of the reciprocal vectors. Fig. 12.5 shows a two-dimensional real lattice and

the corresponding reciprocal lattice vectors.

Exercises

12.1 Show that hermitian operators have real eigenvalues.

12.2 Show that the operator for k is hermitian. Use partial integration of

f — (‚g/‚x) dx; the product f — g vanishes at the integration bound-

aries.

12.3 Show that the expectation of k must vanish when the wave function

is real.

12.4 Show that the expectation value of k equals k0 if the wave function

is a real function, multiplied by exp(ik0 ).

12.5 Derive (12.62) to (12.64) from (12.60) and (12.61) and express ak

and bk in terms of Fk .

Show that ξ = a— · r, · = b— · r, ζ = c— · r.

12.6

12.7 Prove (12.97) and (12.98).

12.8 Derive the Fourier pair (12.72) and (12.73) from the original de¬ni-

tions in Section 12.1.

13

Electromagnetism

13.1 Maxwell™s equation for vacuum

For convenience of the reader and for unity of notation we shall review the

basic elements of electromagnetism, based on Maxwell™s equations. We shall

use SI units throughout. Two unit-related constants ¬gure in the equations:

the electric and magnetic permittivities of vacuum µ0 and μ0 :

1

= 8.854 187 817 . . . — 10’12 F/m,

µ0 = (13.1)

2

μ0 c

μ0 = 4π — 10’7 N/A2 , (13.2)

1

µ0 μ0 = 2 . (13.3)

c

The basic law describes the Lorentz force F on a particle with charge q

and velocity v in an electromagnetic ¬eld:

F = q(E + v — B). (13.4)

Here, E is the electric ¬eld and B the magnetic ¬eld acting on the particle.

The ¬elds obey the four Maxwell equations which are continuum equations

in vacuum space that describe the relations between the ¬elds and their

source terms ρ (charge density) and j (current density):

div E = ρ/µ0 , (13.5)

div B = 0, (13.6)

‚B

curl E + = 0, (13.7)

‚t

1 ‚E

curl B ’ 2 = μ0 j. (13.8)

c ‚t

Moving charges (with velocity v) produce currents:

j = ρv. (13.9)

335

336 Electromagnetism

The charge density and current obey a conservation law, expressed as

‚ρ

div j + = 0, (13.10)

‚t

which results from the fact that charge ¬‚owing out of a region goes at the

expense of the charge density in that region.

13.2 Maxwell™s equation for polarizable matter

In the presence of linearly polarizable matter with electric and magnetic

susceptibilities χe and χm , an electric dipole density P and a magnetic dipole

density (M ) are locally induced according to

P = µ0 χe E, (13.11)

M = χm H. (13.12)

The charge and current densities now contain terms due to the polariza-

tion:

ρ = ρ0 ’ div P , (13.13)

‚P

j = j0 + + curl M , (13.14)

‚t

where ρ0 and j 0 are the free or unbound sources. With the de¬nitions of

the dielectric displacement D and magnetic intensity 1 H and the material

electric and magnetic permittivities µ and μ,

D = µ0 E + P = µE, (13.15)

1 1

B ’ M = B,

H= (13.16)

μ0 μ

the Maxwell equations for linearly polarizable matter are obtained:

div D = ρ0 , (13.17)

div B = 0, (13.18)

‚B

curl E + = 0, (13.19)

‚t

‚D

curl H ’ = j 0. (13.20)

‚t

In older literature H is called the magnetic ¬eld strength and B the magnetic induction.

1

13.3 Integrated form of Maxwell™s equations 337

The time derivative of D acts as a current density and is called the displace-

ment current density. The permittivities are related to the susceptibilities:

µ = (1 + χe )µ0 , (13.21)

μ = (1 + χm )μ0 . (13.22)

µ is often called the dielectric constant, although it is advisable to reserve

that term for the relative dielectric permittivity

µ

µr = . (13.23)

µ0

13.3 Integrated form of Maxwell™s equations

The Maxwell relations may be integrated for practical use. We then obtain:

• the Gauss equation, relating the integral of the normal component of D

over a closed surface to the total charge inside the enclosed volume:

D · dS = q, (13.24)

which leads immediately to the Coulomb ¬eld of a point charge at the

origin:

qr

D(r) = ; (13.25)

4π r3

• Faraday™s induction law, equating the voltage along a closed path with

the time derivative of the total magnetic ¬‚ux through a surface bounded

by the path:

‚

Vind = E · dl = ’ B · dS; (13.26)

‚t

• Ampere™s law, relating the magnetic ¬eld along a closed path to the total

current i through a surface bounded by the path:

‚

H · dl = D · dS.

i+ (13.27)

‚t

13.4 Potentials

It is convenient to describe electromagnetic ¬elds as spatial derivatives of a

potential ¬eld. This can only be done if four quantities are used to describe

the potential, a scalar potential φ and a vector potential A:

‚A

E = ’ grad φ ’ , (13.28)