QuantumPhysics.dvi

for all|φ〉,|ψ〉∈H, while a self-adjoint bounded operator satisfies〈ψ|A|φ〉=〈φ|A|ψ〉∗for all

|φ〉,|ψ〉∈H. A projection operator is always bounded. Unbounded operatorswill, however,

be pervasive in quantum mechanics, and will need to be dealt with.

• An operatorUis aunitary operatorprovided that for all|φ〉,|ψ〉∈H, we have

(U|ψ〉,U|φ〉) = (|ψ〉,|φ〉) (3.36)

for all|φ〉,|ψ〉 ∈ H. Clearly, a unitary operator is a bounded operator withCU = 1, and

is invertible. The inverseU−^1 may be defined by settingU|ψ〉=|u〉, so that (|u〉,U|φ〉) =

(U−^1 |u〉,|φ〉) for all|φ〉,|u〉 ∈ H. Using now the definition of the adjoint ofU, we see that

for a unitary operator, we have

U−^1 =U† U†U=IH (3.37)

Unitary operators will be key ingredients in quantum mechanics because unitary transfor-

mations will preserve transition amplitudes, and represent symmetries.

3.4 Hermitian and unitary operators in finite-dimension

Self-adjoint operators will play a central role in quantum mechanics. We now derive some

of their key properties. In a finite-dimensional Hilbert space, a Hermitian operator is self-

adjoint, and vice versa, and may be represented by a Hermitian matrix.

Theorem 1

(i) The eigenvalues of a self-adjoint operator are real.

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

(iii) A self-adjoint operator may be written as a direct sum of mutuallyorthogonal projection

operators, weighted by the distinct eigenvalues.

Proof

(i) LetAbe a Hermitian matrix with eigenvalueaand associated eigenvector|φ〉6= 0,

A|φ〉=a|φ〉 (3.38)

Taking the†of this equation gives〈φ|A†=a∗〈φ|, and using the fact thatA†=A, simplifies

this equation to〈φ|A=a∗〈φ|. Taking the inner product of this equation with|φ〉and of

the eigenvalue equation with〈φ|, we obtain,

〈φ|A|φ〉=a〈φ|φ〉=a∗〈φ|φ〉 (3.39)

Since|φ〉6= 0, we have〈φ|φ〉6= 0, and hencea∗=a, which proves the first assertion.