QuantumPhysics.dvi

3.4.1 Unitary operators

Unitary operators are very closely related to self-adjoint operators. In fact, one proves the

following theorem by completely analogous methods,

Theorem 2

(i) The eigenvalues of a unitary operator are pure phases.

(ii) Eigenvectors corresponding to two distinct eigenvalues are orthogonal to one another.

(iii) A unitary matrix may be written as a direct sum of mutually orthogonal projection

operators, weighted by the distinct eigenvalues.

Thus, a unitary operatorU admits a decomposition very analogous to a Hermitian op-

erator, but only the nature of the eigenvalues differs,

U=

∑

i

eiθiPi=









eiθ^1 I 1 0 0 ··· 0

0 eiθ^2 I 2 0 ··· 0

0 0 eiθ^3 I 3 ··· 0

···

0 0 0 ··· eiθmIm







 (3.46)

where the angles θi are real and distinct mod 2π, and Ii is the identity matrix in the

eigenspace Ei, representing the orthogonal projection operatorPi onto the eigenspace of

Uwith eigenvalueeiθi.

3.4.2 The exponential map

For anyN×N matrix, and any analytic function functionf(x), we definedf(A) by the

Taylor expansion off,

f(A) =

∑∞

n=0

1

n!

f(n)(0)An (3.47)

wheref(n)(x) is then-th derivative off(x). Using this definition, it is especially easy to

find the functionfevaluated on a Hermitian matrixA, with the decomposition in terms of

orthogonal projectors given in (3.45), and we have

f(A) =

∑

i

f(ai)Pi (3.48)

The relation between Hermitian and unitary operators may be made explicit by using the

exponential function. For any Hermitian matrixA, the matrixU, uniquely defined by

U=eiA (3.49)