The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.
The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.
Abstract We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.
The authors derive a semiclassical formula for the Wigner function W(q,p,t) describing the evolution in the two-dimensional phase space qp of a nonstationary quantum state psi (q,t) for a system with one degree of freedom. The initial state psi (q,0) corresponds to a family of classical orbits represented by a curve C 0 in qp. Under the classical motion C 0 evolves into a curve C t ; it is shown that the region where W is large hugs C t in an adiabatic fashion, and that W has semiclassical oscillations depending only on the geometry of C t and neighbouring curves.
Abstract The probability density Π is calculated for quantum eigenstates near spatial boundaries of classically chaotic regions. By contrast with integrable systems, for which the classical Π diverges on classical boundaries, which are caustics, in chaotic systems the classical Π does not diverge but vanishes abruptly in a way that depends on the number of freedoms N; the boundaries are anticaustics. Quantum mechanics softens anticaustics, to give Π in terms of a set of canonical diffraction patterns, one for each N; these are studied in detail. The appropriate definition of Π involves averaging over eigenstates in an energy range larger than O(h) but smaller than O(h⅔) (where h is Planck’s constant), that is over a range of ∆N states near the Nth, where N1-1/N ≪ ∆N ≪ N1-⅔N.
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.
We present a comprehensive study of semiclassical phase-space propagation in the Wigner representation, emphasizing numerical applications, in particular as an initial-value representation. Two semiclassical approximation schemes are discussed. The propagator of the Wigner function based on van Vleck's approximation replaces the Liouville propagator by a quantum spot with an oscillatory pattern reflecting the interference between pairs of classical trajectories. Employing phase-space path integration instead, caustics in the quantum spot are resolved in terms of Airy functions. We apply both to two benchmark models of nonlinear molecular potentials, the Morse oscillator and the quartic double well, to test them in standard tasks such as computing autocorrelation functions and propagating coherent states. The performance of semiclassical Wigner propagation is very good even in the presence of marked quantum effects, e.g., in coherent tunneling and in propagating Schrodinger cat states, and of classical chaos in four-dimensional phase space. We suggest options for an effective numerical implementation of our method and for integrating it in Monte-Carlo-Metropolis algorithms suitable for high-dimensional systems.
Abstract Semiclassical mechanics is a short-wavelength link between classical and quantum mechanics, similar to that between wave and geometrical optics. Its object is to maintain contact with the structure imposed by classical mechanics, without sacrificing quantum mechanical accuracy, or, in the vivid words of Berry and Mount (1972), to ‘sew quantum mechanical flesh onto classical bones’. The overriding principles are that the classical phase space is divided into ‘accessible’ and ‘inaccessible’ regions characteristic of the type of motion, and that the boundaries of these regions impose a classical structure, which is evidenced in the quantum mechanics by interference in the bright accessible regions and exponential decay into the shadow regions. The strength of the classical skeleton is that this structure is maintained whether the observations are made in coordinate space, quantum number space, or some mixed representation. In all cases the central quantity is the classical action, whose dimensions are those of Planck’s constant; on one hand it guides the classical motion, according to the minimum action principle, and on the other it adds a phase to each classical trajectory, which gives rise both to quantum mechanical interference and, by a mathematical extension, to quantum mechanical tunnelling.