The Aharonov-Bohm effect is reconsidered as a scattering event of an electron by a magnetic field confined in an infinite solenoid of finite radius both in the situation where the solenoid is penetrable as well as impenetrable. We next discuss the validity of the Born approximation for the partial-wave scattering amplitudes and explain why for the cylindrically symmetric ( m = 0 ) partial wave the first Born approximation fails in the long-wavelength limit or as the radius of the solenoid shrinks to zero.
We present here a new approach to nonrelatlvistic perturbation theory. We show that In problems reducible to one dimension, the energy shifts and wave-function corrections, including corrections to the position of the nodes, to any order, can be expressed in quadrature in a hierarchy scheme. The second-order energy shift calculated is explicitly shown to be equivalent to that in the ordinary Rayleigh-Schrodinger theory.
A method previously developed for one-dimensional nonrelativistic perturbation theory is extended to three-dimensional problems. This method essentially consists of performing the perturbation expansion on the logarithm of the wave function instead of on the wave function itself. It is shown that, for the firstorder corrections in problems that are not reducible to one dimension, this method is equivalent to that of Sternheimer and to that of Dalgarno and Lewis. In the present approach, the higher-order corrections can be obtained in a hierarchical scheme and there exists an isomorphisim between the equation for the firstorder correction and the equation for the ith-order correction. As an illustration of the technique developed, the authors consider the hydrogen atom in an external multipole field and in two different spherically symmetric perturbation potentials, 8(r -a) and e '". The last potential is related to the problem of the screened Coulomb potential. By considering the 8(r -a)-type potential, two interesting sum. rules are obtained.
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