Quasiclassical approximation for the Dirac equation

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

A Semiclassical Approach to the Dirac Equation

Jens Bolte, Stefan Keppeler

arXiv:quant-ph/9811025Annals of PhysicsMay 1, 1999135 citations

Quasiclassical approximation for the resonant Kapitza–Dirac scattering

G. A. Muradyan, A. Zh. Muradyan

In a momentum space, the resonant Kapitza–Dirac effect is described by the difference Schrödinger equation the step of which is the two-photon recoil momentum. For the interferometry of a multipath atom, the case of intense counter-propagating waves is of interest, when the number of generated momenta can reach to some tens. After that, in an intermediate stage of calculations, the discrete momentum distribution is regarded as a continuous one, and the Taylor expansion is applicable to it. This approximation preserves the spectrum of a diffraction well, in particular, the formation of a pair of almost monopulse states from the initial Gaussian distribution.In a momentum space, the resonant Kapitza–Dirac effect is described by the difference Schrödinger equation the step of which is the two-photon recoil momentum. For the interferometry of a multipath atom, the case of intense counter-propagating waves is of interest, when the number of generated momenta can reach to some tens. After that, in an intermediate stage of calculations, the discrete momentum distribution is regarded as a continuous one, and the Taylor expansion is applicable to it. This approximation preserves the spectrum of a diffraction well, in particular, the formation of a pair of almost monopulse states from the initial Gaussian distribution.

Journal of Contemporary Physics (Armenian Academy of Sciences)Jul 1, 20171 citation

Multicomponent Wentzel–Kramers–Brillouin approximation on arbitrary symplectic manifolds: A star product approach

C. Emmrich, Hartmann Römer

It is known that in the Wentzel–Kramers–Brillouin approximation of multicomponent systems like the Dirac equation or Born–Oppenheimer approximation, an additional phase appears apart from the Berry phase. So far, this phase was only examined in special cases, or under certain restrictive assumptions, namely, that the eigenspaces of the matrix or endomorphism valued symbol of the Hamiltonian form trivial bundles. We give a completely global derivation of this phase which does not depend on any choice of local trivializing sections. This is achieved using a star product approach to quantization. Furthermore, we give a systematic and global approach to a reduction of the problem to a problem defined completely on the different “polarizations.” Finally, we discuss to what extent it is actually possible to reduce the problem to a really scalar one, and make some comments on obstructions to the existence of global quasiclassical states.It is known that in the Wentzel–Kramers–Brillouin approximation of multicomponent systems like the Dirac equation or Born–Oppenheimer approximation, an additional phase appears apart from the Berry phase. So far, this phase was only examined in special cases, or under certain restrictive assumptions, namely, that the eigenspaces of the matrix or endomorphism valued symbol of the Hamiltonian form trivial bundles. We give a completely global derivation of this phase which does not depend on any choice of local trivializing sections. This is achieved using a star product approach to quantization. Furthermore, we give a systematic and global approach to a reduction of the problem to a problem defined completely on the different “polarizations.” Finally, we discuss to what extent it is actually possible to reduce the problem to a really scalar one, and make some comments on obstructions to the existence of global quasiclassical states.

Journal of Mathematical PhysicsJul 1, 199812 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)