534 Quantum time-dependent statistical mechanics
|E >i
|E >f
Fig. 14.2 A transition from an initial eigenstate ofH 0 with energyEito a final state with
energyEfdue to the external field coupling.
The remainder of this chapter will be devoted to analyzing the process in Fig. 14.2
both as an isolated event and in an ensemble of quantum systems. Wewill examine
the behavior of quantum systems subject to time-dependent perturbations of both
a general nature and specific to an external electromagnetic field-coupling, thereby
providing an introduction to quantum time correlation functions andthe field of linear
spectroscopy. Finally, we will discuss numerical approaches for approximating quantum
correlation functions. We begin with a discussion of the time-dependent Schr ̈odinger
equation whenHˆ 1 (t) is a weak perturbation.
14.2 Time-dependent perturbation theory in quantum mechanics
14.2.1 The interaction picture
For a system described by a HamiltonianHˆ(t) =Hˆ 0 +Hˆ 1 (t), the unperturbed Hamil-
tonianHˆ 0 is taken to describe a physical system of interest, such as a gas, liquid, solid,
or solution.Hˆ 1 (t) represents an arbitrary time-dependent perturbation that induces
transitions between the eigenstates ofHˆ 0.
The state vector of the system|Ψ(t)〉evolves in time from an initial state vector
|Ψ(t 0 )〉according to the time-dependent Schr ̈odinger equation
Hˆ(t)|Ψ(t)〉=(
Hˆ 0 +Hˆ 1 (t))
|Ψ(t)〉=i ̄h∂
∂t|Ψ(t)〉. (14.2.1)Although it might seem that obtaining an appropriate propagator from eqn. (14.2.1)
is straightforward, the presence of operators on the left side ofthe equation, together
with the fact that [Hˆ 0 ,Hˆ 1 (t)] 6 = 0, in general, renders this task nontrivial. However, if
we viewHˆ 1 (t) as a weak perturbation, then we can develop a perturbative approach
to the solution of eqn. (14.2.1). We begin by noting that eqn. (14.2.1)can be cast in
a form more amenable to a perturbative treatment by transforming the state vector
from|Ψ(t)〉to|Φ(t)〉via
|Ψ(t)〉= e−i
Hˆ 0 (t−t 0 )/ ̄h
|Φ(t)〉. (14.2.2)In Section 9.2.6, we introduced the concept of a picture in quantum mechanics and
discussed the difference between the Schr ̈odinger and Heisenberg pictures. The state
vector|Φ(t)〉in eqn. (14.2.2) represents yet another quantum mechanical picture called
theinteraction picture. The interaction picture can be considered as “intermediate”