Topological Effects in Quantum Mechanics by G. N. Afanasiev (auth.), G. N. Afanasiev (eds.)

By G. N. Afanasiev (auth.), G. N. Afanasiev (eds.)

Among the themes coated during this quantity are the topological results of quantum mechanics, together with Bohm-Aharonov and Aharonov-Casher results and their generalisations; the toroidal moments, anapoles and their generalisations; the numerical research of Tonomura experiments trying out the rules of quantum mechanics; the time-dependent Bohm-Aharonov impact, the thorough examine of toroidal solenoids and their use as powerful transmitters of electromagnetic waves; and the topical questions of the Vavilov-Cherenkov radiation. moreover, concrete recommendation is given for the development of magnetic and electrical solenoids and the functionality of experiments at the Bohm-Aharonov impression. additionally, houses of exceptional charge-current configurations and useful functions are studied.
Audience: This quantity can be of curiosity to postgraduate scholars and researchers facing new potent resources of electromagnetic waves.

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Such a spin chain (a magnetised whisker) was used in earlier experiments testing the existence of the B-A effect. 38 CHAPTER 3. 2: Explicit realisation of the magnetic (electric) cylindrical solenoid by means of a linear chain of magnetic (electric) dipoles. , [16]): Be = r~ [r~ r(mr) - m] + 8; mo (r). 3 Sometimes in the physics literature another representation of B is used [16,59]: 1 [3 411" 3 B'ffi="3 "2r(mr)-m ] - -mo (r). r r 3 This difference arises for the following reason [60]. If we identify a magnetic dipole with an electric current flowing in an infinitely small circular coil, then the vector potential is given by 1 cr -3 me= (me ~ f(r X X r), j)dV.

44) :rx Ql/2 (z2 +~: + X2) . 44) that for p > d the argument y = (p2 + Z2 + x 2 )/2ax of the Legendre function Q-l/2 always exceeds 1 for all x in the interval 0 S; x S; a. This means (as the cut of the Legendre functions coincides with the interval (-1, 1» that the function Q' and all its derivatives are continuous functions of z for p > a. For p < a y acquires the value 1 for z = 0, x = p. In this case the function Q' and its derivatives may possess singularities. 45). In fact, the argument of the (Z2 + x 2 + a 2 )/2ax of the Ql/2 always exceeds 1 for all p < a.

DIvA + -£pc -8¢ = O. 46) contain x and t in the combination x - vt. 47) 4rr --p, £ where 82 Ll. = 81i;2 + 82 8y2 . Let us now introduce the elliptic coordinates v, () : where a Ii; a cosh v cos () , fj a sinh v sin (), R sinhvo ' = tanhvo = J. 46) reduces to the form 8(P'-R) = 3.. coshvosinhvo 8(v-vo). - cos 2(} R cosh 21/0 The values 1/ > Vo and v < I/o correspond to points lying outside and inside the solenoid, respectively. In the coordinates 1/ and () the charge and current densities are given by p = sinh 21/0 sin () cR cosh 2vo - cos 2(} l'(3) - -- ) sinh 2vo cos () R cosh 2vo - cos 2(} - « 0 « 01/-1/0 1/ - I/o ) , ) , )1' sinh 2vo sin () « ) 0 v - I/o .

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