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Journal of Physics A Mathematical and General

Separation of variables in path integrals and path integral solution of two potentials on the Poincare upper half-plane

Ch. Grosche

Published Nov 7, 1990• 19 citations
Publisher

Abstract

The author discusses how to separate variables in path integrals. It is assumed that a one-dimensional problem with potential V(x) has an exact solution with energy levels E lambda and wavefunctions Psi lambda . In order to perform the separation of variables, a time transformation is performed back and forth in the path integral which allows one to insert the path integral solution corresponding to the potential V(x). The author illustrates the method by discussing some specific potential problems on the Poincare upper half-plane.

DOI
10.1088/0305-4470/23/21/028

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The author discusses how to separate variables in path integrals. It is assumed that a one-dimensional problem with potential V(x) has an exact solution with energy levels E lambda and wavefunctions Psi lambda . In order to perform the separation of variables, a time transformation is performed back and forth in the path integral which allows one to insert the path integral solution corresponding to the potential V(x). The author illustrates the method by discussing some specific potential problems on the Poincare upper half-plane.
A practical method is developed for calculating numerically one-dimensional path integrals for an arbitrary potential V(x)V(x)V(x). For the oscillator potential V(x)=kx22V(x)=\frac{k{x}^{2}}{2}V(x)=2kx2​ the well-known analytic solution is obtained. To illustrate the numerical convergence of this method the path integral for the Ginzburg-Landau potential, $V(x)=\ensuremath{-}\frac{A{x}^{2}}{2}+\frac{B{x}^{4}}{4}$, is calculated over a range of the positive constants AAA and BBB and compared with numerical solutions of the Schr\"odinger equation.
The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.
Quantum mechanics on the hyperbolic spaces of rank one is discussed using a path interaction technique. Hyperbolic spaces are multi-dimensional generalizations of the hyperbolic plane, i.e. the Poincare upper half-plane endowed with a hyperbolic geometry. The author evaluates the path integral on S 1 equivalent to SO(n, 1)/SO (n) and S 2 equivalent to SU(n,1)/S(U(1)*U(n)) in a particular coordinate system, yielding explicitly the wavefunctions and the energy spectrum. Furthermore, one can exploit a general property of all these spaces, namely that they can be parameterized by a pseudopolar coordinate system.
In this paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half-plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half-plane to the existing one. All the calculations are mainly based on Fourier-expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Pöschl-Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave-functions and the energy-spectrum are derived. The energy-spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p -momentum). The “zero-point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.
In a reasonably self-contained presentation mathematical rigour is supplied to the important ideas of solving certain non-Gaussian path integrals by changes of dimension and/or path-dependent time transformations. The resulting genuine path-integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations. The power of the calculus is illustrated by two standard quantum-physics applications. First, the calculation of the time-dependent propagator corresponding to a particle on the half-line in a harmonic plus inverse-square potential is shown to be a simple exercise. Second, the first rigorous derivation of the energy-dependent Green function of the one-dimensional Morse system is given.