each term “ are unconditionally stable.

The following relation is very useful as starting point to derive the behav-

ior of reduced systems, which can be viewed as projections of the complete

388 Vectors, operators and vector spaces

system onto a reduced space (Chapter 8). For any pair of time-independent,

ˆ ˆ

non-commuting operators A and B we can write

t

ˆˆ ˆ ˆ ˆ ˆ

ˆ

(A+B)t At

eA(t’„ ) Be(A+B)„ d„.

e =e + (14.45)

0

Proof First write

ˆ ˆ ˆ

ˆ

e(A+B)t = eAt Q(t), (14.46)

so that

Q(t) = e’At e(A+B)t .

ˆ ˆˆ

ˆ

By di¬erentiating (14.46), using the di¬erentiation rule (14.33), we ¬nd

ˆ

ˆ dQ

ˆ ˆ ˆˆ ˆˆˆ

(A + B)e(A+B)t = AeAt Q(t) + eAt ,

dt

and using the equality

ˆ ˆ ˆˆ ˆˆˆ ˆ ˆˆ

(A + B)e(A+B)t = AeAt Q(t) + Be(A+B)t ,

we see that

ˆ

dQ

= e’At Be(A+B)t .

ˆˆ ˆ ˆ

dt

ˆ

Hence, by integration, and noting that Q(0) = 1:

t

e’A„ Be(A+B)„ d„.

ˆˆ ˆˆ

ˆ

Q(t) = 1 +

0

ˆ

Inserting this Q in (14.46) yields the desired expression.

There are two routes to practical computation of the matrix exp(A). The

¬rst is to diagonalize A : Q’1 AQ = Λ and construct

eA = Q diag (e»1 , e»2 , . . .) Q’1 . (14.47)

For large matrices diagonalization may not be feasible. Then in favorable

cases the matrix may be split up into a sum of block-diagonal matrices,

each of which is easy to diagonalize, and the Trotter expansion applied to

the exponential of the sum of matrices. It may also prove possible to split

the operator into a diagonal part and a part that is diagonal in reciprocal

space, and therefore solvable by Fourier transformation, again applying the

Trotter expansion.

14.6 Exponential operators and matrices 389

The second method4 is an application of the Caley“Hamilton relation,

which states that every n — n matrix satis¬es its characteristic equation

An + a1 An’1 + a2 An’2 + . . . + an’1 A + an 1 = 0. (14.48)

Here a1 , . . . an are the coe¬cients of the characteristic or eigenvalue equation

det(A ’ »1) = 0, which is a n-th degree polynomial in »:

»n + a1 »n’1 + a2 »n’2 + . . . + an’1 » + an = 0. (14.49)

Equation (14.49) is valid for each eigenvalue, and therefore for the diagonal

matrix Λ; (14.48) then follows by applying the similarity transformation

QΛQ’1 .

According to the Caley“Hamilton relation, An can be expressed as a linear

combination of Ak , k = 0, . . . , n ’ 1, and so can any Am , m ≥ n. Therefore,

the in¬nite sum in (14.25) can be replaced by a sum over powers of A up to

n ’ 1:

eA = μ0 1 + μ1 A + · · · + μn’1 An’1 . (14.50)

The coe¬cients μi can be found by solving the system of equations

μ0 + μ1 »k + μ2 »2 + · · · + μn’1 »n’1 = exp(»k ), k = 1, . . . , n (14.51)

k k

(which follows immediately from (14.50) by transforming exp(A) to diagonal

form). In the case of degenerate eigenvalues, (14.51) are dependent and the

super¬‚uous equations must be replaced by derivatives:

μ1 + 2μ2 »k + · · · + (n ’ 1)μn’1 »n’2 = exp(»k ) (14.52)

k

for a doubly degenerate eigenvalue, and higher derivatives for more than

doubly degenerate eigenvalues.

14.6.1 Example of a degenerate case

Find the exponential matrix for

⎛ ⎞

010

A = ⎝ 1 0 0 ⎠.

001

According to the Caley“Hamilton relation, the exponential matrix can be

expressed as

exp(A) = μ0 1 + μ1 A + μ2 A2 .

4 See, e.g., Hiller (1983) for the application of this method in system theory.

390 Vectors, operators and vector spaces

Note that the eigenvalues are +1, +1, ’1 and that A2 = I. The equations

for μ are (because of the twofold degeneracy of »1 the second line is the

derivative of the ¬rst)

μ0 + μ1 »1 + μ2 »2 = exp(»1 ),

1

μ1 + 2μ2 »1 = exp(»1 ),

μ0 + μ1 »3 + μ2 »2 = exp(»3 ).

3

Solving for μ we ¬nd

1 1

μ0 = μ2 = (e + ),

4 e

1 1

μ1 = (e ’ ),

2 e

which yields the exponential matrix

⎛ ⎞

e + 1/e e ’ 1/e 0

1

eA = ⎝ e ’ 1/e e + 1/e 0 ⎠ .

2

0 0 2e

The reader is invited to check this solution with the ¬rst method.

14.7 Equations of motion

In this section we consider solutions of the time-dependent Schr¨dinger equa-

o

tion, both in terms of the wave function and its vector representations, and

in terms of the expectation values of observables.

14.7.1 Equations of motion for the wave function and its

representation

The time-dependent Schr¨dinger equation

o

‚ iˆ

Ψ(r, t) = ’ HΨ(r, t) (14.53)