The mathematical foundation to the quasi-classical approximation to the Dirac equation is discussed. After constructing the propagator $U^\kappa(t,s)$ and the quasi-classical propagator for the Dirac equation in an external electro-magnetic field, the usual quasi-classical approximation to the wave function $U^\kappa(t,s)(\exp (iS(x)/h)f)$ is obtained with error estimates in the Hilbert $L^2(R^3,C^4)$ .The mathematical foundation to the quasi-classical approximation to the Dirac equation is discussed. After constructing the propagator $U^\kappa(t,s)$ and the quasi-classical propagator for the Dirac equation in an external electro-magnetic field, the usual quasi-classical approximation to the wave function $U^\kappa(t,s)(\exp (iS(x)/h)f)$ is obtained with error estimates in the Hilbert $L^2(R^3,C^4)$.

Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, MathematicsMar 31, 19826 citations

QUASI-CLASSICAL APPROXIMATION

WORLD SCIENTIFIC eBooksDec 1, 19981 citation

Quasi-Classical Transformation Theory

Ralph Schiller

The quasi-classical theory is the theory of the WKB approximation as defined by Van Vleck. This approximate wave theory is invariant under canonical transformations. We derive the transformation laws for wave functions and operators in this theory. The connection problem of the ordinary configuration space is discussed from the point of view of Van Vleck. The usual method of WKB quantization is then contrasted with another method of quantization in a classical configuration space of creation and destruction operators. In this new configuration space we find that, for some problems, the quasi-classical solutions are exact solutions of the Schr\"odinger equation. Finally, we show that the canonical transformations of the quasi-classical theory may be put into one-to-one correspondence with a group of approximate unitary transformations.The quasi-classical theory is the theory of the WKB approximation as defined by Van Vleck. This approximate wave theory is invariant under canonical transformations. We derive the transformation laws for wave functions and operators in this theory. The connection problem of the ordinary configuration space is discussed from the point of view of Van Vleck. The usual method of WKB quantization is then contrasted with another method of quantization in a classical configuration space of creation and destruction operators. In this new configuration space we find that, for some problems, the quasi-classical solutions are exact solutions of the Schr\"odinger equation. Finally, we show that the canonical transformations of the quasi-classical theory may be put into one-to-one correspondence with a group of approximate unitary transformations.

Physical ReviewFeb 1, 196241 citations

Semi-Classical Approximations

WORLD SCIENTIFIC eBooksJan 1, 20033 citations

Quasi-Classical Limit

Nicolaas P. Landsman

Compendium of Quantum PhysicsJan 1, 20093 citations

The mathematical foundation to the quasi-classical approximation to the Dirac equation is discussed. After constructing the propagator

U^\kappa(t,s)

and the quasi-classical propagator for the Dirac equation in an external electro-magnetic field, the usual quasi-classical approximation to the wave function

U^\kappa(t,s)(\exp (iS(x)/h)f)

is obtained with error estimates in the Hilbert

L^2(R^3,C^4)