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A semiclassical analysis of the Efimov energy spectrum

Rajat K. Bhaduri, M. Brack, M. V. N. Murthy

Published Mar 9, 2011• 0 citations
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Abstract

We demonstrate that the (s-wave) geometric spectrum of the shallow Efimov energy levels is generated by the radial motion of a primitive periodic orbit of the corresponding classical system and its harmonics. The action of the primitive orbit is given by the logarithm of the energy. This is shown to be consistent with the leading term of an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized at the classical turning point using Langer’s connection formula, are practically indistinguishable from the exact wave functions even for the lowest (n = 0) state, apart from a tiny shift of its zeros that remains constant for large n.

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We demonstrate that the (s-wave) geometric spectrum of the shallow Efimov energy levels is generated by the radial motion of a primitive periodic orbit of the corresponding classical system and its harmonics. The action of the primitive orbit is given by the logarithm of the energy. This is shown to be consistent with the leading term of an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized at the classical turning point using Langer’s connection formula, are practically indistinguishable from the exact wave functions even for the lowest (n = 0) state, apart from a tiny shift of its zeros that remains constant for large n.
We demonstrate that the (sss-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cutoff radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wave functions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest (n=0n=0n=0) state, apart from a tiny shift of its zeros that remains constant for large nnn.
First, two conditions are specified for the lowest order Wentzel–Kramers–Brillouin quantization rule to yield exact results. These rules are related to the periodic orbit decomposition of the quantum density of states. This approach is then applied to supersymmetric quantum mechanics. It leads to a new derivation of the result that shape-invariant potentials give exact results when the classical action is calculated with the square of the super potential, but without the Maslov index or the Langer correction.PACS Nos.: 03.65.Sq, 12.60.Jv
We investigate the classical nonrelativistic dynamics of three bodies in Efimov's $\ensuremath{-}1/{R}^{2}$ potential in three spatial dimensions, with a view towards semiclassical quantization and insight into the geometry of Efimov states. Without the short-distance cutoff, these dynamics are superintegrable, which allows an exact integration of the equations of motion for arbitrary initial conditions. We show that periodic orbits necessarily lead to exactly vanishing binding energy of the bound states, in disagreement with Efimov's quantum-mechanical results. A scaling anomaly demands that the quantum dynamics of three bodies in this potential be augmented by additional boundary conditions affecting all three particles at the short-distance cutoff point, i.e., near the triple collision. We discuss the inherent difficulties in the definition of appropriate three-body boundary conditions in three spatial dimensions and briefly discuss their consequences for (quasi)periodic orbits. Consequently, the classical orbits corresponding to Efimov states cannot be exactly periodic, but must have a finite timescale (lifetime), associated with the time it takes the system's hyperradius to fall to zero, or to the cutoff value, which is typically much longer than the (quasi)period of the hyperangular motion. The scaling properties of the lifetime are in agreement with the quantum-mechanical predictions of the half-life (width) of Efimov states surrounded by an ultracold gas. Detailed spatiotemporal evolution of the system is generally unpredictable beyond the three-body collision point, even though global conservation laws ensure that the system's hyperradius must be periodic.
In this article we extend previous semiclassical studies by including more general perturbative potentials of the harmonic oscillator in arbitrary spatial dimensions. Our starting point is a radial harmonic potential with an arbitrary even monomial perturbation, which we use to study the resulting U(D)\mathrm{U}(D)U(D) to O(D)\mathrm{O}(D)O(D) symmetry breaking. We derive the gross structure of the semiclassical spectrum from periodic orbit theory, in the form of a perturbative (ℏ→0\hbar \rightarrow 0ℏ→0) trace formula. We then show how to apply the results to even order polynomial potentials, possibly including mean-field terms. We have drawn the conclusion that the gross structure of the quantum spectrum is determined from only classical circular- and diameter-orbits for this class of systems.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
The wave function for a state of definite angular momentum of the three-dimensional isotropic harmonic oscillator is expressed exactly in terms of the corresponding classical trajectories. In particular, the three-dimensional wave function as well as the radial wave function and the spherical harmonics are obtained as integrals over quantities determined entirely by the classical motion. The expression for the harmonic-oscillator radial wave function is shown also to yield the exact radial wave function for the free particle. The expressions are cast in forms suitable for use as uniform semiclassical approximations for wave functions of other systems. Numerical examples confirm that such wave functions obey boundary conditions appropriate for spherical coordinates and that they are free of caustic singularities. The wave functions obtained by this technique can be quite accurate even for low-energy states where semiclassical approximations are expected to be poor.