13 Time-dependent Processes
So far, we have studied problems in which the Hamiltonian is time-independent. This is suitable for
any isolated system, in which the total energy is conserved. In many physical problems, however,
it is inconvenient to treat the entire closed system, and views part of the system, instead, as an
external source, which may be time dependent. The Hamiltonian for such systems then often takes
the form of a sum of a time-independent unperturbed HamiltonianH 0 , and a time time-dependent
perturbationV(t),
H=H 0 +V(t) (13.1)and it will be assumed again that the spectrum and eigenstates ofH 0 is exactly known. Very few
such time dependent problems can be solved exactly, and often one will have to resort to carrying
out perturbation theory in the strength of the potentialV(t).
13.1 Magnetic spin resonance and driven two-state systems
The problem of magnetic spin resonance is exactly solvable and is also of substantial practical
significance. Actually, the problem is more general than magnetic spin would suggest, and the
same methods may be applied for any two-state system driven by an external periodic oscillation.
We deal with this problem first and then develop perturbationtheory. The Hamiltonian is of the
form
H 0 = ω 0 Sz
V(t) = ω 1 cos(ωt)Sx+ω 1 sin(ωt)Sy (13.2)Here,ωis the driving frequency. As a magnetic problem, this Hamiltonian arises from a particle
with spinSand magnetic momentM=γSin the presence of a constant magnetic fieldB 0 along
thez-axis, and an oscillating magnetic fieldB 1 (t) =B 1 cos(ωt)nx+B 1 sin(ωt)nyin thexy-plane.
The frequencies are then given by
ω 0 = γB 0
ω 1 = γB 1 (13.3)But the problem may be considered generally, without referring to the magnetic system.
Denote an orthonormal basis of the two state system by the vectors|+|>and|−〉. In this
basis, the Hamiltonian is given by
H=
̄h
2(
ω 0 ω 1 e−iωt
ω 1 e+iωt −ω 0)
(13.4)A general state|ψ(t)〉may be decomposed onto|+〉and|−〉,
|ψ(t)〉=a+(t)|+〉+a−(t)|−〉 (13.5)