the same reason mercury is a much poorer conductor (by a factor of 14)

than cadmium. For further reading on this subject the reader is referred to

Norrby (1991) and Pyykk¨ (1988).

o

4 Jensen (1999), p. 216.

2.4 Electrodynamic interactions 31

2.4 Electrodynamic interactions

¿From the relation E = p2 /2m and the correspondence relations between

energy or momentum and time or space derivatives we derived the non-

relativistic Schr¨dinger equation for a non-interacting particle (2.24). How

o

is this equation modi¬ed if the particle moves in an external potential?

In general, what we need is the operator form of the Hamiltonian H,

which for most cases is equivalent to the total kinetic plus potential energy.

When the potential energy in an external ¬eld is a function V (r) of the

coordinates only,such as produced by a stationary electric potential, it is

simply added to the kinetic energy:

2

‚Ψ

=’ ∇2 Ψ + V (r)Ψ.

i (2.60)

‚t 2m

In fact, electrons feel the environment through electromagnetic interactions,

in general with both an electric and a magnetic component. If the electric

¬eld is not stationary, there is in principle always a magnetic component.

As we shall see, the magnetic component acts through the vector potential

that modi¬es the momentum of the particle. See Chapter 13 for the basic

elements of electromagnetism.

In order to derive the proper form of the electromagnetic interaction of a

particle with charge q and mass m, we must derive the generalized momen-

tum in the presence of a ¬eld. This is done by the Lagrangian formalism

of mechanics, which is reviewed in Chapter 15. The Lagrangian L(r, v) is

de¬ned as T ’V , where T is the kinetic energy and V is the potential energy.

In the case of an electromagnetic interaction, the electrical potential energy

is modi¬ed with a velocity-dependent term ’qA · v, where A is the vector

potential related to the magnetic ¬eld B by

B = curl A, (2.61)

in a form which is invariant under a Lorentz transformation:

V (r, v) = qφ ’ qA · v. (2.62)

Thus the Lagrangian becomes

L(r, v) = 1 mv 2 ’ qφ + qA · v. (2.63)

2

The reader should verify that with this Lagrangian the Euler“Lagrange equa-

tions of motion for the components of coordinates and velocities

d ‚L ‚L

= (2.64)

dt ‚vi ‚xi

32 Quantum mechanics: principles and relativistic e¬ects

lead to the common Lorentz equation for the acceleration of a charge q in

an electromagnetic ¬eld

mv = q(E + v — B),

™ (2.65)

where

‚A

def

E = ’∇φ ’ (2.66)

‚t

(see Chapter 13). The generalized momentum components pi are de¬ned as

(see Chapter 15)

‚L

pi = , (2.67)

‚vi

and hence

p = mv + qA, (2.68)

or

1

(p ’ qA).

v= (2.69)

m

For the Schr¨dinger equation we need the Hamiltonian H, which is de¬ned

o

as (see Chapter 15)

1

def

H = p·v’L= (p ’ qA)2 + qφ. (2.70)

2m

Thus the non-relativistic Schr¨dinger equation of a particle with charge q

o

and mass m, in the presence of an electromagnetic ¬eld, is

2

2

‚Ψ iqA

ˆ =’ ∇’

i = HΨ + qφ(r) Ψ. (2.71)

‚t 2m

Being non-relativistic, this description ignores the magnetic e¬ects of spin

and orbital momentum, i.e., both the spin-Zeeman term and the spin-orbit

interaction, which must be added ad hoc if required.5

The magnetic ¬eld component of the interaction between nuclei and elec-

trons or electrons mutually is generally ignored so that these interactions

are described by the pure Coulomb term which depends only on coordinates

and not on velocities. If we also ignore magnetic interactions with exter-

nal ¬elds (A = 0), we obtain for a N -particle system with masses mi and

5 The Dirac equation (2.52) in the presence of an external ¬eld (A, φ) has the form:

‚Ψ

= [c± · (ˆ ’ qA) + βmc2 + qφ1]Ψ = HΨ.

p ˆ

i

‚t

This equation naturally leads to both orbital and spin Zeeman interaction with a magnetic

¬eld and to spin-orbit interaction. See Jensen (1999).

2.4 Electrodynamic interactions 33

charges qi the time-dependent Schr¨dinger equation for the wave function

o

Ψ(r 1 , . . . , r N , t):

‚Ψ ˆ

i = HΨ

‚t

2

qi qj

∇2 + 11

= [’ + Vext (r 1 , . . . , r N , t)]Ψ,(2.72)

i i,j rij

2 4πµ0

2mi

i

where the 1/2 in the mutual Coulomb term corrects for double counting in

the sum, the prime on the sum means exclusion of i = j, and the last term,

if applicable, represents the energy in an external ¬eld.

Let us ¬nally derive simpli¬ed expressions in the case of external electro-

magnetic ¬elds. If the external ¬eld is a “slow” electric ¬eld, (2.72) su¬ces.

If the external ¬eld is either a “fast” electric ¬eld (that has an associated

magnetic ¬eld) or includes a separate magnetic ¬eld, the nabla operators

should be modi¬ed as in (2.71) to include the vector potential:

iq

∇i ’ ∇i ’ A(r i ). (2.73)

For simplicity we now drop the particle index i, but note that for ¬nal results

summation over particles is required. Realizing that

∇ · (AΨ) = (∇ · A)Ψ + A · (∇Ψ), (2.74)

the kinetic energy term reduces to

2

q2

iq iq 2iq

∇’ =∇ ’ (∇ · A) ’ A·∇’

2