Vector Wave Fields

http://link.springer.com/content/pdf/10.1007/978-3-642-84031-9_4.pdf

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

Quantal phase factors accompanying adiabatic changes

The crossing of energy levels has been a matter of considerable discussion. The essential features may be illustrated in the crossing of a polar and homopolar state of a molecule. Let ψ1 ( x /R), ψ2 ( x /R) be two electronic eigenfunctions of a molecule with stationary nuclei. Let these eigenfunctions have the property that for R≫R, ψ1 has polar characteristics, ψ2 homopolar; while at R≪R, ψ2 has polar characteristics, ψ1 homopolar. In the region R=R these two eigenfunctions may be said to exchange their characteristics.

Non-Adiabatic Crossing of Energy Levels

By L. D. Landau and E. M. Lifshitz. Translated from the Russian by J. B. Sykes and J. S. Bell Oxford: Pergamon Press Ltd. Pp. x + 417. Price 84s. As the various volumes of this great work come out in English translation the precise measure of its greatness is revealed more and more. One can only repeat over and over again that there is nothing to compare with it inmodern times, not only in its scope but in the unity of the conception of theoretical physics lying behind it.

Electrodynamics of Continuous Media

Asymptotic expansions for ordinary differential equations , Asymptotic expansions for ordinary differential equations , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

Asymptotic expansions for ordinary differential equations

I Quantization of Velocity Field (the Canonical Operator).- 1. The method of Stationary phase. The Legendre Transformation.- 2. Pseudodifferential Operators.- 3. The Hamilton-Jacobi Equation. The Hamilton System.- 4. The Lagrangian Manifolds and Canonical Transformations.- 5. Fourier Transformation of a ?-Pseudo-differential Operator (the Transition to p-Representation).- 6. The Precanonical Operator (Quantization of the Velocity Field in the Small).- 7. The Index of a Curve on a Lagrangian Manifold.- 8. The Canonical Operator (Global Quantization of the Velocity Field).- 9. Global Quantization of the Velocity Field. Higher Approximations.- II Semi-Classical Approximation for Non-Relativistic and Relativistic Quantum Mechanical Equations.- 10. The Cauchy Problem with Rapidly Oscillating Initial Data for Scalar Hamiltonians.- 11. Matrix Hamiltonians.- 12. The Semi-Classical Asymptotics of the Cauchy Problem for the Schroedinger Equation.- 13. The Asymptotic Series for the Eigenvalues (Bohr's Quantization Rule).- 14. Semi-Classical Approximations for the Relativistic Dirac Equation.- References.- Index of Assumptions, Theorems, Etc..

Semi-classical approximation in quantum mechanics

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Vector Wave Fields

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References

Quantal phase factors accompanying adiabatic changes

Non-Adiabatic Crossing of Energy Levels

Electrodynamics of Continuous Media

Asymptotic expansions for ordinary differential equations

Semi-classical approximation in quantum mechanics

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