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|V_i> + β|V_j>)= αΩ|V_i> + βΩ|V_j>.
• <V|αΩ = α<V|Ω,  (<V_i|α + <V_j|β)Ω = α<V_i|Ω + β<V_j|Ω.

• 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 D_x =  ∂/∂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.
  D_xψ(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 = ∑_iv_i|i>,  Ω|V> = Ω ∑_iv_i|i> =  ∑_iv_iΩ|i> = ∑_iv_i|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.

• The projection operator P_ψ has no inverse.  Let |ψ> be a ket with unit norm, <ψ|ψ> = 1. 
  Define the projection operator through P_ψ = |ψ><ψ|.  Then
  P_ψ|Φ> = |ψ><ψ|Φ> =  ket times complex #,
  P_ψ²|Φ> =  P_ψ|ψ><ψ|Φ> = |ψ><ψ|ψ><ψ|Φ> =  ket times 1 times complex #  = P_ψ|Φ>.
  P_ψ  =  P_ψ² is a property of every projection operator.

• Let {|Φ_i>}^q be a set of orthonormal vectors in V, <Φ_i|Φ_j> = δ_ij.  Let V_q be the subspace of V spanned by these vectors.
  Define P_q = ∑_i|Φ_j><Φ_j|.
  Let |ψ> ∉ V.
  Then P_q|ψ> =  ∑_i|Φ_j><Φ_j|ψ>  ∉ V_q.
  P_q projects |ψ> into the subspace E_q.  It is a projector into a subspace.  P_q²  =  P_q is a property of any projector.

Rules:

(A^†)^† = A,  (λA)^† = λ^*A^†,  (A + B)^† = A^† + B^†,   (AB)^† = B^†A^†, (|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.

• 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.

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

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>.

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.

Example:

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

The corresponding function of an operator A is defined as F(A) = ∑_nf_nAⁿ.
For example   e^A = ∑_nAⁿ/n! = I + A + A²/2 + ... .
Let |Φ_a> be an eigenvector of A with eigenvalue a.  Then
F(A)|Φ_a> = ∑_nf_nAⁿ|Φ_a> = ∑_nf_naⁿ|Φ_a> = F(a)|Φ_a> = number*|Φ_a>.
|Φ_a> is also an eigenvector of F(A).

Note:   In general e^Ae^B ≠ e^Be^A ≠ e^{(A + B)}.
The order of the operators matters, unless the operators commute.

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