quantum origin manifest themselves in many different situations and hence the study of dissipative systems and their quantization is of a great theoretical and practical value. Many dissipative systems can be modelled as a linearly damped harmonic oscillator.
A ready comparison of the two methods noted in the title is obscured by the very different formalisms that they employ. Our sojourn-time approach [Jaworski and Wardlaw, Phys. Rev. A 45, 292 (1992)] is based on a Hermitian sojourn-time operator in the framework of time-dependent scattering theory whereas Sokolovski and co-workers [Sokolovski and Baskin, Phys. Rev. A 36, 4604 (1987); Sokolovski and Connor, ibid. 42, 6512 (1990)] have developed an approach based on the Feynman path-integral formulation of quantum mechanics. By translating the path-integral approach into the language of wave functions and linear operators, we reveal the similarities of the two approaches and show that the results for the interaction times are closely related, becoming equivalent in the asymptotic limit.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.
A new formulation of nonrelativistic scattering theory is developed which expresses the SSS matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the SSS matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving $\ensuremath{\delta}$ function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path-integral expression for the TTT matrix. The introduction of the concept of the classical interaction picture provides considerable physical insight into this formulation. This formulation also suggests a successionof improvements to the eikonal approximation, the first of which is discussed explicitly.