A 2

Let us consider two examples: a stationary homogeneous magnetic ¬eld B

and an electromagnetic plane wave.

2.4.1 Homogeneous external magnetic ¬eld

Consider a constant and homogeneous magnetic ¬eld B and let us ¬nd a

solution A(r) for the equation B = curl A. There are many solutions

(because any gradient ¬eld may be added) and we choose one for which

∇ · A = 0 (the Lorentz convention for a stationary ¬eld, see Chapter 13):

A(r) = 1 B — r. (2.76)

2

The reader should check that this choice gives the proper magnetic ¬eld

while the divergence vanishes. The remaining terms in (2.75) are a linear

term in A:

A · ∇ = 1 (B — r) · ∇ = 1 B · (r — ∇), (2.77)

2 2

34 Quantum mechanics: principles and relativistic e¬ects

which gives a term in the Hamiltonian that represents the Zeeman interac-

tion of the magnetic ¬eld with the orbital magnetic moment:

e ˆ

ˆ B·L ,

Hzeeman = (2.78)

2m

ˆ

L = r — p = ’i r — ∇,

ˆ (2.79)

ˆ

where L is the dimensionless orbital angular momentum operator, and a

quadratic term in A that is related to magnetic susceptibility.6

The Zeeman interaction can be considered as the energy ’μ·B of a dipole

in a ¬eld; hence the (orbital) magnetic dipole operator equals

eˆ ˆ

μ=’ L = ’μB L,

ˆ (2.80)

2m

where μB = e /2m is the Bohr magneton. In the presence of spin this

modi¬es to

ˆ

μ = ’gμB J ,

ˆ (2.81)

where

ˆ ˆˆ

J = L + S, (2.82)

and g is the Lande g-factor, which equals 1 for pure orbital contributions,

2.0023 for pure single electron-spin contributions, and other values for mixed

ˆ

states. The total angular momentum J is characterized by a quantum

number J and, if the spin-orbit coupling is small, there are also meaningful

quantum numbers L and S for orbital and spin angular momentum. The

g-factor then is approximately given by

J(J + 1) + S(S + 1) ’ L(L + 1)

g =1+ . (2.83)

2J(J + 1)

2.4.2 Electromagnetic plane wave

In the case of perturbation by an electromagnetic wave (such as absorption

of light) we describe for simplicity the electromagnetic ¬eld by a linearly

polarized monochromatic plane wave in the direction k (see Chapter 13):

E = E 0 exp[i(k · r ’ ωt)], (2.84)

1k

—E ,

B= (2.85)

ck

ω = kc. (2.86)

6 For details see Jensen (1999).

2.4 Electrodynamic interactions 35

These ¬elds can be derived from the following potentials:

i

A= (2.87)

E,

ω

φ = 0, (2.88)

∇ · A = 0. (2.89)

Note that physical meaning is attached to the real parts of these complex

quantities.

As in the previous case, the Hamiltonian with (2.75) has a linear and a

quadratic term in A. The quadratic term is related to dynamic polarization

and light scattering and (because of its double frequency) to “double quan-

tum” transitions. The linear term in A is more important and gives rise

to ¬rst-order dipole transitions to other states (absorption and emission of

radiation). It gives the following term in the Hamiltonian:

iq q

ˆ A · ∇ = ’ A · p.

ˆ

Hdip = (2.90)

m m

If the wavelength is large compared to the size of the interacting system,

the space dependence of A can be neglected, and A can be considered

as a spatially constant vector, although it is still time dependent. Let us

consider this term in the Hamiltonian as a perturbation and derive the form

of the interaction that will induce transitions between states. In ¬rst-order

perturbation theory, where the wave functions Ψn (r, t) are still solutions

ˆ

of the unperturbed Hamiltonian H0 , transitions from state n to state m

ˆ

occur if the frequency of the perturbation H1 matches |En ’ Em |/h and the

corresponding matrix element is nonzero:

∞

—ˆ

def

ˆ

m|H1 |n = ψm H1 ψn dr = 0. (2.91)

’∞

Thus we need the matrix element m|ˆ |n , which can be related to the

p

matrix element of the corresponding coordinate:

m

m|ˆ |n = (Em ’ En ) m|r|n (2.92)

p

i

(the proof is given at the end of this section). The matrix element of the

ˆ

perturbation Hdip (see (2.90)), summed over all particles, is

i

ˆ

m|Hdip |n = (Em ’ En )A · m| qi r i |n

i

(Em ’ En )

=’ E 0 · μmn , (2.93)

ωmn

36 Quantum mechanics: principles and relativistic e¬ects

where we have made use of (2.87). The term between angular brackets is the

transition dipole moment μmn , the matrix element for the dipole moment

operator. Note that this dipolar interaction is just an approximation to the

total electromagnetic interaction with the ¬eld.

Proof We prove (2.92). We ¬rst show that

mˆ

ˆ

p = [H0 , r], (2.94)

i

which follows (for one component) from

12

ˆ

[H0 , x] = ’ [ˆ , x],

p

2m