Part I, Chapter 2 | Page 1 of 9

Linear Operators

Observables as operators on Hilbert space

2.1 Definition of Linear Operators

In quantum mechanics, physical observables (position, momentum, energy, spin) are represented by linear operators acting on the Hilbert space.

Definition

An operator $\hat{A}: V \to V$ is linear if:

$$\hat{A}(\alpha|\psi\rangle + \beta|\phi\rangle) = \alpha\hat{A}|\psi\rangle + \beta\hat{A}|\phi\rangle$$

for all $|\psi\rangle, |\phi\rangle \in V$ and $\alpha, \beta \in \mathbb{C}$.

Matrix Representation

In a basis $\{|n\rangle\}$, an operator is represented by a matrix with elements:

$$A_{mn} = \langle m | \hat{A} | n \rangle$$

Acting on a state $|\psi\rangle = \sum_n c_n |n\rangle$ gives:

$$\hat{A}|\psi\rangle = \sum_{m,n} A_{mn} c_n |m\rangle$$

Examples of Operators in QM:

• Position: $\hat{x}$
• Momentum: $\hat{p} = -i\hbar \frac{d}{dx}$
• Hamiltonian (energy): $\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x})$
• Angular momentum: $\hat{L} = \hat{r} \times \hat{p}$
• Spin: $\hat{S}_x, \hat{S}_y, \hat{S}_z$

2.2 Adjoint Operator

The adjoint (or Hermitian conjugate) $\hat{A}^\dagger$ of an operator$\hat{A}$ is defined by:

$$\langle \psi | \hat{A}^\dagger | \phi \rangle = \langle \phi | \hat{A} | \psi \rangle^*$$

Or equivalently:

$$\langle \hat{A}^\dagger \psi | \phi \rangle = \langle \psi | \hat{A} \phi \rangle$$

Properties of the Adjoint

$(\hat{A}^\dagger)^\dagger = \hat{A}$
$(\alpha\hat{A} + \beta\hat{B})^\dagger = \alpha^*\hat{A}^\dagger + \beta^*\hat{B}^\dagger$
$(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger\hat{A}^\dagger$ (reverse order!)

Matrix Representation of Adjoint

If A represents $\hat{A}$, then $\hat{A}^\dagger$ is represented by:

$$(A^\dagger)_{mn} = A_{nm}^* = \overline{A^T}$$

(complex conjugate transpose, also called Hermitian transpose)

2.3 Hermitian Operators

An operator is Hermitian (or self-adjoint) if:

$$\hat{A}^\dagger = \hat{A}$$

Equivalently, for all states:

$$\langle \psi | \hat{A} | \phi \rangle = \langle \phi | \hat{A} | \psi \rangle^*$$

🌟 Fundamental Postulate of QM:

Every observable in quantum mechanics is represented by a Hermitian operator.This ensures that measured values (eigenvalues) are real numbers!

Key Property: Real Eigenvalues

Hermitian operators have real eigenvalues:

Proof: Let $\hat{A}|a\rangle = a|a\rangle$. Then:

$$\langle a | \hat{A} | a \rangle = a \langle a | a \rangle$$

Since $\hat{A}$ is Hermitian:

$$\langle a | \hat{A} | a \rangle = \langle a | \hat{A} | a \rangle^*$$

Therefore $a = a^*$, so a is real! □

Examples of Hermitian Operators

Position: $\hat{x}$

Eigenvalues are all real numbers (positions)

Momentum: $\hat{p} = -i\hbar\frac{d}{dx}$

Eigenvalues are all real numbers (momenta)

Hamiltonian: $\hat{H}$

Eigenvalues are energy levels (real)

Pauli matrices: $\sigma_x, \sigma_y, \sigma_z$

Eigenvalues are ±1 (spin measurements)

2.4 Unitary Operators

An operator $\hat{U}$ is unitary if:

$$\hat{U}^\dagger \hat{U} = \hat{U} \hat{U}^\dagger = \hat{I}$$

Equivalently, $\hat{U}^\dagger = \hat{U}^{-1}$.

Key Property: Preserves Inner Products

Unitary operators preserve inner products (and hence norms):

$$\langle \hat{U}\psi | \hat{U}\phi \rangle = \langle \psi | \hat{U}^\dagger \hat{U} | \phi \rangle = \langle \psi | \phi \rangle$$

Physical Significance:

Time evolution in quantum mechanics is unitary! The time evolution operator$\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ is unitary, ensuring probability conservation. Symmetry transformations are also unitary.

Examples

Time evolution:

$$\hat{U}(t) = e^{-i\hat{H}t/\hbar}$$

Rotation operator:

$$\hat{R}(\theta) = e^{-i\theta\hat{J}_z/\hbar}$$

Key Concepts (Page 1)

• Observables are represented by Hermitian operators
• Hermitian operators have real eigenvalues (measurable values)
• Unitary operators preserve inner products and norms
• Time evolution is unitary: probability conservation
• Adjoint: $(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger\hat{A}^\dagger$

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Chapter 2: Linear Operators

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