Linear Operators

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 an a and b are elements of F then

Examples:

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 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

Examples: 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:

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:

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.