QuantumPhysics.dvi

A= 0. Nonetheless, we can use this property to solve the Schr ̈odinger equation, by setting

ψ(r) =ψ 0 (r) exp

{

−iθ

eΦB

2 π ̄h

}

(7.40)

whereψ 0 (r) now satisfies the Schr ̈odinger equation in the absence of any gauge fields,

1

2 m

(

−i ̄h∇

) 2

ψ(r) +V(r)ψ(r) =Eψ(r) (7.41)

Imagining now two localized wave packets, with wave functionsψ±(r) traveling parallel into

the solenoid from infinity (where we shall choose θ= 0), one packet traveling on the left

and one traveling on the right of the solenoid. After passing the solenoid and proceeding

far away from the solenoid, the left wave packet is characterized byθ=π, which the right

packet is characterized byθ=−π. Thus asymptotically we have

ψ±(r)→ψ 0 (r) exp

{

∓i

eΦB

2 ̄h

}

(7.42)

The total wave functionψ++ψ−exhibits interference, which is constructive when

eΦB= 2π ̄hn (7.43)

for any integern, and destructive thenn− 1 /2 is an integer. This is a measurable outcome,

which has been confirmed by experiment.

7.4.2 The bound state Aharonov-Bohm effect

In this set-up a spinless charged particle is constrained to move between two impenetra-

ble concentric cylinders with radiiR+> R−, impenetrability being again enforced by the

presence of a potentialV(r) which vanishes forR−<|r| < R+and is infinite otherwise.

Inside the small cyclinder, we have a magnetic flux ΦB, and in between the two cylinders,

the magnetic field vanishes (see figure 8).

We shall show that, although the particle is always in a region withB= 0, nonetheless

its energy spectrum depends on ΦB. The gauge potential Ais as in (7.37) and the wave

function satisfies (7.38). This time, we work out the Schr ̈odinger equation is cyclindrical

coordinatesr,θ. The gradient in these coordinates is given by

∇=nr

∂

∂r

+

nθ

r

∂

∂θ

(7.44)

where nr and nθ are unit vectors in ther andθ directions respectively. The set-up is

invariant under rotations (i.e. translations inθ), so that the angular momentum operator