Q— Qjm φi |φj = (Q† )ni Qim ,

φn |φm = in

ij i

it follows that

(Q† )ni Qim = δnm ,

i

or

Q† Q = 1,

which implies that Q is unitary.

The representations c and c of a vector are related as

c = Qc , (14.20)

c = Q’1 c, (14.21)

c = Q† c, (14.22)

where (14.22) is only valid for unitary transformations.

ˆ

Let A be a hermitian operator with eigenvalues »1 , »2 , . . . and with or-

thonormal eigenvectors c1 , c2 , . . .. Then, if we construct a matrix U with

columns formed by the eigenvectors, it follows that

AU = ΛU, (14.23)

where Λ is the diagonal matrix of eigenvalues. Since all columns of U are

orthonormal, U is a unitary matrix, and thus

U† AU = Λ. (14.24)

In other words: U is exactly the coordinate transformation that diagonalizes

A.

14.6 Exponential operators and matrices

ˆ

ˆ

We shall often encounter operators of the form exp(A) = eA , e.g., as formal

solutions of ¬rst-order di¬erential equations. The de¬nition is

∞

1 ˆk

ˆ

A

e= A. (14.25)

k!

k=0

Exponential matrices are similarly de¬ned.

From the de¬nition it follows that

ˆˆ ˆˆ

AeA = eA A, (14.26)

386 Vectors, operators and vector spaces

and

ˆ ˆ ˆ

eA (f + g) = eA f + eA g. (14.27)

ˆ

The matrix representation of the operator exp(A) is exp(A):

∞ ∞

1 1

(Ak )nm = (eA )nm .

ˆ ˆ

n|e |m = n|Ak |m =

A

(14.28)

k! k!

k=0 k=0

The matrix element (exp A)nm is in general not equal to exp(Anm ), unless

A is a diagonal matrix Λ = diag(»1 , . . .):

(eA )nm = e»n δnm . (14.29)

ˆ

From the de¬nition follows that exp(A) or exp(A) transforms just like any

other operator under a unitary transformation:

∞ ∞

1 † ˆk 1 †ˆ

†A

(U† AU)k = eU AU .

ˆ ˆ

U e U= U A U= (14.30)

k! k!

k=0 k=0

This transformation property is true not only for unitary transformations,

but for any similarity transformation Q’1 AQ.

Noting that the trace of a matrix is invariant for a similarity transforma-

tion, it follows that

det(eA ) = Πn e»n = exp »n = exp( tr A). (14.31)

n

Some other useful properties of exponential matrices or operators are

’1

eA = e’A , (14.32)

and

d At

e = AeAt = eAt A (t is a scalar variable). (14.33)

dt

Generally, exp(A + B) = exp(A) exp(B), unless A and B commute. If A

and B are small (proportional to a smallness parameter µ), the ¬rst error

term is of order µ2 and proportional to the commutator [A, B]:

eµ(A+B) = eµA eµB ’ 1 µ2 [A, B] + O(µ3 ), (14.34)

2

= eµB eµA + 1 µ2 [A, B] + O(µ3 ). (14.35)

2

We can approximate exp(A + B) in a series of approximations, called the

Lie“Trotter“Suzuki expansion. These approximations are quite useful for

the design of stable algorithms to solve the evolution in time of quantum or

classical systems (see de Raedt, 1987, 1996). The basic equation, named the

14.6 Exponential operators and matrices 387

Trotter formula after Trotter (1959), but based on earlier ideas of Lie (see

Lie and Engel, 1888), is

m

eA/m eB/m

e(A+B) = lim . (14.36)

m’∞

Let us try to solve the time propagator

U („ ) = e’i(A+B)„ , (14.37)

where A and B are real matrices or operators. The ¬rst-order solution is

obviously

U1 („ ) = e’iA„ e’iB„ + O(„ 2 ). (14.38)

Since

† ’1

U1 („ ) = eiB„ eiA„ = U1 („ ), (14.39)

the propagator U1 („ ) is unitary. Suzuki (1991) gives a recursive recipe to

derive higher-order products for the exponential operator. Symmetric prod-

ucts are special cases, leading to algorithms with even-order precision. For

second order precision Suzuki obtains

U2 („ ) = e’iB„ /2 e’iA„ e’iB„ /2 + O(„ 3 ). (14.40)

Higher-order precision is obtained by the recursion equation (for symmetric

products; m ≥ 2)

U2m („ ) = [U2m’2 (pm „ )]2 U2m’2 ((1 ’ 4pm )„ )[U2m’2 (pm „ )]2 + O(„ 2m+1 ),

(14.41)

with

1

pm = . (14.42)

4 ’ 41/(2m’1)

For fourth-order precision this works out to

U4 („ ) = U2 (p„ )U2 (p„ )U2 ((1 ’ 4p)„ )U2 (p„ )U2 (p„ ) + O(„ 5 ), (14.43)

with

1

p= = 0.4145. (14.44)

4 ’ 41/3

All of the product operators are unitary, which means that algorithms based