The Bohm – Aharonov experiment

Advanced EM - Master in Physics 2011-2012

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The Bohm – Aharonov experimentThe Bohm – Aharonov experiment

Bohm and Aharonov, Physical Review 115, 485; 123,150 R.G. Chambers, Phys. Rev. Letters 5,3Feynman II, 15.5

A source emits and sends electrons of well defined energy against an absorber with two very near slits, at distance “d”. Behind the absorber at a certain distance “l” is an electron detector who counts the rate of arrival of electrons.The detector can be moved and therefore measures the intensity of the incoming electrons as a function of position. The electron beams passing through the 2 slits (well, rather the amplitudes of the electrons passing through one or the other slit and arriving at the detector position) interfere, and a maximum intensity is observed in normal condition at the “central” position.


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This position is determined by the equality of the phases of the two amplitudes for an electron passing through the two slits. Then, a very thin iron “whisker” is inserted just behind the absorber, and is magnetized with B along it axis.It has therefore the field configuration of a solenoid: magnetic field only inside the iron, vector potential everywhere in space.

If we turn on the magnetic field in the whisker an electron taking one of the two trajectories will have a phase increased by

line

dq sA

And the phase difference between the two trajectories corresponding to the two slits will be:

sAsAsA ddqdq

lineline 21

A phase difference δ between the two trajectories will introduce a lateral displacement of the interference pattern of :

dlx

The circular integral introduced before is also, of course, equal to the flow of B through the surface enclosed by the 2 trajectories, which includes the whisker. But the point is that where the trajectories pass there is no B, only A.


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The wave equations

JAActc

tc

41

41

2

2

22

2

2

22

These are the wave equations for the scalar and the vector potentials. They have been derived by the EofM with the extra condition of the Lorentz gauge. We have seen in the previous lesson the recipe for getting the fields once the potentials are known.

In vacuum, of course, these equations still hold even if in large regions of space ρ and J are equal to zero. In vacuum, they are of the type:

012

2

22

tc

Everybody knows these equations and everybody knows that they are the wave equations. As a matter of fact, these equations only tell us that the functional dependence of ψ on x is equal to that from (±ct). They do not say anything about the function itself. We can therefore write that a general solution is any function, as long as it is a function of (r-ct) or of (r+ct) [or other forms of “distance –ct”] . It is well known that these represent “functions of space coordinates” which travel in time. The simplest example is plane waves, i

x


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Plane waves depend on only one space coordinates (x in the example). A function ψ(x) moves forward (f) or backwards (g) in space.Of course, if we have charge or current distributions independent of time, these equations also admit static solutions. Now we disregard them, will only concern ourselves with the time_dependent fields.

)()(),( ctxgctxft r

E At time t=0 At time

t=x/c

Plane wavesPlane waves are waves whose formulas only depend, beside time, on only one space coordinate. They are shown to be travelling in the direction of that coordinate.


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To start with, let us consider the solutions in charge- and current-free space. There the homogeneous equations apply. But, of course, they are contained in the general, non-homogeneous ones. We have identified that the solutions there are “travelling waves”. These waves are waves of A and of Φ, but also of E and B.…It can be demonstrated that these quantities also satisfy the wave equation in vacuum. Some properties of the wave can be easily understood with the simplest example, that of “plane waves”.Plane waves are defined as those waves which depend only on time and only one space coordinate, let us choose “x”. And, let us study how the fields E and B behave.From the definition of plane wave we have that dE/dy, dE/dz and analog for B are all zero; then the values of physical quantities are all constant along the yz plane: this is the base of the concept of wave front. Moreover, from the 1st EofM we find that in a region of no charges

0 also as , B0E 0dz

dEdy

dEdx

dE zyx

Since in this formula the 2 derivatives wrt y and z are zero, the divergence equation states that (dEx/dx)=0. This translates into saying that, since derivatives wrt y and z are all zero, also (dEx/dx)=0 i.e., then Ex, (Bx as well!) have constant components along x. Which is to say, the varying fields are orthogonal to the direction of motion. Or, in other words, electromagnetic waves are transverse waves. If we take the solutions in which the vector E has always the same direction, let us say along y, then Ez =Ex=0; and the curl(E) 2nd EofM becomes (dBx/dt)=(dBy/dt)=0; dBz/cdt=-(dEy/dx). Conclusion:Only d(Bz)/dt is different from 0. Therefore only Bz is dependent of time – and belongs to the family of fields dependent on time. Therefore E and B are perpendicular not only to the direction of propagation, but also to each other.


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We have now determined – using a simplifiede example (Feynman II, ch.18 and 20) – some important properties of the travelling fields – because such are the “fields varying with time”. Of course, some are really static fields moving with uniform velocity… they can be seen as static fields as seen by a travelling observer.

What we have found so far is that 1. The fields move in a certain direction as a “travelling

wave” , which is any function of space, does NOT need to be a sine wave.

2. The waves are transverse , i.e. the fields are perpendicular to the direction of propagation of the wave.

3. The fields are perpendicular not only to the direction of travel but also to each other (I mean: B and E fields, of course).

E

x

At time t=0

At time t=x/c


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Spherical wavesA special – and especially interesting – class of solutions of the wave equation is that of the spherical waves. They are waves (solutions) which depend only on the (spherical) coordinate “r”, i.e. on the distance from a specific “point” in space (which is also chosen as the origin of the coordinate system). Much in the same way as the plane waves have just been introduced. It all starts by writing the laplacian in spherical coordinates and only keeping the derivatives wrt r. And the wave equation in vacuum becomes:

scoordinate sphericalin laplacian theofpart -r theis This )(12

22

rr

r

0)(1)(

:get wer""by gMultiplyin 01)(1

2

2

22

2

2

2

22

2

tr

crr

tcrr

r

This last equation has precisely the same aspect as that of the plane waves. We write down the same solutions.

)]()([1)()(

ctrgctrfr

ctrgctrfr

Where the “f” and the “g” are two waves propagating from the origin outside and from outside to the origin respectively.

The last equation is valid everywhere apart from r=0, where it faces infinity, because to write it we had to divide by r.


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In a way, the spherical wave is a solution of the wave equation for r≠0 in the same way as q/r is a solution of the Laplace equation for a point charge only for r≠0 . We may say that

)(4)(2 rqrq

using the Dirac δ function. This comparison has some benefit: it reminds us that after all what the electrostatic potential looks like at any distance from the source is determined by the value of “source” (charge in electrostatics). [Note, at this point, that we will deal with “outgoing” waves only]. And, what we have done so far is to examine the solution of the wave equation where there are no charges. And what these equation can tell us is how the fields propagate, NOT what they look like! In fact, they have told us that in vacuum waves propagate, and that they are transverse waves. What they look like, is determined by how the sources move. And to know it, we need to solve the inhomogeneous equations.Before attacking that problem, there is another important information hidden in the formula for the spherical waves. It is the 1/r factor sort of hidden in the general formula of an (outgoing) spherical wave. We have already said that both the potentials and the fields satisfy a wave equation. This means that they can ALL be expressed as a spherical wave – with a 1/r general dependence from the distance. Let us compare this result with what be obtained from electrostatics: there the potential Φ of a source limited in space decreases with distance as 1/r. The field, which is the gradient of Φ, has a 1/r2 dependence. This remark is also valid for the magnetic field, which is also the derivative of a potential. In THIS (waves) case instead the fields go as 1/r.

)(1 ctrfr


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This remark is not just mathematics: We insisted in previous lessons on the formula for calculating the energy of the electromagnetic field; and found that the general formula for the energy density is

)(81 22 BEU

Now, if the fields decrease with distance as 1/r then, well, it means that a wave which occupies a certain solid angle, and carries a certain energy, when we increase the distance from the source will go through a larger surface. But, if its fields go like 1/r, then by increasing r their energy density will go as 1/r2 (because U=E2+B2) but the surface they go through goes like r2, so that the total energy that goes with that beam is the same as r changes, as the wave moves on: the fields carry their own energy. But.... How is it possible that the simple adding to the field equations (to the lagrangians) the term

2

2

2

1tc

changes so strongly the r dependence of the fields? We will get an answer later. Let us remember, for the time being, that an electromagnetic wave travels carrying its energy with itself, and dispersing nothing around (in vacuum!).


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The Bohm – Aharonov experiment

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