Lehner G. Electromagnetic Field Theory for Engineers and Physicists

Подождите немного. Документ загружается.

594 Appendices

magnetic field, and thus, also describes the action of the field on the particle in the

sense of classical mechanics in a complete manner. This describes a local

interaction and not an action at a distance.

In quantum mechanics – for non-relativistic particles – Schrödinger’s

equation takes the place of the classical equation of motion. It results from the

Hamiltonian of classical mechanics. For an arbitrary system, the Hamiltonian is a

function of the canonical momentum and location coordinates and ,

(A.3.4)

The Hamiltonian is usually referenced by the letter H and should not be confused

with the magnetic field. In this terminology, the classical equation of motion is

expressed by Hamilton’s differential equations

(A.3.5)

. (A.3.6)

The Hamiltonian for a particle of mass m, located in a force field with the potential

(A.3.7)

where the

(A.3.8)

are the components of the momentum. Then

(A.3.9)

and

(A.3.10)

thus

(A.3.11)

i.e. the classical (Newton’s) equation of the motion.

For a particle in an electromagnetic field we have

(A.3.12)

where A and ϕ are the electromagnetic potentials which allow to calculate E and B:

(A.3.13)

HHp

,()=

--------

∂H

∂q

--------–=

--------

∂H

∂p

--------=

,,()

------- U+

------------------------------ Ux

,,()+==

-------

∂U

∂x

---------–=

-------

----=

-------

∂U

∂x

-------–=

p QA–()

------------------------- Qϕ+=

E ∇ϕ–

∂A

∂t

-------–=

595

(A.3.14)

p is the canonical momentum,

(A.3.15)

which should not be confused with the ordinary momentum mv, into which it

transforms in the limit . In this case, Hamilton’s differential equations result

in the afore mentioned equation of motion (A.3.3), which we will not prove here.

Replacing the physical quantities by operators, as we have done before in

Sect. A.1, we obtain:

(A.3.16)

. (A.3.17)

This allows to write Schrödinger’s equation in the following form

(A.3.18)

In particular, for a particle in an electromagnetic field, we obtain from (A.3.12)

(A.3.19)

Particular care is advised when calculating the square because the momentum

operator and the location operator, or the location dependent quantities,

respectively (here ), do not commute. Written more explicitly, we

obtain:

(A.3.20)

A.3.2 The Role of Fields and Potentials

The equation of motion which applies to classical electromagnetic field theory is

(A.3.21)

Its counterpart in quantum mechanics is the Schrödinger equation

(A.3.22)

One equation contains the fields E and B, while the other uses the potentials A and

ϕ. Within the classical theory, the potentials were introduced formally as auxiliary

quantities, whose task was to simplify the solution of Maxwell’s equations, and

they live up to their task. Two of the four of Maxwell’s equations are automatically

solved by (A.3.13) and (A.3.14), while the other two result in the inhomogeneous

wave equations (7.187) and (7.188). It is easily possible to eliminate the fields from

the equation of motion (A.3.21) and replace them with the potentials. The question

BA∇×=

p mv QA+=

A 0=

⇒ iÑ

∂

∂t

----

⇒ iÑ∇–=

iÑ

∂ψ

∂t

-------

iÑ∇– q,()ψ=

iÑ

∂ψ

∂t

-------

iÑ∇– QA–()

------------------------------------ Qϕ+ ψ=

AAr()=

iÑ

∂ψ

∂t

-------

∇

– iÑQA ∇• iÑQ∇ A• Q

+++

--------------------------------------------------------------------------------------------------- Qϕ+ ψ=

F m

--------

QE QvB ×+==

iÑ

dψ

-------

ih∇– QA–()

------------------------------------ Qϕ+ ψ=

A.3 On the Significance of Electromagnetic Fields and Potentials

596 Appendices

arises, if it is also possible to eliminate the potentials in Schrödinger’s equation in

favor of the actual fields E and B. This is not possible, at least not without

difficulties. If one insists to do so, the best place to start are the inhomogeneous

wave equations (7.187) and (7.188). Their solutions are the retarded potentials

(7.195) and (7.196). Furthermore, one may express ρ and g by E and B.

and

This allows to write the retarded potentials (7.195), (7.196) in the following form

(A.3.23)

. (A.3.24)

It does not appear to be too beneficial to substitute these expressions into the

Schrödinger equation. The thereby emerging formula would be very complicated,

without carrying any particular benefit. Furthermore, they have the inconvenient

property that the wave function ψ(r) does not only depend on the fields E(r) and

B(r) at their respective location, but also on the integrals of these fields in the entire

space, that is, the advantage of the local interaction is lost. This would defeat the

original intent for introducing these potentials in the classical theory. In contrast,

the local interaction remains in place when we keep the potentials A and ϕ in the

Schrödinger equation and regard them as real, not replaceable fields (in contrast to

simply auxiliary quantities). Our subsequent discussion will show that this is also

necessary based on different, even more fundamental reasons.

For our later use, we will analyze two special cases of Schrödinger’s equation

(A.3.22).

a) The first case covers

(A.3.25)

If is a solution of the equation for ,

(A.3.26)

then

(A.3.27)

is a solution to (A.3.25) because

ρε

∇ E•=

------

∇ B×ε

∂E

∂t

-------

–=

ϕ r t,()

∇ E• r' t

rr'–

---------------–,





4π rr'–

-------------------------------------------------

dτ'

∫

Art,()

∇ B× r' t

rr'–

---------------–,





-----

∂

∂t

----

Er' t

rr'–

---------------–,





–

4π rr'–

-----------------------------------------------------------------------------------------------------------

dτ'

∫

ϕ 0=

iÑ

∂ψ

∂t

-------

iÑ∇– QA–()

------------------------------------

ψ=

A 0=

iÑ

∂ψ

∂t

----------

iÑ∇–()

--------------------

ψψ

----

Asd•

∫

exp=

597

and

b) The second case covers and (i.e. ϕ is independent of the

location and thus, ):

(A.3.28)

If is a solution of equation (A.3.26), then a solution of eq. (A.3.28) is

(A.3.29)

The premise that is satisfied e.g., for a particle that moves inside a

Faraday cage whose surface has a time dependent potential.

The potentials cause, in both cases, an additional phase factor with the phase

shift of and , respectively.

A.3.3 The Ehrenfest Theorems

Despite the significant differences between classical mechanics and quantum

mechanics, they do not mutually contradict themselves. Stated somewhat

simplified, a result of quantum mechanics is that the averages of physical quantities

behave like classical mechanics predicts. This is expressed in Ehrenfest’s

Theorems. Using the notation to express the average of the physical quantity

g, allows to obtain the following relations from Schrödinger’s equation:

(A.3.30)

(A.3.31)

or when combining the two

(A.3.32)

This is the classical equation of motion, which occurs here as a consequence of

Schrödinger’s equation. Deviations from these averages are very improbable when

dealing with macroscopic systems, which renders the difference between (A.3.32)

and the classical results negligible. However, this is not true for microscopic

systems.

iÑ∇– QA–()ψ

----

Asd•

∫

exp i

----

Asd•

∫

exp iÑ∇–()ψ

iÑ∇– QA–()

----

Asd•

∫

exp i

----

Asd•

∫

exp iÑ∇–()

A 0= ϕϕt()=

E 0=

iÑ

∂ψ

∂t

-------

∇

– Qϕ+





ψ=

ψψ

----

ϕ t'()t'd

∫

–exp=

ϕϕt()=

Q Ñ⁄()Ard•

∫

Q Ñ⁄–()ϕt() td

∫

g〈〉

r〈〉

p〈〉

--------=

p〈〉 U r() ∇〈〉–=

-------

r〈〉 U r() ∇〈〉–=

A.3 On the Significance of Electromagnetic Fields and Potentials

598 Appendices

Looking at the motion of a particle in an electromagnetic field, one obtains

from Schrödinger’s equation (A.3.32), after some tedious mathematics, Ehrenfest’s

theorem in the following form:

(A.3.33)

We conclude that quantum mechanics also permits to calculate the “mean” particle

trajectory by means of E and B, even in a form analogous to the classical equation

of the motion. This applies only to the average and does not describe the exact

motion of any particular particle in an electromagnetic field. The form of (A.3.33)

may initially look peculiar, but results from the fact that velocity in quantum

mechanics is an operator related with the momentum, what arises from (A.3.15)

and (A.3.17). This operator does not commute with the location operator or any of

the location dependent operators, e.g. or . The classical

expression is of course:

(A.3.34)

In quantum mechanics, these expressions may not be equated. However, one can

see that in the limit of classical mechanics, Ehrenfest’s theorem in the form

(A.3.33), just yields the Lorentz force.

A.3.4 Magnetic Field and Vector Potential in an infinitely long Coil

Consider an infinitely long coil as shown in Fig. A.3.1. Its magnetic field can be

expressed by the vector potential A, where

(A.3.35)

The magnetic flux through an arbitrary surface is:

(A.3.36)

B is gauge invariant, i.e., is not affected by a transition from one vector potential to

another of different gauge, where both vector potentials may only differ by the

gradient of an arbitrary function. This makes the flux gauge invariant as well, as

(A.3.36) makes obvious, because can not change in this case

( ). Fig. A.3.1 shows the graph of and . Under the

-------

r〈〉 QE

----

vB× Bv×–() +〈〉=

AAr()= BBr()=

---

vB× Bv×–()

---

vB× vB×+()vB×()==

Fig. A.3.1

r()

---

∼

r∼

r()

B ∇ A×=

φ Bad•

∫

∇ A× ad•

∫

A ds•

∫

== =

A ds•

∫

∇fds•

∫

0= B

r() A

r()

599

currently assumed gauge, A possesses only a ϕ-component . Now, (see also

Sect. 5.2.3):

(A.3.37)

and

(A.3.38)

Interesting to realize is that the magnetic field outside of the coil vanishes, while

the vector potential does not. This fact will be of our interest next. We will discuss

the question whether the vector potential outside of the coil might influence, in any

way, charged particles whose behavior is described by Schrödinger’s equation. For

this purpose, we will study experiments where electron beams interfere at a double

slit, as was described by Bohm and Aharonov [34].

A.3.5 Interference of Electron Beams at Double Slit

A double slit as illustrated in Fig. A.3.2 shall be used for interference experiments

with electron beams. Starting without a coil and its magnetic field behind the slit,

we obtain an interference pattern on the screen with intensity maxima and minima

due to the impinging electron rays. The quantities (a, d, L, r

, r

, x), defined in

Fig. A.3.2 allow to calculate the difference in the geometric path length.

(A.3.39)

If , then

(A.3.40)

This corresponds to a phase difference of

(A.3.41)

if denotes the wave length of matter associated with the electrons, which is a

result of the de Broglie relation

(A.3.42)

where p represents the momentum of the electron.

Repeating this experiment with the coil and its magnetic field in place, the

electrons from each of the interfering rays travel within the range of the vector

r()

for rr

≤

0 for rr







r()

---

for rr

≤

-----

for rr

.≥











– L

---





+ L

---

–





+–==

xL«

------

≈

2π

------

⋅=

---=

A.3 On the Significance of Electromagnetic Fields and Potentials

600 Appendices

potential created by the coil. The coil shall be assumed to be ideal, i.e. has no

fringing fields. The ray of electrons shall not be able to penetrate the inside of the

coil. Expressed differently, the wave function of the electrons and the magnetic

induction of the field inside the coil shall not overlap. This results in additional

phase differences between the rays along the paths C

and C

, respectively:

(A.3.43)

and

(A.3.44)

Material for the interference is the phase difference

(A.3.45)

where is the flux contained inside the coil. This result is very peculiar. As long

as we only regard the phase difference , then only the overall flux is relevant, not

the specific spatial distribution of the magnetic field which produces this flux. The

flux causes a shift of the maxima and minima of the interference picture on the

screen by the distance

(A.3.46)

∆x

uniform

field

magnetic

∆x

coil with

flux

Fig. A.3.2 Fig. A.3.3

----

Asd•

∫

----

Asd•

∫

ββ

–

----

Asd•

∫

----

φ== =

∆x

Lλ

2πd

----------

β⋅

Lλ

2πd

----------

----

φ⋅

LλQφ

---------------== =

601

which results from (A.3.41) when replacing α by β and x by ∆x. Fig. A.3.2 shows

the interference pattern picture a) without and b) with magnetic field.

Precise measurements show [35], that although the maxima and minima shift

by ∆x as expressed by (A.3.36) and is depicted in Fig. A.3.2, the envelope remains

unchanged, however. This is the result predicted by Bohm and Aharonov and

meanwhile also verified experimentally, most recently and most clearly by [36].

This effect is not explainable by the B field alone. From the perspective of classical

mechanics, the coil should not influence those passing rays (at least under the

condition that the B field of the coil does not overlap with the wave function ,

which is not easy to achieve experimentally and gave rise to many controversies on

the validity of the Bohm-Aharonov effect).

Interesting is also to consider a variation of the experiment of Fig. A.3.2. This

is described by Fig. A.3.3. Here, the coil and its magnetic field is replaced by a

region, carrying a uniform field (perpendicular to the paper plane). The width of

this region is w. Of course, without magnetic field, the result is just as before. With

the magnetic field, we obtain a flux, which is what is relevant here. The flux is

approximately (i.e., if )

(A.3.47)

and the thereby caused shift of the maxima and minima of the interference pattern

is then with (A.3.36)

(A.3.48)

Fig. A.3.3 illustrates, just as Fig. A.3.2, the interference pattern a) without and b)

with a magnetic field. However, in contrast to the case of Fig. A.3.2, now the entire

interference pattern, including the envelope, shifts by in x-direction. This is

plausible and fits well with the classical understanding. For , the Lorentz

force causes

(A.3.49)

where represents the time which it takes for a particle to pass through the region

of the uniform magnetic field. If v is its velocity, then

(A.3.50)

Furthermore,

(A.3.51)

After multiplication by L, this gives exactly the previous result, eq. (A.3.48), which

we have derived there in a rather different way. All electrons are deflected by

exactly the same angle ( ), i.e., the entire interference pattern is shifted by

xL«

φ B

wd=

∆x

LλQ

-----------

LλQB

----------------------==

∆x

xL«

∆p

QvB

τ QvB

----

w== =

----=

∆x

------

∆p

---------

≈

---

---------------

λQB

-------------------==

∆xL⁄≈

A.3 On the Significance of Electromagnetic Fields and Potentials

602 Appendices

the distance . Regarding the sign, we note that for electrons ( ) and for

magnetic fields which point out of the paper plane, the shift is to the right.

The difference between the two variants of the experiment (namely one where

the coil that has no outside magnetic field, and the other where a uniform magnetic

field is applied), is rather peculiar. Nevertheless, on the basis of Ehrenfest’s

theorems, on one hand, and the phase difference caused by the vector potential on

the other, it should be conceptually clear. In order to determine the root cause of the

maxima and minima, one knows that this only depends on the phase difference, and

this means it solely depends on the enclosed flux ( ). The average of a

particle trajectory is determined by the Lorentz force, and this means the magnetic

induction, whereby the particular form of the Lorentz force given in eq. (A.3.33)

has to be considered. All this is true from a quantum mechanical perspective as

well. Consequently, the center of mass of the particle trajectory remains unchanged

in the case of the coil without fringe field, and therefore, the envelope remains

unchanged as well. In contrast in the case of the uniform field, the electrons are

deflected both, classically and quantum mechanically by the same angle. The

averages shift accordingly. Therefore, in this case, both the envelope and the

interference pattern are shifted by the same distance . This corresponds to a

shift of the center of mass (of the average) by exactly this distance . With this,

we have gained a clear and easily understandable picture of the process.

An entirely different, nevertheless related experiment (which also goes back

to the mentioned work of Bohm and Aharonov) is sketched in Fig. A.3.4.

Each of the two rays travels through a shielded cavity in form of a tube

and , respectively. Then the potentials and are applied to the pipes

during the time when the wave packets travel through a tube but are not too close to

the ends (where there may be fringe fields). This creates a phase difference

(A.3.52)

and

(A.3.53)

If , then an interference pattern with maxima and minima corresponding to

the difference

(A.3.54)

∆xQ0<

A ds•

∫

∆x

Fig. A.3.4

t()

t() ϕ

t()

----

t()td

∫

–=

----

t()td

∫

–=

≠

ββ

–=

A.4 Liénard-Wiechert Potentials 603

is visible at a screen, where as before, eq. (A.3.46) applies. This is as peculiar as

the previous experiment, involving the magnetic field of a coil. Even though inside

the pipes we have

(A.3.55)

the potential still influences the electrons which pass through it. Similarly to the

previous explanation, it is important to realize that the electric field is not sufficient

to describe the interaction. True also in this case is that the envelope of the

interference pattern (i.e., the “center of mass” of the incident electrons) does not

shift if . Merely the position of the maxima and minima is affected by the

potentials and . Conversely, when electrons pass through non-

vanishing electric fields, then with shift of the center of mass of the rays, the entire

pattern, including envelope shifts. Both obey the proposition of Ehrenfest’s

theorems.

A.3.6 Conclusions

We have found that quantum mechanics provides a new perspective on the reality

of the fields E and B, on one side, and the potentials A and ϕ on the other.

Maxwell’s equations are thereby unaffected. They can be accepted as correct even

in light of quantum mechanics. However, the interaction of charged particles with

electromagnetic fields exhibits effects which can not be explained classically by

the equation of the motion, and the fields of Maxwell’s equations can not

sufficiently describe these effects. Particularly, the potentials A and ϕ are

indispensable and one needs to consider them as separate fields. It would be easier

to dispense of the classical fields E and B because the potentials are, anyway, a

means within the classical theory to eliminate E and B, while the reverse in

quantum mechanics is not possible, at least not without major difficulties.

A.4 Liénard-Wiechert Potentials

A very interesting, special case of the retarded potentials as of eqs. (7.195) and

(7.196) results, if we consider a charged particle travelling on an arbitrarily

prescribed trajectory . This case is described by:

(A.4.1)

. (A.4.2)

Q represents the charge of the particle and

(A.4.3)

is its velocity. The potentials are

E ∇ϕ–0==

E 0=

t() ϕ

t()

ρ Qδ rr

t()–[]=

g Qr

δ rr

t()–[]=

δ rr

t()–[]=

t()

--------------- v

t()==