36
B. Sutcliffe and R.G. Woolley
The only non-zero commutators of the operators are
[ˆ x, ˆ
p] = i,
[ ˆ
X, ˆ
P] = i.
Following the conventional discussion of electron-nuclear separation outlined in
Sects. 1.3.1, 1.3.2, define
ˆ
H o = ˆ
p
2
+ ˆ
x
2
+ a ˆ
x ˆ
X + ˆ
X
2
(1.56)
ˆ
H 1 = ˆ
P
2
(1.57)
so that
ˆ
H = ˆ
H o + κ
4 ˆ
H 1
(1.58)
with Schrödinger equation
( ˆ
H − E)Ψ = 0.
(1.59)
We note that
[ ˆ
H o , ˆ
H 1 ] ] = 0
(1.60)
so the two parts cannot be simultaneously diagonalized. A principal axis transformation of the whole expression ˆ
H brings it to separable form, but we do not need to
pursue explicitly the full solution here.
On the other hand
[ ˆ
H o , ˆ
X] = 0
(1.61)
so these two operators may be simultaneously diagonalized, and consider ˆ
H o at a
definite eigenvalue of ˆ
X, say X
ˆ
K o = ˆ
p
2
+ ˆ
x
2
+ a ˆ
xX + X
2 .
(1.62)
This is the Hamiltonian (in the variables ˆ
x, ˆ
p) of a displaced oscillator in which X is
a (c-number) parameter, with Schrödinger equation in position representation
ˆ
K o ϕ(x, X) n = E n (X)ϕ(x, X) n
(1.63)
ˆ
K o is the analogue in this model of the ‘clamped-nuclei’ electronic Hamiltonian.
The solution is immediate; we make a unitary transformation by introducing a
displaced coordinate involving X
ˆ
x
= ˆ
x −
1
2
aX,
ˆ
p
= ˆ
p
(1.64)
ˆ
U = e
iaX ˆ
p/2 ,
ˆ
K
o = ˆ
U
−1 ˆ
K 0 ˆ
U
(1.65)
so that the transformed ˆ
K o in the new variables is
ˆ
K
o = ˆ
p
2
+ ˆ
x
2
+
1 −
a 2
4
X
2