Solution of the previously posted problem on Aharonov-Bohm effect.

The physical model described in the question is illustrated on Fig. 1.

Fig. 1: Electron in the spherical shell confined to a circular track around an infinite solenoid.

Two types of motion could be distinguished in the model: motion of the electron inside the spherical shell and motion of the spherical shell on the circular track. Therefore it is convenient to describe the system using angleto define the position of the spherical shell center on the circular track and spherical coordinatesto describe the point inside the spherical shell.

Given that the solenoid is ideal, the electric and magnetic field vectors equal zero outside of the solenoid, while inside of the solenoid there is a constant magnetic filed . This makes two types of motion mentioned above topologically inequivalent: trajectories of the electron inside of the spherical shell do not encompass any regions containing the field and therefore are simply connected, in contrary, the trajectory of the sphere on the track encompasses section of the solenoid with a magnetic field and therefore is not simply connected. A charged particle traversing through the latter type of trajectory is known to be subject to Aharonov-Bohm effect, which is an example of non-local interaction between the field and a charged particle. In the case of the magnetic field the effect is called magnetic Aharonov-Bohm effect. It arises from the fact that even if magnetic field (B) equals zero along a trajectory, but the trajectory encompasses an area containing magnetic field (with a non zero flux) the vector field potential (A) along the trajectory is not zero and therefore affects motion of a charged particles.

By the definition magnetic field and vector field potential are related through the following equation:

.	(1)

Taking the surface integral of the both sides we obtain:

,	(2)

whereis the surface encompassed by the trajectory.

Applying Stokes' theorem to the right hand side gives:

,	(3)

where L is the trajectory.

Due to the symmetry of the problem is constant along the trajectory therefore:

,	(4)

Where A is the component of the vector potential tangent to the trajectory (the component parallel to the magnetic filed is zero due to the symmetry of the problem and another component could be made 0 by a gauge transformation).

Taking into account that the trajectory is a circle with radius R we obtain:

	(5)

Combining equations 3-5 we obtain an expression for the vector filed potential:

	(6)

The numerator of the ratio in the equation (6) is the flux of the magnetic field through the encompassed surface. Since B is not zero only inside of the solenoid and is constant:

	(7)

The remaining surface integral is just the area of the surface enclosed by the solenoid with radius a:

	(8)

Combining equations (7) and (8) results in an explicit expression for :

	(9)

Since the radius of the circular track (R₀ = 50 Å) is by an order of magnitude bigger then the radius of the spherical shell (r₀ = 5 Å) the value ofinside the shell may be assumed to be equal to the value in the center of the shell:

	(10)

Also the timescale of motion of the sphere on the track is slower then that for an electron in the sphere. Therefore we may assume that electron moving in the sphere experiences constant in time. Within these assumptions we may now separate two types of motion:

1. Motion of the sphere on the track becomes motion of a particle in a ring with a non-zero magnetic flux.

2. Motion of the electron in the sphere becomes motion of a particle in a sphere with an infinite square potential.

Now we solve these two problems independently and then combine their solutions.

Particle in a ring with a non-zero magnetic flux.

The Hamiltonian describing the motion of a charged particle in the presence of the non-zero magnetic vector potential is given by [1]:

.	(11)

Taking into account the specifics of the problem:

,	(12)

where is the angle describing the position on the ring. Then in spherical coordinates using atomic units (m=1, q=-1) and the fact that A is tangent to the trajectory, the equation of motion is:

	(13)

Applying a substitutionto this differential equation leads to a quadratic equation:

	(14)

From which x could be found:

	(15)

Due to the boundary condition of the particle on a ring problem:

	(16)

the value of x must satisfy:

	(17)

where N must be an integer. Therefore levels of energy are quantized as follows:

	(18)

As one can notice presence of the magnetic vector potential has removed the degeneracy of the states with N=k and N=-k quantum numbers. Combining equations (15) with (18) and recalling the substitution gives the eigenstates of the system:

,	(19)

and

.	(20)

Particle in sphere with an infinite square potential.

Motion of the electron in a spherically symmetric potential given by [2,3]:

and	(21)

is described by the following Hamiltonian:

	(22)

Due to symmetry of the problem in spherical coordinates radial and angular variables are separable and solutions are described in terms of the free spherical waves:

and	(23)

Where is a normalization coefficient, is spherical Bessel function, is a spherical harmonics, is n-th root of equation and n, l, m are quantum numbers. The energy eigenvalues of these wavefunctions are:

	(24)

Now we can combine solutions for two types of motions together and describe the overall eigenstates of the system as:

and	(25)

	(26)

with energies:

	(27)

Squares of the wavefunctions given by equations 25 and 26 are the probabilities of finding center of the sphere in a point of track defined by anglewith an electron inside of the sphere in the point defined by.

Wavefunction 25 is a combination of the free standing spherical wave for the electron inside of the sphere and a plane wave circling around the track with the speed defined by the wave vector N. Wavefunction 26 has similar interpretation, with an exception that electron moves in different direction with the speed defined by the wave vector N + AR₀. For a given set of quantum numbers (N, n, m, l ) the general wavefunction is a linear combination of 25 and 26:

	(28)

By substitution the magnitude of A from equation 10 we obtain:

A more precise solution will require to take into account the fact values of the magnetic vector potential are different for different points inside of the spherical shell and also take into consideration coupling between motion of the sphere and motion of the electron inside of it (analogous to the corrections to the Born-Oppenheimer approximation).

[1] C. Wittig, Molecular Dynamics lectures (2006).

[2] G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, Academic Press, 1995.

[3] G. Salvador, Phys. Rev. 67, 12102 (2003)