The adiabatic approximation can be regarded as the first term in an adiabatic series for t...

The adiabatic approximation can be regarded as the first term in an adiabatic series for the coefficients cm(t) in Equation 10.12. Suppose the system starts out in the nth state; in the adiabatic approximation, it remains in the nth state, picking up only a time-dependent geometric phase factor (Equation 10.21):

(a) Substitute this into the right side of Equation 10.16 to obtain the "first correction" to adiabaticity:

  (1)

This enables us to calculate transition probabilities in the nearly adiabatic regime. To develop the "second correction," we would insert Equation 1 on the right side of Equation 10.16, and so on.

(b) As an example, apply Equation 1 to the driven oscillator (Problem 10.9). Show that (in the near-adiabatic approximation) transitions are possible only to the two immediately adjacent levels, for which

(The transition probabilities are the absolute squares of these, of course.)