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A two-dimensional analogue of Levinson's theorem for nonrelativistic quantum mechanics is established, which relates the phase shift at threshold(zero momentum) for the $m$th partial wave to the total number of bound states with angular momentum $m\hbar(m=0,1,2,...)$ in an attractive central field.Comment: LaTeX, no figur

We calculate the probability of electron-positron pair creation in vacuum in 3+1 dimensions by an external electromagnetic field composed of a constant uniform electric field and a constant uniform magnetic field, both of arbitrary magnitudes and directions. The same problem is also studied in 2+1 and 1+1 dimensions in appropriate external fields and similar results are obtained.

Neutral fermions of spin 1 2 with magnetic moment can interact with electromagnetic fields through nonminimal coupling. In 2+1 dimensions the electromagnetic field strength plays the same role to the magnetic moment as the vector potential to the electric charge. This duality enables one to obtain physical results for neutral particles from known ones for charged particles. We give the probability of neutral particle-antiparticle pair creation in the vacuum by non-uniform electromagnetic fields produced by constant uniform charge and current densities.

The authors quantize the Abelian and nonAbelian Chern-Simons theories in (2+1) dimensions by the Dirac quantization formulation. Gauge theories with a topological mass term in both the Abelian and nonAbelian cases are also quantized by the same method.

In a recent paper the magnetostatic boundary-value problem for a magnetic dipole with transverse direction in the presence of a superconducting sphere was solved in both cases when the London penetration depth is zero and finite. It was concluded that the levitation force on the transverse magnetic dipole is exactly half that for a magnetic dipole with radial direction. We show that this conclusion is incorrect in either case. In the former case it is due to an incorrect boundary condition. In the latter case it is caused by calculational errors. Corrected results are presented. The distribution of supercurrent and the associated magnetic moment are also calculated.In a recent paper Coffey solved the magnetostatic boundary-value problem for a magnetic dipole with transverse direction in the presence of a superconducting sphere in both cases when the London penetration depth is zero and finite. 1 The result for the former case was also published in a separate paper. 2 The latter case involves some mathematical difficulty and is interesting. From these studies it was concluded that the levitation force on the transverse magnetic dipole is exactly half that for a magnetic dipole with radial direction. Unfortunately, this conclusion appears to be incorrect in either case. In the former case it is due to an incorrect boundary condition employed. In the latter case it is caused by calculational errors. Because the conclusion is impressive it deserves some clarification. In addition to discussions of the errors, the corrected results are presented here. We also calculate the distribution of supercurrent and the associated magnetic moment.For the convenience of comparison, we will use similar notations as in Ref. 1. We use both the rectangular coordinates ͑x , y , z͒ and the spherical ones ͑r , , ͒. The unit vectors of these coordinate systems are denoted by ͑e x , e y , e z ͒ and ͑e r , e , e ͒, respectively. The position vector is denoted by r. As in Ref. 1 we use MKS units.Consider a superconducting sphere with radius b, whose center is located at the origin of the coordinate system. There is a point magnetic dipole located at the position d = de z where d Ͼ b ͑d is denoted by a in Ref. 1͒, the magnetic dipole moment being m 0 = e x m 0 cos 0 + e y m 0 sin 0 . Here we are considering a somewhat more general case; when 0 = 0 it reduces to the case in Ref. 1. ͑m 0 is denoted by m in Ref. 1. We change the notation to avoid confusion with the angular eigenvalue in the spherical harmonics.͒ The problem is to find the magnetic induction in the whole space.The magnetic induction outside the sphere is B = B 1 + B 2 , where B 1 is the field of m 0 in free space, and B 2 is the induced field produced by the supercurrent in the sphere. B 1 can be described by a scalar potential B 1 =− 0 ١⌽ 1 , whereThe Maxwell equation for B 2 is obviouslyTherefore, B 2 can also be described by a scalar potential B 2 =− 0 ١⌽ 2 , where ⌽ 2 satisfies the Laplace equation ١ 2 ⌽ 2 = 0, and thus can be expanded aswhere A lm Ͼ are constants to be dete...

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It is well known that any cyclic solution of a spin 1/2 neutral particle moving in an arbitrary magnetic field has a nonadiabatic geometric phase proportional to the solid angle subtended by the trace of the spin. For neutral particles with higher spin, this is true for cyclic solutions with special initial conditions. For more general cyclic solutions, however, this does not hold. As an example, we consider the most general solutions of such particles moving in a rotating magnetic field. If the parameters of the system are appropriately chosen, all solutions are cyclic. The nonadiabatic geometric phase and the solid angle are both calculated explicitly. It turns out that the nonadiabatic geometric phase contains an extra term in addition to the one proportional to the solid angle. The extra term vanishes automatically for spin 1/2. For higher spin, however, it depends on the initial condition. We also consider the valence electron of an alkaline atom. For cyclic solutions with special initial conditions in an arbitrary strong magnetic field, we prove that the nonadiabatic geometric phase is a linear combination of the two solid angles subtended by the traces of the orbit and spin angular momenta. For more general cyclic solutions in a strong rotating magnetic field, the nonadiabatic geometric phase also contains extra terms in addition to the linear combination.