Quantum Mechancial Operators

Definition

An operator $\hat{A}$ , acting on a state, is the mathematical rule that $\hat{A} | ψ ⟩ = | ψ^{'} ⟩$ and $⟨ φ | A = ⟨ φ^{'} |$ .

Linear Operators

We will use linear operators for PH651.

Forbidden Quantities

$A ⟨ φ |$
$| φ ⟩ A$

Properties

Expectation values (mean)

$\overset{―}{A} = ⟨ ψ | \hat{A} | ψ = ⟨ \hat{A} ⟩$ for normalized $| ψ ⟩$ .

For non-normalized, $⟨ \hat{A} ⟩ = \frac{⟨ ψ | A | ψ ⟩}{⟨ ψ | ψ ⟩}$ .

Outer Product Operator

$| φ ⟩ ⟨ ψ |$

Applying: $ψ ⟩ φ | ψ^{'} ⟩ = ⟨ φ | ψ ⟩ | ψ ⟩ = | ψ^{″} ⟩$

Hermitian Adjoint (conjugate)

$α^{†} = α^{*}$

${\hat{A}}^{†} = A^{T *}$

$⟨ ψ | {\hat{A}}^{†} | φ ⟩ = ⟨ φ | A | ψ ⟩^{*}$

Hermitian Operator

$A^{†} = A$ . $⟨ ψ | \hat{A} | φ ⟩ = ⟩ φ | A | ψ ⟩^{*}$ .

Antihermitian Operator

$B^{†} = - B$ . $⟨ ψ | \hat{A} | φ ⟩ = - ⟩ φ | A | ψ ⟩^{*}$ .

Examples

${\hat{A}}^{† †}$
$(α \hat{A})^{†} = α^{*} A^{†}$
$(A + B + C + D)^{†} = (A^{†} + B^{†} + C^{†} + D^{†})$
$(A B C D)^{†} = D^{†} C^{†} B^{†} A^{†}$
$(A B C D | φ ⟩)^{†} = ⟨ φ | D^{†} C^{†} B^{†} A^{†}$

Hermicity

$\hat{A} = \hat{X}$ is it Hermitian? $⟨ ψ | \hat{X} | φ ⟩ = \int ψ^{*} (x) x φ (x) d x = (\int ψ (x) x^{*} φ^{*} (x) d x)^{*} = (\int φ^{*} (x) x ψ (x) d x)^{*} = ⟨ φ | \hat{X} | φ ⟩^{*}$ , Yes.
Is $\hat{A} = \frac{d}{d \hat{X}}$ Hermitian? $⟨ ψ | \frac{d}{d \hat{X}} | φ ⟩ = \int ψ^{*} (x) \frac{d}{d x} φ (x) d x = 0 - \int φ^{*} (x) \frac{d ψ}{d x} d x^{*} = - ⟨ φ | \hat{A} | ψ ⟩^{*}$ , No, anti-hermititan.
$- i ℏ \frac{d}{d x}$ is it Hermitian? Yes.

Commutator

Functions of Operators

Inverse

If it exists, then $A^{- 1} A = A A^{- 1} = I$ .

If $A | ψ ⟩ = a | ψ ⟩$ then $A^{- 1} a | ψ ⟩ = | ψ ⟩$ , or $A^{- 1} | ψ ⟩ = \frac{1}{a} | ψ ⟩$ .

Unitary Operators

Projection Operator

$P_{n} = | φ_{n} ⟩ ⟨ φ_{n} |$ . $P_{n} | ψ ⟩ = | φ_{n} ⟩ ⟨ φ_{n} | ψ ⟩ = c_{n} | φ_{n} ⟩$ .

Thus, $I = \sum_{i} P_{i}$ .

Matrix Representations of Kets, Bras, Operators

Operators: Matrix - Product of Column Vector with Row Vector $A_{j}^{i} = ⟨ i | A | j ⟩, (A_{j}^{i})^{†} = ⟨ j | A^{†} | i ⟩$
Kets: Column Vector. $| ψ ⟩ = \sum_{n} c_{n} | φ_{n} ⟩ = (\begin{matrix} ⟨ φ_{1} | ψ ⟩ \\ \dots \end{matrix}) = (\begin{matrix} c_{1} \\ c_{2} \\ \dots \end{matrix})$
Bras: Row Vector. $⟨ ψ | = (⟨ ψ | φ_{1} ⟩ \dots) = (c_{1}^{*} \dots)$
Inner Product: $⟨ ψ | ψ ⟩ = \sum_{n} c_{n}^{*} d_{n}$

Trace

Types

Real Matrix

$A = A^{*}$

Imaginary Matrix

$A = - A^{*}$

Symmetric Matrix

$A^{T} = A$

Anti-Symmetric Matrix

$A^{T} = - A$

N.B. The diagonal of an Anti-Symmetric Matrix must be zero.

Orthogonal Matrix

$A^{T} = A^{- 1}$

Theorem

A Real-Orthogonal Matrix is Unitary.

Expectation Values

$⟨ A ⟩ = \sum_{n, m} ⟨ ψ | φ_{n} ⟩ ⟨ φ_{n} | A | φ_{m} ⟩ ⟨ φ_{m} | ψ ⟩ = \sum_{n, m} a_{n}^{*} a_{m} A_{m}^{n}$ . Weighted average.

For Diagonal Operator, $⟨ A ⟩ = \sum_{n} A_{n} | a_{n} |^{2}$ . (In this case, we expect diagonalization since $| φ_{n} ⟩$ are orthonormal bases for $A$ )

Quantum Mechanical Measurements

Quantum Mechanical Observables

Uncertainty Principle

$(Δ A)^{2} (Δ B)^{2} = ⟨ (Δ \hat{A})^{2} ⟩ ⟨ (Δ \hat{B})^{2} ⟩ \geq \frac{1}{4} | ⟨ [A, B] ⟩ |^{2}$

$Δ \hat{A} = \hat{A} - ⟨ \hat{A} ⟩$ .
$Δ A = \sqrt{⟨ {\hat{A}}^{2} ⟩ - ⟨ \hat{A} ⟩^{2}}$
$(Δ \hat{A})^{2} = (\hat{A} - ⟨ \hat{A} ⟩)^{2}$ .
$⟨ (Δ A)^{2} ⟩ = ⟨ {\hat{A}}^{2} ⟩ - ⟨ \hat{A} ⟩^{2}$ .
$| a ⟩ = Δ \hat{A} | ψ ⟩$
$| b ⟩ = Δ \hat{B} | ψ ⟩$
$| ⟨ a | b ⟩ |^{2} \leq ⟨ a | a ⟩ ⟨ b | b ⟩ = ⟨ ψ | Δ {\hat{A}}^{†} Δ \hat{A} | ψ ⟩ ⟨ ψ | Δ {\hat{B}}^{†} Δ \hat{B} | ψ ⟩ = ⟨ {\hat{A}}^{2} ⟩ ⟨ {\hat{B}}^{2} ⟩ = (Δ A)^{2} (Δ B)^{2}$
$| ⟨ a | b ⟩ = ⟨ (Δ \hat{A} Δ \hat{B}) ⟩$
$| ⟨ (Δ \hat{A}) (Δ \hat{B}) ⟩ |^{2} \leq (Δ A)^{2} (Δ B)^{2}$
$A B = \frac{1}{2} ([A, B] + {A, B}) \Rightarrow Δ \hat{A} Δ \hat{B} = \frac{1}{2} ([Δ \hat{A}, Δ \hat{B}] + {Δ \hat{A}, Δ \hat{B}}) = \frac{1}{2} ([\hat{A}, \hat{B}] + {Δ \hat{A}, Δ \hat{B}})$
Anti-commutator has real expectation value and expectation value of commutator is imaginary.
$⟨ Δ \hat{A} Δ \hat{B} ⟩^{2} = \frac{1}{4} (| ⟨ [\hat{A}, \hat{B}] ⟩ |^{2} + | ⟨ {Δ \hat{A}, Δ \hat{B}} |^{2}) \geq \frac{1}{4} | ⟨ [\hat{A}, \hat{B}] ⟩ |^{2}$
$(Δ A) (Δ B) \geq \frac{1}{2} | ⟨ [\hat{A}, \hat{B}] ⟩ |$