reads as vector equation in Hilbert space on a stationary orthonormal basis

set:

i

c = ’ Hc.

™ (14.54)

In these equations the Hamiltonian operator or matrix may itself be a funct-

ion of time, e.g., it could contain time-dependent external potentials.

14.7 Equations of motion 391

These equations can be formally solved as

t

i ˆ

Ψ(r, t) = exp ’ H(t ) dt Ψ(r, 0), (14.55)

0

t

i

c(t) = exp ’ H(t ) dt (14.56)

c(0),

0

which reduce in the case that the Hamiltonian does not depend explicitly

on time to

iˆ

Ψ(r, t) = exp ’ Ht Ψ(r, 0), (14.57)

i

c(t) = exp ’ Ht c(0). (14.58)

These exponential operators are propagators of the wave function in time,

to be written as

ˆ

Ψ(r, t) = U (t)Ψ(r, 0), (14.59)

c(t) = U(t)c(0). (14.60)

The propagators are unitary because they must keep the wave function

normalized at all times: c† c(t) = c(0)† U† Uc(0) = 1 for all times only if

U† U = 1. We must agree on the interpretation of the role of the time in the

exponent: the exponential operator is time-ordered in the sense that changes

at later times act subsequent to changes at earlier times. This means that,

for t = t1 + t2 , where t1 is ¬rst, followed by t2 , the operator factorizes as

iˆ iˆ iˆ

exp ’ Ht = exp ’ Ht2 exp ’ Ht1 . (14.61)

Time derivatives must be interpreted as

i dt ˆ

ˆ ˆ

U (t + dt) = ’ H(t) U (t), (14.62)

ˆ ˆ

even when U and H do not commute.

14.7.2 Equation of motion for observables

The equation of motion for the expectation A of an observable with oper-

ˆ

ator A,

A = Ψ|A|Ψ , (14.63)

392 Vectors, operators and vector spaces

is given by

d i ˆˆ ‚A

A = [H, A] + . (14.64)

dt ‚t

Proof

d i ‚A i

Ψ— AΨ dt = (H — Ψ— )AΨ d„ + Ψ— AHΨ d„.

ˆ ˆ ˆ ˆˆ

’

dt ‚t

ˆ

Because H is hermitian:

(H — Ψ— )AΨ d„ = Ψ— H AΨ d„,

ˆ ˆ ˆˆ

and (14.64) follows.

Instead of solving the time-dependence for several observables separately

by (14.64), it is more convenient to solve for c(t) and derive the observables

from c. When ensemble averages are required, the method of choice is to

use the density matrix, which we shall now introduce.

14.8 The density matrix

Let c be the coe¬cient vector of the wave function Ψ(r, t). on a given

orthonormal basis set. We de¬ne the density matrix ρ by

ρnm = cn c— , (14.65)

m

or, equivalently,

ρ = cc† . (14.66)

The expectation value of an observable A is given by

c— φ— A

ˆ

A= cn φn d„ = ρnm Amn = (ρA)nn (14.67)

mm

m n n,m n

so that we obtain the simple equation5

A = tr ρA. (14.68)

So, if we have solved c(t) then we know ρ(t) and hence A (t).

The evolution of the density matrix in time can also be solved directly

from its equation of motion, called the Liouville“von Neumann equation:

i

ρ=

™ [ρ, H]. (14.69)

5 The trace of a matrix is the sum of its diagonal elements.

14.8 The density matrix 393

Proof By taking the time derivative of (14.66) and applying (14.54), we see

that

i i

ρ = cc† + cc† = ’ Hcc† + c(Hc)† .

™™ ™

Now (Hc)† = c† H† = c† H because H is hermitian, so that

i i i

ρ = ’ Hρ + ρH = [ρ, H].

™

This equation also has a formal solution:

i i

ρ(t) = exp ’ Ht ρ(0) exp + Ht , (14.70)

where, if H is time-dependent, H(t) in the exponent is to be replaced by

t

0 H(t ) dt .

Proof We prove that the time derivative of (14.70) is the equation of motion

(14.69):

i i i

ρ = ’ H exp ’ Ht ρ(0) exp + Ht

™ (14.71)

i i i

= + exp ’ Ht ρ(0)H exp + Ht (14.72)

i

= [ρ, H]. (14.73)

Here we have used the fact that H and exp ’ i Ht commute.