Wave functions for the Coulomb, Morse, Rosen-Morse, and Eckart systems are expressed exactly and explicitly in terms of the corresponding classical trajectories. This is accomplished by a technique that maps previously obtained classical representations for wave functions of the three-dimensional isotropic harmonic oscillator to the wave functions for the present systems. By design, the classical behavior enters these expressions in a manner that allows them to be used as uniform semiclassical approximations for other systems. The construction of a hydrogen orbital from the corresponding classical orbits is illustrated for a particular case. The classical representation of the Eckart wave function is analyzed to understand how ordinary, real, classical trajectories that are confined to one side of a barrier are able to describe tunneling through the barrier.
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.
Tunneling in the one-dimensional Eckart system is treated by a semiclassical method that describes the S-matrix in terms of an integral over the initial momenta of real-valued classical trajectories. The results are found to be sensitive to a certain parameter γ which is expected to be essentially arbitrary for classically allowed processes. Analysis of the semiclassical error allows formulation of conditions for the validity of the tunneling treatment. This, in turn, leads to an explanation for the sensitivity of the results to γ and an understanding of how this parameter should be chosen. With an optimized choice, the semiclassical method is found to yield very accurate tunneling results even for probabilities as small as 10−10. The relationship between the present method and the conventional uniform semiclassical treatment of barrier tunneling is discussed.
Classical exact
(CE) wave functions are certain integral representations of energy eigenfunctions that are parameterized in terms of the motion of the corresponding classical system in a semiclassically relevant way. When applied to systems for which they are not exact, such expressions serve as semiclassical approximations. Previous work identified
CE wave functions for a number of specific systems and established their semiclassical usefulness. This paper explores the degree to which such representations can be found for more general systems. It is shown that
CE wave functions exist, in principle, for bound states of an arbitrary integrable system that are confined to a single classically allowed region. Evidence is presented that
CE representations also exist for more general states of such a system that are unbound, or that extend over more than one allowed region. The
CE expressions are not unique: an innumerable variety exists for each such system. The existence proof provides a formal method for constructing
CE expressions by Fourier transforming certain superpositions of energy eigenstates. The parameterization in terms of the classical motion is achieved by identifying certain quantities in these superpositions as classical action and angle variables. The semiclassical relevance of this identification is ensured by imposing some mild conditions on the coefficients in the superposition. This procedure for parameterizing exact wave functions in terms of classical variables indicates a basic relationship between the quantum and classical descriptions of states. The method of constructing
CE wave functions introduced in the proof is shown to be consistent with a number of previously obtained
CE formulas and is used to derive two new, closed-form,
CE expressions. A simple numerical example is presented to illustrate the semiclassical application of one of these expressions and to further verify the physical significance of the classical parameterization.
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.
We investigate the semiclassical Kepler/Coulomb problem using the classical constants of the motion in the framework of Nelson's stochastic mechanics. This is done by considering the eigenvalue relations for a family of coherent states (known as the atomic elliptic states) whose wave functions are concentrated on the elliptical orbit corresponding to the associated classical problem. We show that these eigenvalue relations lead to identities for the semiclassical energy, angular momentum and Hamilton–Lenz–Runge vectors in the elliptical case. These identities are then extended to include the cases of circular, parabolic and hyperbolic motions. We show that in all cases the semiclassical wave function is determined by our identities and so our identities can be seen as defining a semiclassical Kepler/Coulomb problem. The results are interpreted in terms of two dynamical systems: one a complex valued solution to the classical mechanics for a Coulomb potential and the other the drift field for a semiclassical Nelson diffusion.