Kiefer C. Quantum gravity

166 QUANTUM GEOMETRODYNAMICS
naturally available in quantum gravity). Inserting the ansatz (5.124) into the
Schrödinger equation and projecting on ψ m (q, Q) yields
∣ ∣
∑ ∣∣∣
〈ψ m − 2 ∂ 2 〉 ∣∣∣
2M ∂Q 2 ψ n χ n + V (Q)χ m + ∑ 〈ψ m |h|ψ n 〉χ n = Eχ m , (5.125)
n
n
where the Q-derivative acts on everything to the right. We now introduce the
‘Born–Oppenheimer potentials’
ɛ mn (Q) ≡〈ψ m |h|ψ n 〉 , (5.126)
which for eigenstates of the ‘light’ variable, h|ψ n 〉 = ɛ n |ψ n 〉, just read: ɛ mn (Q) =
ɛ n (Q)δ mn . In molecular physics, this is often the case of interest. We shall, however,
keep the formalism more general. We also introduce the ‘connection’
〈 ∣ 〉 ∣∣∣ ∂ψ n
A mn (Q) ≡ i ψ m
∂Q
(5.127)
and a corresponding momentum
P mn ≡ i
(
δ mn
∂
∂Q − i )
A mn
. (5.128)
Making use of (5.126) and (5.128), one can write (5.125) in the form 18
∑
n
( )
P
2
mn
2M + ɛ mn(Q) χ n (Q)+V (Q)χ m (Q) =Eχ m (Q) . (5.129)
The modification in the momentum P mn and the ‘Born–Oppenheimer potential’
ɛ mn express the ‘back reaction’ from the ‘light’ part onto the ‘heavy part’. Inserting
the ansatz (5.124) into the Schrödinger equation without projection on
ψ m ,onegets
∑
χ n (Q)
n
[
h(q, Q) −
(E − V (Q)+ 2 ∂ 2 )
χ n
2Mχ n ∂Q 2
− 2
2M
∂ 2
∂Q 2 −
2 ∂χ n
Mχ n ∂Q
∂
∂Q
]
ψ n (q, Q) =0. (5.130)
We emphasize that (5.129) and (5.130) are still exact equations, describing the
coupling between the ‘heavy’ and the ‘light’ part.
18 Here, P 2 mn ≡ P k P mkP kn .