913 effect of an external magnetic field when an atom

9.1.3 Effect of an external magnetic field
When an atom is placed in a magnetic field, the wavelengths of lines in
its spectrum change slightly. Much of quantum mechanics emerged from
attempts to understand this phenomenon. We now use perturbation theory
to explain it.
In § 3.3 we discussed the motion of a free particle in a uniform magnetic
field. Our starting point was the Hamiltonian (3.30), which governs the mo-
tion of a free particle of mass m and charge Q in the magnetic field produced
by the vector potential A . This is the Hamiltonian of a free particle, p
2
/ 2 m ,
with p replaced by p − Q A . Hence we can incorporate the effects of a mag-
netic field on a hydrogen atom by replacing p
n
and p
e
in the gross-structure
Hamiltonian (8.1) with p
p
− e A and p
e
+ e A , respectively. With Z = 1 the
kinetic energy term in the Hamiltonian then becomes
H
KE
≡
( p
p
− e A ) 2
2 m
p
+
( p
e
+ e A ) 2
2 m
e
=
p
2
p
2 m
p
+
p
2
e
2 m
e
+
1
2
e
braceleftbiggparenleftbigg
p
e
m
e
−
p
p
m
p
parenrightbigg
· A + A ·
parenleftbigg
p
e
m
e
−
p
p
m
p
parenrightbiggbracerightbigg
+ O( A
2
)
(9 . 19)
We neglect the terms that are O( A 2 ) on the grounds that when the field is
weak enough for the O( A ) terms to be small compared to the terms in the
gross-structure Hamiltonian, the O( A 2 ) terms are negligible.
Equation (8.4) and the corresponding equation for ∂/∂ x
p
imply that
p
e
=
m
e
m
e
+ m
p
p
X
+ p
r
and p
p
=
m
p
m
e
+ m
p
p
X
− p
r
, (9 . 20)
where p
X
is the momentum associated with the centre of mass coordinate
X , while p
r
is the momentum of the reduced particle. From the algebra that
leads to equation (8.6a) we know that the first two terms on the right of
the second line of equation (9.19) reduce to the kinetic energy of the centre-
of-mass motion and of the reduced particle. Using equations (9.20) in the
remaining terms on the right of equation (9.19) yields
H
KE
=
p
2
X
2( m
e
+ m
p
)
+
p
2
r
2 μ
+
e
2 μ
( p
r
· A + A · p
r
) , (9 . 21)
where μ is the mass of the reduced particle (eq. 8.6b). It follows that an
external magnetic field adds to the gross-structure Hamiltonian of a hydrogen
atom a perturbing Hamiltonian
H
B
=
e
2 μ
( p
r
· A + A · p
r
) . (9 . 22)

9.1 Time-independent perturbations 187
On the scale of the atom the field is likely to be effectively homogeneous, so
we may take A =
1
2
B × r (page 48). Then H
B
becomes
H
B
=
e
4 m
e
( p · B × r + B × r · p ) , (9 . 23)
where we have approximated μ by m
e
and dropped the subscript on p .
The two terms in the bracket on the right can both be transformed into
B · r × p = ¯ h B · L because (i) these scalar triple products involve only
products of different components of the three vectors, and (ii) [ x
i
,p
j
] = 0
for i negationslash = j . Hence, we do not need to worry about the order of the r and
p operators and can exploit the usual invariance of a scalar triple product
under cyclic interchange of its vectors.