130_notes.dvi

Now we want to calculate transition rates. To first order, all theck(t) are small compared to
ci(t)≈= 1, so the sum can be neglected.

∂c(1)n (t)

∂t

=

1

i ̄h

Vni(t)eiωnit

c(1)n(t) =

1

i ̄h

∫t

0

eiωnit

′

Vni(t′)dt′

This is theequation to use to compute transition probabilities for a general time de-
pendent perturbation. We will also use it as a basis to compute transition rates for the specific
problem of harmonic potentials. Again we are assumingtis small enough thatcihas not changed
much. This is not a limitation. We can deal with the decrease of the population of the initial state
later.

Note that, if there is a large energy difference between the initial and final states, a slowly varying
perturbation can average to zero. We will find that the perturbation will need frequency components
compatible withωnito cause transitions.

If the first order term is zero or higher accuracy is required, the second order term can be computed.
In second order, a transition can be made to an intermediate stateφk, then a transition toφn. We

just put the first orderc(1)k (t) into the sum.

∂cn(t)

∂t

=

1

i ̄h



Vni(t)eiωnit+

∑

k 6 =i

Vnk(t)c(1)k (t)eiωnkt





∂cn(t)

∂t

=

1

i ̄h



Vni(t)eiωnit+

∑

k 6 =i

Vnk(t)

1

i ̄h

eiωnkt

∫t

0

eiωkit

′

Vki(t′)dt′





c(2)n (t) =

− 1

̄h^2

∑

k 6 =i

∫t

0

dt′′Vnk(t′′)eiωnkt

′′

∫t′′

0

dt′eiωkit

′

Vki(t′)

c(2)n(t) =

− 1

̄h^2

∑

k 6 =i

∫t

0

dt′′Vnk(t′′)eiωnkt

′′

∫t′′

0

dt′eiωkit

′

Vki(t′)

• See Example 28.3.1:Transitions of a 1D harmonic oscillator in a transient E field.*

28.2 Sinusoidal Perturbations

An important case is a pure sinusoidal oscillating (harmonic) perturbation. We canmake up
any time dependence from a linear combinationof sine and cosine waves. We define our