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Physical Review A

An exact propagator and its correspondence limits at a caustic

Lawrence S. Schulman

Published Jun 1, 1987• 5 citations
Publisher

Abstract

The time-dependent propagator for the potential V(x)=1/2m(${\ensuremath{\omega}}^{2}$${x}^{2}$+${\ensuremath{\rho}}^{2}$/${x}^{2}$), x\ensuremath{\in}${\mathrm{scrR}}^{+}$ is known. It is used in this article to see how certain general ideas about the semiclassical approximation for the propagator are in fact realized. The phenomena demonstrated include caustics at classical focal points, analytic structure of the classical action, complex trajectories for forbidden regions (not forbidden in this instance by energy), and the Langer correction for approximating the centrifugal potential. We also provide an unusual derivation of the solution to a classical problem from its quantum version.

DOI
10.1103/physreva.35.4956

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The time-dependent propagator for the potential V(x)=1/2m({\ensuremath{\omega}}^{2}$${x}^{2}+\ensuremathρ2{\ensuremath{\rho}}^{2}\ensuremathρ2/x2{x}^{2}x2), x\ensuremath{\in}scrR+{\mathrm{scrR}}^{+}scrR+ is known. It is used in this article to see how certain general ideas about the semiclassical approximation for the propagator are in fact realized. The phenomena demonstrated include caustics at classical focal points, analytic structure of the classical action, complex trajectories for forbidden regions (not forbidden in this instance by energy), and the Langer correction for approximating the centrifugal potential. We also provide an unusual derivation of the solution to a classical problem from its quantum version.
We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential.The models studied in detail have potentials proportional to x α for x > 0; the limit α → ∞ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases α = 1 and 2, for which exact solutions in terms of well known functions are available for comparison.We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action S, the amplitude function A and the Laplacian of A. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.)The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion.We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential.Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.
We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to xαx^αxα for x>0x>0x>0; the limit α→∞α\to\inftyα→∞ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases α=1α=1α=1 and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action SSS, the amplitude function AAA and the Laplacian of AAA. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.
Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.
The semiclassical limit of the coherent state propagator $⟨{\mathbf{z}}^{\ensuremath{''}}\ensuremath{\mid}{e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}T∕\ensuremath{\hbar}}\ensuremath{\mid}{\mathbf{z}}^{\ensuremath{'}}⟩$ involves complex classical trajectories of the Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}(\mathbf{u},\mathbf{v})=⟨\mathbf{v}\ensuremath{\mid}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\mid}\mathbf{u}⟩$ satisfying $\mathbf{u}(0)={\mathbf{z}}^{\ensuremath{'}}$ and $\mathbf{v}(T)={\mathbf{z}}^{\ensuremath{''}}*$. In this work we study mostly the case ${\mathbf{z}}^{\ensuremath{'}}={\mathbf{z}}^{\ensuremath{''}}$. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as $\ensuremath{\hbar}$ goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.
Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.
The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V ( x ) = 1/ x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.