QuantumPhysics.dvi

13.5 Sinusoidal perturbation

Another very important special case, analogous to the problem of spin magnetic resonance, is when
the time dependence of the perturbing potential is sinusoidal,

V(t) =Veiωt+V†e−iωt (13.51)

whereVis a time independent operator. We again use the perturbative formula for the coefficients
cn, given in (13.34), and assume that at timet= 0, the system has been prepared in one of the
eigenstates|i〉ofH 0. We obtain,

cn(t) = δn,i−

i

̄h

∫t

0

dt′

(

Veiωt

′

+V†e−iωt

′

)

eiωnit

′

= δn,i+

1 −ei(ωni+ω)t

̄h(ωni+ω)

Vni+

1 −ei(ωni−ω)t

̄h(ωni−ω)

Vni† (13.52)

We see that this formula is very similar to the switched on interaction, except thatωni→ωni±ω.
Thus, the transition rate is supported not atωni= 0 as in the case of the switched on interaction,
but rather atωni±ω= 0. Thus the rates are given by

ωi→E=

d^2 P(E,t)

dE dt

=

2 π

̄h

ρ(E)|VE,Ei|^2 δ(E−Ei± ̄hω) (13.53)

The two cases correspond to absorption or emission a quantumwith energy ̄h.