4.2 Lecture 17: Outer Product, Adjoint oper- ators

CHAPTER 4. QUANTUM SYSTEMS 70

4.2 Lecture 17: Outer Product, Adjoint oper-
ators

4.2.1 Outer product

Another convenient representation of linear operators is an outer product
representation defined by,

|ψ ϕ| (|ϕ ) ≡ |ψ ϕ|ϕ = ϕ|ϕ |ψ . (4.15)

For example an identity operator can be represented as a sum of outer prod-
uct of an orthonormal basis, i.e.

n (4.16)

Iˆ = |i i|

i=1

This is known as a completeness relation which can be used to obtain an
outer product representations of operators,

nn

Aˆ = IˆAˆIˆ = i|Aˆ|j |i j|. (4.17)

i=1 j=1

Eigenvectors |i and their respective eigenvalues λi of a linear operator Aˆ

are defined by Aˆ|i = λi|i .

(4.18)

In a matrix representation the eigenvalues can be determined from a charac-

teristic equation

det Aˆ − λIˆ = 0. (4.19)

Diagonalizable representation of an operator (also known as a orthonormal
decomposition) is given by

n (4.20)

Aˆ = λi|i i|

i=1

where the eigenvectors |i form an orthonormal set.

4.2.2 Adjoint operators

For every linear operator Aˆ on a Hilbert space there exist an adjoint (or
Hermitian conjugate) operator defined as

(|ψ , Aˆ|ϕ ) ≡ (Aˆ†|ψ , |ϕ ). (4.21)

CHAPTER 4. QUANTUM SYSTEMS 71

It follows that (AB)† = B†A† (4.22)
and (4.23)
† (4.24)
Aˆ|ψ ψ|Aˆ†,
= (4.25)

where

|ψ † ≡ ψ|.

In a matrix representation an adjoint operator is defined as

Aˆ† ≡ Aˆ∗ T

where ()∗ is a complex conjugation and ()T is a transpose operation.
Some useful definitions:

• Aˆ is a positive definite operator if (|ψ , Aˆ|ψ ) is a positive real number
for all |ψ = 0.

• Aˆ is a self-adjoint (or Hermitian) operator if Aˆ† = Aˆ. Any positive
definite operator is Hermitian.

• Aˆ is a normal operator if AˆAˆ† = Aˆ†Aˆ. For example, any Hermitian
operator is normal.

• Pˆ is a projection operator if Pˆ = k |i i|, where |i is an orthonormal
i=1
basis and k ≤ n.

• Uˆ is a unitary operator if Uˆ †Uˆ = Iˆ. Any pair of orthonormal basis |ψi
and |ϕi can be used to define unitary operators, i.e. Uˆ = n
i=1 |ψi ϕi|.

Some useful results:

• Unitary operators preserve the inner product, (4.26)
(Uˆ |ψ , Uˆ |ϕ ) = ψ|Uˆ †Uˆ |ϕ = ψ|Iˆ|ϕ = ψ|ϕ .

• Normal operators have a diagonal representation,

n (4.27)

Aˆ = λi|i i|.

i=1

It follows that the positive definite operators as well as Hermitian op-
erators have diagonal representations.

Homework problem: Prove the spectral decomposition theo-
rem which states that any normal operator is diagonal with re-
spect to some orthonormal basis.

CHAPTER 4. QUANTUM SYSTEMS 72

4.2.3 Operator functions (4.28)

If Aˆ has a diagonal decomposition (4.27), then (4.29)

n (4.30)
(4.31)
f (Aˆ) ≡ f (λi)|i i| (4.32)

i=1 (4.33)

for an arbitrary function f .
Another important matrix function is the trace defined as

n

tr(Aˆ) ≡ Aii

i=1

and has the following properties:

tr(AˆBˆ) = tr(BˆAˆ)
tr(Aˆ + Bˆ) = tr(Aˆ) + tr(Bˆ)

tr(zAˆ) = z tr(Aˆ).

These imply an invariance under similarity transformations

tr Uˆ AUˆ † = tr Uˆ †Uˆ A = tr (A) .

Because of this invariance one can define a trace of an operator as a trace of
any of its matrix representations.