Multiply by ψ
0* and integrate:
(12.57)
The operators used in Schrödinger’s equation are called Hermetian operators and have the property:
(12.58)
and
(12.59)
Resulting in the expression for the first-order energy correction:
(12.60)
For the n 1 = 1, n 2 = 1 state this is:
(12.61)
dτ 1 = r 1
2 sin θ 1 dr 1 dθ 1 dϕ 1
(12.62)
dτ 2 = r 2
2 sin θ 2 dr 2 dθ 2 dϕ 2
The term r 12 must be rewritten in spherical coordinates, where the > or
< signs denote the larger or smaller values of r 1 or r 2 :
(12.63)
Experimentally, a value of −79.01 eV is determined for the n 1 = 1 and
n 2 = 1 state of helium. Substitution of these values into E
0 gives −108.8 eV.
The first-order term has a value of +34.0 eV, giving −74.8 eV, and using
up to the third order gives −78.9 eV.
SPIN–ORBITAL COUPLING
The electron spin influences the energies of the electronic states because
associated with the spin is a magnetic dipole moment. Likewise, the orbital
angular momentum, for the states with l greater than 0, will possess
a dipole moment due to the angular momentum. The interaction of
the spin magnetic moment with the magnetic field arising from orbital
1
4
2 1
12
1
1 1
r
l
r
r
Y
Y
m
l
l
l
l
m
l
m
l
l
l
( )
=
+
∑
∑
<
>
+
π
θ ϕ *
( (
)
θ ϕ
2 2
E
Z
a
e
e
Zr a
Zr a
1
2
6
0
6
2
1 2
2
1
1 0
2
/
,
/
=
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
−
−
∫∫
π
0 0
2
0 12
1
2
4
1
e
r
πε
τ τ
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ d d
E
H
1
0
1 0
*
= ∫ ψ ψ τ
d
ψ
ψ τ
ψ ψ τ
0
1 0
1
0
0
1
*
*
E
E
E
∫
∫
=
=
d
d
ψ
ψ τ
ψ
ψ
τ
ψ
ψ
τ
0
0 1
1
0 0
1
0
0
*
(
)
(
)
H
H
E
∫
∫
∫
=
=
=
d
* d
* d
ψ ψ
ψ τ
0
0 1
* E
∫
d
ψ
ψ τ
ψ
ψ τ
ψ
ψ τ
ψ
0
0 1
0
0 1
0
1 0
0
*
*
*
*
∫
∫
∫
−
=−
+
H
E
H
d
d
d
∫ ∫ E
1 0
ψ τ
d
264
PART 2
QUANTUM MECHANICS AND SPECTROSCOPY
9781405124362_4_012.qxd 4/29/08 9:11 Page 264