A linear operator is an instruction for transforming any given vector |V> in V into another vector |V’> in V while obeying the following rules:

If Ω is a linear operator and a and b are elements of F then

• Ωα|V> = αΩ|V>,  Ω(α|Vi> + β|Vj>)= αΩ|Vi> + βΩ|Vj>.
• <V|αΩ = α<V|Ω,  (<Vi|α + <Vj|β)Ω = α<Vi|Ω + β<Vj|Ω.
Examples:
• The simplest linear operator is the identity operator I.
I|V> = |V>,  <V|I = <V|.
• The parity operator ∏, operating on elements ψ(x,y,z) of L2, is a linear operator.
∏ψ(x,y,z) = ψ(-x,-y,-z).
• The operator Dx =  ∂/∂x, which differentiates with respect to x, is a linear operator if it operates on elements of the subspace L2 for which ∂ψ(x,y,z)/∂x is square integrable.
Dxψ(x,y,z) = ∂ψ(x,y,z)/∂x.

If the action of a linear operator on the basis vectors is known, then the action on any vector in the vector space is determined.  Let {|i>} be a basis and let Ω|i> = |i’>.
V = ∑ivi|i>,  Ω|V> = Ω ∑ivi|i> =  ∑iviΩ|i> = ∑ivi|i'>.
(Note: This is not true if the operator is not a linear operator.)

The product of two linear operators A and B, written AB, is defined by AB|ψ> = A(B|ψ>).  The order of the operators is important.  The commutator [A,B] is by definition [A,B] = AB - BA.
Two useful identities using commutators are
[A,BC] = B[A,C] + [A,B]C and [AB,C] = A[B,C] + [A,C]B.
Proof:  [A,BC] = ABC - BCA + (BAC - BAC) = ABC + B[A,C] - BAC = B[A,C] + [A,B]C.
The inverse operator of A, denoted by A-1 satisfies AA-1 = A-1A = I.  Not every operator has an inverse.

Examples:
• The projection operator Pψ has no inverse.  Let |ψ> be a ket with unit norm, <ψ|ψ> = 1.
Define the projection operator through Pψ = |ψ><ψ|.  Then
Pψ|Φ> = |ψ><ψ|Φ> =  ket times complex #,
Pψ2|Φ> =  Pψ|ψ><ψ|Φ> = |ψ><ψ|ψ><ψ|Φ> =  ket times 1 times complex #  = Pψ|Φ>.
Pψ  =  Pψ2 is a property of every projection operator.
• Let {|Φi>}q be a set of orthonormal vectors in V, <Φij> = δij.  Let Vq be the subspace of V spanned by these vectors.
Define Pq = ∑ij><Φj|.
Let |ψ> ∉ V.
Then Pq|ψ> =  ∑ij><Φj|ψ>  ∉ Vq.
Pq projects |ψ> into the subspace Eq.  It is a projector into a subspace.  Pq2  =  Pq is a property of any projector.

The adjoint of a linear operator
To every ket a|V> = |aV> corresponds a bra <aV| = a*<V|.  Let Ω be a linear operator.  To every ket Ω|V> = |ΩV> corresponds a bra <ΩV| = <V|Ω.  This defines Ω.  Ω is called the adjoint of Ω.
<ψ|Ω|Φ> = <Ωψ|Φ> = <Φ|Ωψ>* = <Φ|Ω|ψ>*.

Rules:

(A) = A,  (λA) = λ*A,  (A + B) = A + B,   (AB) = BA, (|u><v|) = |v><u|.

To every expression corresponds an adjoint expression.  To take the adjoint or Hermitian conjugate of an expression involving constants, kets, bras, and operators

• replace constants by their complex conjugates, (a --> a*),
• replace kets (bras) by bras (kets), (|ψ> --> <ψ|, <ψ| --> |ψ>),
• replace operators by their adjoints, (Ω --> Ω),
• reverse the order of the factors.
Examples:
• a<Φ|A|ψ>|r><ζ|  is an operator.
|ζ><r|<ψ|A|Φ>a* = a*<ψ|A|Φ>|ζ><r| is its adjoint.
• a|ψ><ζ|Φ>  is a ket,
a*<Φ|ζ><ψ|  is the corresponding bra.
Hermitian operators
An operator A is Hermitian if A = A.  A Hermitian operator satisfies <ψ|A|Φ> = <Φ|A|ψ>*.  An operator A is anti Hermitian if A = -A.

Unitary operators
An operator U is unitary if UU = UU = I.  An unitary operator preserves the norm.
<Uψ|Uψ> = <ψ|UU|ψ> = <ψ|ψ>.

Problem:

Let Ω be the operator defined bψ Ω = |Φ><ψ| where |Φ> and |ψ> are two vectors in a vector space V.
(a)  Under what conditions is Ω Hermitian?
(b)  Calculate Ω2.  Under what conditions is Ω a projector?

Solution:

• Concepts:
Mathematical foundations of quantum mechanics
• Reasoning:
An operator A is Hermitian if A = A. A Hermitian operator satisfies <ψ|A|Φ> = <Φ|A|ψ>*.
A projector is a Hermitian operator.  If Ω is a projector, then Ω2 = Ω.
• Details of the calculation:
(a)  Ω = |Φ><ψ|, Ω = |ψ><Φ|.
Ω = Ω  -->  Ω|ζ> = Ω|ζ>  for any |ζ> in V.
We need |Φ><ψ|ζ> = |ψ><Φ|ζ>.  This implies |Φ> = a|ψ>.
Then a|ψ><ψ|ζ> = |ψ><Φ|ζ> or a<ψ|ζ> =  <Φ|ζ>.
But we also have <Φ|ζ> = a*<ψ|ζ>.
Therefore a = a*.
We need |Φ> = a|ψ>, with a real.
(b) Ω2 = |Φ><ψ|Φ><ψ|.  We need <ψ|Φ> = 1 and |Φ> = a|ψ>.
This implies a<ψ|ψ> = 1, a = 1/<ψ|ψ>.
Summary
Linear operators can operate on bras and kets.  They map one ket onto another or one bra onto another, obeying certain rules.  If Ω is the operator transforming |V> into |V'>, then Ω is the operator transforming <V| into <V'|.  Ω is the adjoint of Ω.

The beauty of the Dirac notation:
We have defined rules for taking the adjoint of expressions consisting of bras, kets, operators, and complex numbers.  If we put together these elements in any order and use Dirac notation, then whenever a bracket is complete, it becomes a complex number and can be moved or complex conjugated.

Functions of operators
Consider a function F(z) which can be expanded in a power series in z, F(z) = ∑nfnzn.

Example:

• Any function that can be expanded in a Taylor series.

The corresponding function of an operator A is defined as F(A) = ∑nfnAn.
For example   eA = ∑nAn/n! = I + A + A2/2 + ... .
Let |Φa> be an eigenvector of A with eigenvalue a.  Then
F(A)|Φa> = ∑nfnAna> = ∑nfnana> = F(a)|Φa> = number*|Φa>.
a> is also an eigenvector of F(A).

Note:   In general eAeB ≠ eBeA ≠ e(A + B).
The order of the operators matters, unless the operators commute.

If A is a Hermitian operator, then eiA is a unitary operator.
Let T = exp(iA) then T = exp(-iA) = exp(-iA) and TT = I.