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( 1 B0 r^ for r<R

'

A= 2 (7.18)

2 r^

1 B0 R2 ' for r>R

7.5. THE AHARONOV-BOHM EFFECT 67

z

R

y

x

Figure 7.7: Geometry of the magnetic eld.

The important point of this expression is that although the magnetic eld is only

nonzero for r < R, the vector potential (and its gradient) is nonzero everywhere in space!

The vector potential is thus much more non-local than the magnetic eld. This leads to

a very interesting e ect in quantum mechanics the Aharonov Bohm e ect.

Before introducing this e ect we need to know more about quantum mechanics. As

you have seen in section (6.4) microscopic \particles" such as electrons behave more like

a wave than like a particle. Their wave properties are described by Schrodinger's equa-

tion (6.13). When di erent waves propagate in the same region of space, interference can

occur. In some parts of space the waves may enhance each other (constructive interfer-

ence) while in other parts the waves cancel each other (destructive interference). This is

observed for \particle waves" when electrons are being send through two slits and where

the electrons are detected on a screen behind these slits, see the left panel of gure (7.8).

You might expect that the electrons propagate like bullets along straight lines and that

they are only detected in two points after the two slits. However, this is not the case,

in experiments one observes a pattern of fringes on the screen that are caused by the

constructive and destructive interference of the electron waves. This interference pattern

is sketched in gure (7.8) on the right side of the screens. This remarkable con rmation of

the wave-property of particles is described clearly in ref. 21]. (The situation is even more

remarkable, when one send the electrons through the slits \one-by-one" so that only one

electron passes through the slits at a time, one sees a dot at the detector for each electron.

However, after many particles have arrived at the detector this pattern of dots forms the

interference pattern of the waves, see ref. 54].)

CHAPTER 7. THE THEOREM OF STOKES

68

P P

1 1

B

P P

2 2

Figure 7.8: Experiment where electrons travel through two slits and are detected on a

screen behind the slits. The resulting interference pattern is sketched. The experiment

without magnetic eld is shown on the left, the experiment with magnetic eld is shown

on the right. Note the shift in the maxima and minima of the interference pattern between

the two experiments.

Let us now consider the same experiment, but with a magnetic eld given by equation

(7.15) placed between the two slits. Since the electrons do not pass through this eld

one expects that the electrons are not in uenced by this eld and that the magnetic

eld does not change the observed interference pattern at the detector. However, it is

an observational fact that the magnetic eld does change the interference pattern at the

detector, see ref. 54] for examples. This surprising e ect is called the Aharonov-Bohm

e ect.

In order to understand this e ect, we should note that a magnetic eld in quantum

mechanics leads to a phase shift of the wavefunction. If the wavefunction in the ab-

sence is given by (r), the wavefunction in the presence of the magnetic eld is given by

ie R

(r) exp hc P A dr , see ref. 52]. In this expression h is Planck's constant (divided

by 2 ), c is the speed of light and A is the vector potential associated with the magnetic

eld. The integration is over the path P from the source of the particles to the detec-

tor. Consider now the waves that interfere in the two-slit experiment in the right panel

of gureR (7.8). The wave that travels through the upper slit experiences a phase shift

exp hc P1 A dr , where the integration is over the path P1 through the upper slit. The

ie

ie R

wave that travels through the lower slit obtains a phase shift exp hc P2 A dr where the

path P2 runs through the lower slit.

Problem d: Show that the phase di erence ' between the two waves due to the presence

of the magnetic eld is given by

e I A dr

' = hc (7.19)

P

where the path P is the closed path from the source through the upper slit to the

detector and back through the lower slit to the source.

7.6. WINGTIPS VORTICES 69

This phase di erence a ects the interference pattern because it is the relative phase be-

tween interfering waves that determines whether the interference is constructive or de-

structive.

Problem e: Show that the phase di erence can be written as

'=e (7.20)

hc

where is the magnetic ux through the area enclosed by the path P.

This expression shows that the phase shift between the interfering waves is proportional

to the magnetic eld enclosed by the paths of the interfering waves, despite the fact that

the electrons never move through the magnetic eld B. Mathematically the reason for

this surprising e ect is that the vector potential is nonzero throughout space even when

the magnetic eld is con ned to a small region of space, see expression (7.18) as an

example. However, this explanation is purely mathematical and does not seem to agree

with common sense. This has led to speculations that the vector potential is actually a

more \fundamental" quantity than the magnetic eld 54].

7.6 Wingtips vortices

Figure 7.9: Vortices trailing form the wingtips of a Boeing 727.

CHAPTER 7. THE THEOREM OF STOKES

70

If you have been watching aircraft closely, you may have noticed that sometimes a little

stream of condensation is left behind by the wingtips, see gure (7.9). This is a di erent

condensation trail than the thick contrails created by the engines. The condensation trails

that start at the wingtips is due to a vortex (a spinning motion of the air) that is generated

at the wingtips. This vortex is called the wingtip-vortex. In this section we will use Stokes'

law to see that this wingtip-vortex is closely related to the lift that is generated by the

air ow along a wing.

C

Figure 7.10: Sketch of the ow along an airfoil. The wing is shown in grey, the contour C

is shown by the thick solid line.

Let us rst consider the ow along a wing, see gure (7.10). A wing can only generate

lift when it is curved. In gure (7.10) the air traverses a longer path along the upper

part of the wing than along the lower part. The velocity of the airstream along the upper

part of the wing is therefore larger than the velocity along the lower part. Because of

Bernoulli's law this is the reason that a wing generates lift. (For details of Bernoulli's law

and other aspects of the ow along wings see ref. 60].)

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