QuantumPhysics.dvi

is a unitary matrix. The statement is readily checked by computingU† =e−iA

†

=e−iA.

Conversely, any unitary matrixUmay be written as the exponential of a Hermitiam matrix

A, as in (3.49), but the matrixAis not unique, as various shifts by 2πinAwill produce

the sameU. The easiest way to prove this statement is to decompose U into orthogonal

projection operators as in (3.46); it is then manifest that the matrixAis then given by (3.45)

withai=φimod 2π.

3.5 Self-adjoint operators in infinite-dimensional Hilbert spaces

The statements of Theorem 1 have been written in such a way that they essentially also hold

for infinite dimensional Hilbert spaces, though the proofs will now differ. In particular, self-

adjoint operators will still have real eigenvalues. In statement (iii), the sum over projection

operators weighed by eigenvalues need not be a discrete sum,even in a separable Hilbert

space, but can have discrete and continuous parts, corresponding to the discrete and contin-

uous parts of the spectrum of an operator. Actually, this situation should be familiar from

analyzing the spectra of various Hamiltonians in quantum mechanics,such as the Hydrogen

atom.

The continuous spectrum arises because all the eigenvectors of an operator need not be

normalizable. The simplest case would be the Hamiltonian of the free particle onR, and its

associated eigenvalue problem,

H=−

̄h^2

2 m

d^2

dx^2

HψE(x) =EψE(x) (3.50)

whose eigenfunctions areeikxwithE= ̄h^2 k^2 / 2 m. AlthoughL^2 (R) is a perfectly separable

Hilbert space, and the operatorHis perfectly self-adjoint, the eigenvectorsψE(x) are not

normalizable onR. This situation is characteristic of the continuous part of a spectrum.

The relation between self-adjoint and unitary operators remains valid for infinite-dimensional

Hilbert spaces.