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3.1.2 Effects of a Time Dependent Potential
To find the behavior of a system with a time dependent potential, Ĥ is broken
up into a constant and a time dependent part:
Ĥ = Ĥ0 + ˆV (t) (3.11)
The eigenstates are then found for ˆV (t) = 0 and used as the basis for the evolving
state ψ (r, t) by allowing the superposition weights a n to vary with time. The
Schrödinger equation gives the evolution of the weights:
i ∂ t
( ∑
m
a m (t) e −iEmt/ | m 〉
)
= Ĥ ∑ m
a m (t) e −iEmt/ | m 〉 (3.12)
Projecting both sides of the equation onto eigenstate | n 〉 following the convention
established in Equation 3.10 gives:
〈 n | i ∂ t
( ∑
m
a m (t) e −iEmt/ | m 〉
)
=
) ∑
〈 n |
(Ĥ0 + V (t)
m
a m (t) e −iEmt/ | m 〉
i ∂ t
( ∑
m
)
a m (t) e −iEmt/ 〈 n | m 〉 = ∑ a m (t) e −iEmt/ 〈 n | Ĥ0 + V (t) | m 〉
m
= ∑ (
)
a m (t) e −iE mt/
〈 n | Ĥ0 | m 〉 + 〈 n | V (t) | m 〉
m
= ∑ m
a m (t) e −iE mt/ (E m 〈 n | m 〉 + 〈 n | V (t) | m 〉) (3.13)
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