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
L
^{2}, 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 L^{2}for which ∂ψ(x,y,z)/∂x is square integrable.

D_{x}ψ(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 = ∑_{i}v_{i}|i>, Ω|V> = Ω ∑_{i}v_{i}|i>
= ∑_{i}v_{i}Ω|i> = ∑_{i}v_{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^{-1}A = 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_{ψ}^{2}|Φ> = P_{ψ}|ψ><ψ|Φ> = |ψ><ψ|ψ><ψ|Φ> = ket times 1 times complex # = P_{ψ}|Φ>.

P_{ψ}= P_{ψ}^{2}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_{q}^{2}= P_{q}is a property of any projector.

<ψ|Ω

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

<Uψ|Uψ> = <ψ|U

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 Ω^{2}. 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 Ω^{2}= 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) Ω^{2}= |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) = ∑_{n}f_{n}A^{n}.

For example e^{A} = ∑_{n}A^{n}/n! = I +
A + A^{2}/2 + ... .

Let |Φ_{a}> be an eigenvector of A
with eigenvalue a. Then

F(A)|Φ_{a}> = ∑_{n}f_{n}A^{n}|Φ_{a}>
= ∑_{n}f_{n}a^{n}|Φ_{a}> = F(a)|Φ_{a}>
= number*|Φ_{a}>.

|Φ_{a}> is also an eigenvector of F(A).

Note: In general e^{A}e^{B} ≠
e^{B}e^{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.