φ— φm dr

ˆ

D = 0. (5.39)

n

Hence it follows that

(Dφ— )φm dr + φ— Dφm dr = 0

ˆn ˆ (5.40)

n

or, using the fact that D— = ’D,

—

φ— Dφn dr φ— Dφm dr = 0,

ˆ ˆ

’ + (5.41)

m n

meaning that

D† = D. (5.42)

Since D is imaginary, Dnm = ’Dmn and Dnn = 0.

Proof We prove (5.35). Realizing that ρ = cc† and hence ρ = cc† + cc† , we

™™ ™

¬nd that

i i

ρ = ’ (H + R · D)ρ + ρ(H† + R · D† )

™ ™

™

. Using the fact that both H and D are Hermitian, (5.35) follows.

5.2 Quantum dynamics in a non-stationary potential 123

5.2.4 The two-level system

It is instructive to consider a quantum system with only two levels. The

extension to many levels is quite straightforward. Even if our real system

has multiple levels, the interesting non-adiabatic events that take place under

the in¬‚uence of external perturbations, such as tunneling, switching from

one state to another or relaxation, are active between two states that lie

close together in energy. The total range of real events can generally be

built up from events between two levels. We shall use the density matrix

formalism (see Section 14.8),4 as this leads to concise notation and is very

suitable for extension to ensemble averages.

We start with a description based on a diagonal zero-order Hamiltonian

H0 plus a time-dependent perturbation H1 (t). The perturbation may arise

from interactions with the environment, as externally applied ¬elds or ¬‚uc-

tuating ¬elds from thermal ¬‚uctuations, but may also arise from motions of

the nuclei that provide the potential ¬eld for electronic states, as described

in the previous section. The basis functions are two eigenfunctions of H0 ,

0 0

with energies E1 and E2 . The equation of motion for the density matrix is

(5.29):

i i

[ρ, H0 + H1 (t)].

ρ=

™ [ρ, H] = (5.43)

Since tr ρ = 1, it is convenient to de¬ne a variable:

z = ρ11 ’ ρ22 , (5.44)

instead of ρ11 and ρ22 . The variable z indicates the population di¬erence

between the two states: z = 1 if the system is completely in state 1 and

z = ’1 if it is in state 2. We then have the complex variable ρ12 and the

real variable z obeying the equations:

i

[ρ12 (H22 ’ H11 ) + zH12 ],

ρ12 =

™ (5.45)

2i

[ρ12 H12 ’ ρ— H12 ],

—

z=

™ (5.46)

12

where we have used the Hermitian property of ρ and H. Equations (5.45)

and (5.46) imply that the quantity 4ρ12 ρ— + z 2 is a constant of the motion

12

since the time derivative of that quantity vanishes. Thus, if we de¬ne a real

4 See Berendsen and Mavri (1993) for density-matrix evolution (DME) in a two-level system; the

theory was applied to proton transfer in aqueous hydrogen malonate by Mavri et al. (1993).

The proton transfer case was further extended to multiple states by Mavri and Berendsen (1995)

and summarized by Berendsen and Mavri (1997). Another application is the perturbation of a

quantum oscillator by collisions, as in diatomic liquids (Mavri et al., 1994).

124 Dynamics of mixed quantum/classical systems

three-dimensional vector r with components x, y, z, with:

x = ρ12 + ρ— , (5.47)

12

y = ’i(ρ12 ’ ρ— ), (5.48)

12

then the length of that vector is a constant of the motion. The motion of r

is restricted to the surface of a sphere with unit radius. The time-dependent

perturbation causes this vector to wander over the unit sphere. The equation

of motion for r(t) can now be conveniently expressed in the perturbations

when we write the latter as three real time-dependent angular frequencies:

1 —

def

ωx (t) = (H12 + H12 ), (5.49)

1 —

def

(H12 ’ H12 ),

ωy (t) = (5.50)

i

def 1 —

ωz (t) = (H11 ’ H22 ), (5.51)

yielding

x = yωz ’ zωy ,

™ (5.52)

y = zωx ’ xωz ,

™ (5.53)

z = xωy ’ yωx .

™ (5.54)

These equations can be summarized as one vector equation:

r = r — ω.

™ (5.55)

Equation (5.55) describes a rotating top under the in¬‚uence of a torque.

This equivalence is in fact well-known in the quantum dynamics of a two-spin

system (Ernst et al., 1987), where r represents the magnetization, perturbed

by ¬‚uctuating local magnetic ¬elds. It gives some insight into the relaxation

behavior due to ¬‚uctuating perturbations.

The o¬-diagonal perturbations ωx and ωy rotate the vector r in a verti-

cal plane, causing an oscillatory motion between the two states when the

perturbation is stationary, but a relaxation towards equal populations when

the perturbation is stochastic.5 In other words, o¬-diagonal perturbations

cause transitions between states and thus limit the lifetime of each state. In

the language of spin dynamics, o¬-diagonal stochastic perturbations cause

longitudinal relaxation. Diagonal perturbations, on the other hand, rotate

the vector in a horizontal plane and cause dephasing of the wave functions;

they cause loss of phase coherence or transverse relaxation. We see that the

5 In fact, the system will relax towards a Boltzmann equilibrium distribution due to a balance

with spontaneous emission, which is not included in the present description.

5.2 Quantum dynamics in a non-stationary potential 125

e¬ect of the non-adiabatic coupling vector (previous section) is o¬-diagonal:

it causes transitions between the two states.

In a macroscopic sense we are often interested in the rate of the transition

process from state 1 to state 2 (or vice versa). For example, if the two states

represent a reactant state R and a product state P (say a proton in the

left and right well, respectively, of a double-well potential), the macroscopic

transfer rate from R to P is given by the rate constant k in the “reaction”

k