1549380323-Statistical Mechanics Theory and Molecular Simulation

Time-dependent systems 533

E

B

k

λ

Fig. 14.1Sketch of an electromagnetic wave described by eqn. (14.1.7).

and massesm 1 ,...,mN, including the external electromagnetic field, is specified in
terms of the vector and scalar potentials and is given by

Hˆ(t) =

∑N

i=1

1

2 mi

(

pˆi−

qi

c

A(ˆri,t)

) 2

+

∑N

i=1

qiφ(ˆri,t) +U(ˆr 1 ,...,ˆrN). (14.1.8)

When the square in the kinetic energy is expanded out, the Hamiltonian takes the
general form
Hˆ(t) =Hˆ 0 +Hˆ 1 (t), (14.1.9)

whereHˆ 0 is the pure system Hamiltonian in the absence of the field

Hˆ 0 =

∑N

i=1

pˆ^2 i

2 mi

+U(ˆr 1 ,...,ˆrN), (14.1.10)

andHˆ 1 (t) involves the coupling to the field

Hˆ 1 (t) =−

∑N

i=1

qi

2 mic

[pˆi·A(ˆri,t) +A(ˆri,t)·pˆi]

+

∑N

i=1

qi^2

2 mic^2

A^2 (ˆri,t) +

∑N

i=1

qiφ(ˆri,t). (14.1.11)

For a Hamiltonian of the form given in eqn. (14.1.9), the eigenstates ofHˆ 0 , which
satisfy
Hˆ 0 |Ek〉=Ek|Ek〉, (14.1.12)

are no longer eigenstates ofHˆ(t), which means that they are not stationary states.
Thus, the effect ofHˆ 1 (t) is to induce transitions among the eigenstates ofHˆ 0 as
functions of time. If the system absorbs energy from the field, it can be excited from
an initial state|Ei〉with energyEito a final state|Ef〉with energyEfas depicted in
Fig. 14.2. When the system returns to its initial state, the energy emitted is detected,
providing information about the eigenvalue spectrum ofHˆ 0. This is one facet of the
experimental technique known asspectroscopy.