The Semi-classical Van-Vleck Formula.: Application to the Aharonov-Bohm Effect

Alexander L. Kholmetskii, Tolga Yarman

W. F. Kao, Yi-Chun Kao, De‐Hone Lin

A useful and universal formula for the expectation value of the radial operator in the presence of the Aharonov-Bohm flux and the Coulomb Field is established. We find that the expectation value $< r^λ>$ $(-\infty \leq λ\leq \infty)$ is greatly affected due to the non-local effect of the magnetic flux although the Aharonov-Bohm flux does not have any dynamical significance in classical mechanics. In particular, the quantum fluctuation increases in the presence of the magnetic flux due to the Aharonov-Bohm effect. In addition, the Virial theory in quantum mechanics is also constructed for the spherically symmetric system under the Aharonov-Bohm effect.

Herman Erlichson

Christiane Martin

Edwin K. Gora

Abstract Canonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so‐called Van Vieck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. It will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. The Cayley‐Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase‐space. Explicit results have been obtained for several two‐ to four‐dimensional problems.

François Nicoleau

We study an electromagnetic physical system: with the classical B.K.W method, we show that the unitary group of this system is a Fourier integral operator; then, we write this group as an integral operator with smooth kernel. In particular, we establish, in this case, the well-known Feynman's rule, wich explains the Aharonov-Bohm effect