0198506961.pdf

36 The hydrogen atom

Fig. 2.3The representation of (a) spin-
up and (b) spin-down states as vectors
precessing around thez-axis.

(a) (b)

a structureless elementary particle with no measurable size. So we are

left with the experimental fact that the electron has an intrinsic spin

angular momentum of
/2 and these half-integer values are perfectly

acceptable within the general theory of angular momentum in quantum

mechanics.

2.3.2 The spin–orbit interaction

The Schr ̈odinger equation is non-relativistic, as can readily be seen by

looking at the kinetic-energy operator that is equivalent to the non-

relativistic expressionp^2 / 2 me. Some of the relativistic effects can be

taken into account as follows. An electron moving through an electric

fieldEexperiences an effective magnetic fieldBgiven by

B=−

1

c^2

v×E. (2.45)

This is a consequence of the way an electric field behaves under a Lorentz

transformation from a stationary to a moving frame in special relativity.

Although a derivation of this equation is not given here, it is certainly

plausible since special relativity and electromagnetism are intimately

linked through the speed of lightc=1/

√

 0 μ 0. This equation for the

speed of electromagnetic waves in a vacuum comes from Maxwell’s equa-

tions; 0 being associated with the electric field andμ 0 with the magnetic

field. Rearrangement to giveμ 0 =1/

(

 0 c^2

)

suggests that magnetic

(^45) The Biot–Savart law for the magnetic fields arise from electrodynamics and relativity. 45
field from a current flowing along a
straight wire can be recovered from the
Lorentz transformation and Coulomb’s
law (Griffiths 1999). However, this
link can only be made in this direction
for simple cases and generally the phe-
nomenon of magnetism cannot be ‘de-
rived’ in this way.
We now manipulate eqn 2.45 into a convenient form, by substituting
for the electric field in terms of the gradient of the potential energyV
and unit vector in the radial direction:
E=

1

e

∂V

∂r

r

r

. (2.46)

The factor ofecomes in because the electron’s potential energyVequals

its charge−etimes the electrostatic potential. From eqn 2.45 we have

B=

1

mec^2

(

1

er

∂V

∂r

)

r×mev=

mec^2

(

1

er

∂V

∂r

)

l, (2.47)

where the orbital angular momentum is
l=r×mev. The electron

has an intrinsic magnetic momentμ=−gsμBs, where the spin has a

magnitude of|s|=s=1/2 (in units of
)andgs2, so the moment