The classical correspondence of the S matrix for scattering processes is examined with a view to obtain the classical approximation and also to clarify the nature of the approximations required for the classical formula. In this discussion the Schrödinger equation in the momentum representation is employed instead of the usual coordinate representation in order to provide a more direct route to the classical limit of the S matrix. The S matrix thus obtained essentially corresponds to the WKB approximation, and, with a further approximation, leads to Miller's expression except for a multiplicative factor and an additional phase which are in general not equal to 1 and 0, respectively. The classical limit is also discussed in terms of Wigner-Moyal phase space distribution function, and when the classical limit of this function is taken, an additional factor and phase appear in the S matrix.
The classical correspondence of the S matrix for scattering processes is examined with a view to obtain the classical approximation and also to clarify the nature of the approximations required for the classical formula. In this discussion the Schrödinger equation in the momentum representation is employed instead of the usual coordinate representation in order to provide a more direct route to the classical limit of the S matrix. The S matrix thus obtained essentially corresponds to the WKB approximation, and, with a further approximation, leads to Miller's expression except for a multiplicative factor and an additional phase which are in general not equal to 1 and 0, respectively. The classical limit is also discussed in terms of Wigner-Moyal phase space distribution function, and when the classical limit of this function is taken, an additional factor and phase appear in the S matrix.
The classical-limit scattering matrix (CLSM) formalism was developed by Miller as a simple semiclassical procedure for the analyses of complex scattering and reaction processes. It invokes simple classical dynamics to define trajectories and superposition procedures to obtain interference effects. In one-dimensional systems, Miller (1974) has shown that the CLSM is equivalent to the standard WKB approximation. In this paper, it is shown that the equivalence between the CLSM and WKB extends to general systems. This equivalence between the WKB and CLSM formalisms is attributed to the general equivalence of the h(cross)-expansion procedure in the WKB with the stationary phase approximation used in the CLSM.
This paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian Hc to H is such that it associates aPiQ j +(1−a)Q jPi to piq j, then the formula gives KWKB=KVVexp {( 1/2 −a)ℱT(∂2Hc/∂qi∂pi )(qc(t),pc(t),t) dt}, where KVV≡(2πih/)−n/2 (det M)1/2exp(iSc/h/) is Van Vleck’s well-known formula, Sc being the action functional evaluated at the classical path (qc,pc) and Mij ≡−∂2Sc/∂qa i∂qbj. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].
A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representation is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulation of the Wigner dynamics in the presence of an external magnetic field is investigated.
In (1) the high energy limit of the Wigner transform W1(p,x,n) is investigated for the quantum oscillator. The WKB semiclassical approximation is used and the high energy wavefunction approximated by: 1/ p(x) { exp(i p(x) x) + exp(-i p(x) x) } leading to the result of delta ( PP/2m+kXX/2 - E) where p(x)= sqrt(2m(E-.5kxx)), P,X are classical momentum and position and PP/2m +kXX/2 energy. In this note, we first observe that the WKB approximation may be used for any potential V(x) to find a similar result. We then focus on obtaining the result without using the WKB approximation, but instead utilize: H(p,x) ** W1(p,x,n) = En W1(p,x,n) (from (2)) where H(p,x) is the classical Hamiltonian, ** the Moyal star product involving exp(ihbar/2 Operator) and W(p,x,n) the Wigner function (as used in (2)). The main idea is that in the classical limit, hbar—>0 so exp(ihbar/2 Operator) —>1. Thus H(p,x)W1(p,x,n→large) = En W1n(p,x,n -> large). In (2) it is shown that Integral dp dx W1(p,x,n) = 1, so Integral H(p,x) W1(p,x,n->large) = E. As a result, W1(p,x,n->large) acts as a delta function enforcing pp/2m + V(x)=E.
It is shown that if the wave function of a nonrelativistic particle satisfies the Schr\"odinger equation with a velocity-independent potential, then its scattering amplitudes (and the S matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the S matrix, the complete S matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the S matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.
The time-dependent theory of scattering is reformulated. In §1, it is developed on the basis of a new limiting process which is self-consistent, and the equivalence is shown of Heisenberg's S-matrix and Dyson's one, when the total Hamiltonian permits the existence of bound states. In §2, a theory of scattering of wave packet is proposed in conformity with the physical picture. The damping factor e−ε|t| is derived from the amplitude of the wave packet. In §3, the rearrangement scattering is treated on the basis of the wave packet formalism.