14 Quantum time-dependent statistical mechanics
14.1 Time-dependent systems in quantum mechanics
In this chapter, we will explore how the physical properties of a quantum system
described by a HamiltonianHˆ 0 can be probed by applying a small external time-
dependent perturbation. As we showed in Section 10.4, if the energy levels and energy
eigenfunctions ofHˆ 0 are known, then using the rules of quantum statistical mechan-
ics, all of the thermodynamic and equilibrium properties can be computed from eqns.
(10.4.4) and (10.4.5). This fact emphasizes the importance of techniques that can
provide information about the eigenvalue structure ofHˆ 0. The essence of the experi-
mental technique known asspectroscopyis to employ an external electromagnetic field
to induce transitions between the different eigenstates ofHˆ 0 ; from the frequencies of
photons needed to induce these transitions, information about different parts of the
eigenvalue spectrum can be gleaned for different electromagnetic frequency ranges (in-
frared, visible, ultraviolet, etc.) As we will see in the discussion to follow, the rules of
quantum statistical mechanics help us both to interpret such experiments and to con-
struct approximate computational procedures for calculating quantum spectral and
transport properties of a condensed-phase system.
An external electromagnetic field is described by its electric and magnetic field
componentsE(r,t) andB(r,t), respectively. The term “electromagnetic” arises from
the notion that the electric and magnetic fields can beunifiedinto a single classical
field theory, an idea that was demonstrated by James Clerk Maxwell(1831–1879)
between the years 1864 and 1873. Maxwell’s theory of electromagnetism is embodied
in a set of field equations known asMaxwell’s equations, which describe how the
electric and magnetic fields are coupled in such a unified theory. In the absence of
external sources, i.e., for free fields, Maxwell’s equations specify how the divergence
and curl of the electric and magnetic fields are related to their time rates of change
fields.^1 Once the divergence and curl of a vector field are specified, then these, together
with a knowledge of the time derivatives of the fields, are sufficient todetermine the
(^1) The “divergence” of a vector field is a measure of the extent towhich the field expands or contracts
at a point while the “curl” tells us the degree to which the field circulates around a point; these modes
of behavior are about all a vector field can do. The importanceof the divergence and curl of a vector
field can be understood using a theorem known as Helmholtz’s theorem, which states that ifC(r) and
D(r) are, respectively, smooth scalar and vector functions that decay faster than 1/|r|^2 as|r|→∞,
then there exists a unique vector fieldFsuch that∇·F=C(r) and∇×F=D(r). Consequently,
the divergence and curl ofFare sufficient to determineF(r), and conversely,Fcan be decomposed
in terms of its divergence and curl.