Hermitian operators are the backbone of quantum mechanics, representing physical observables like position and momentum. They ensure that measurement outcomes are always real numbers, aligning with our everyday experience of the physical world.

These operators have unique properties that make them crucial in quantum theory. Their eigenvalues correspond to possible measured values, while their eigenfunctions form a complete set, allowing us to express any quantum state as a superposition of these eigenstates.

Hermitian Operators and Properties

Definition and Mathematical Properties

• Hermitian operators satisfy the condition A† = A, where A† represents the adjoint (conjugate transpose) of A
• Matrix representation of a Hermitian operator equals its own conjugate transpose
• Real-valued expectation values for any state of the system
• Product of two Hermitian operators remains Hermitian only when the operators commute
• Bounded operators with finite eigenvalues within a specific range
• Sum and difference of Hermitian operators yield Hermitian operators
• Preserve inner product of vectors in Hilbert space ensuring conservation of probability in quantum mechanics
• Examples of Hermitian operators include position operator $\hat{x}$ and momentum operator $\hat{p}$

Physical Significance and Applications

• Represent physical observables in quantum mechanics
• Eigenvalues correspond to possible measured values of associated physical observable
• Expectation value represents average value of corresponding observable over many measurements
• Non-commutativity of certain Hermitian operator pairs (position and momentum) leads to uncertainty principle
• Time evolution of expectation values governed by Ehrenfest's theorem
• Ensure measurement outcomes of physical observables are always real numbers
• Commutation relations determine compatibility of simultaneous measurements of corresponding observables
• Applications in various quantum systems (hydrogen atom, harmonic oscillator)

Hermitian Operators and Observables

Relationship Between Operators and Observables

• Every physical observable represented by a Hermitian operator in quantum mechanics
• Eigenvalues of Hermitian operator correspond to possible measured values of associated observable
• Expectation value $\langle A \rangle = \langle \psi | A | \psi \rangle$ represents average value of observable A in state $|\psi\rangle$
• Non-commutativity of certain operator pairs leads to uncertainty relations (position-momentum, energy-time)
• Time evolution of expectation values governed by Ehrenfest's theorem: $\frac{d}{dt}\langle A \rangle = \frac{1}{i\hbar}\langle [A,H] \rangle + \langle \frac{\partial A}{\partial t} \rangle$
• Measurement outcomes of physical observables always real numbers due to Hermitian nature of operators

Examples and Applications

• Position operator $\hat{x}$ and momentum operator $\hat{p}$ as fundamental Hermitian operators
• Angular momentum operators ($\hat{L}_x$, $\hat{L}_y$, $\hat{L}_z$) representing rotational motion
• Hamiltonian operator $\hat{H}$ representing total energy of a system
• Spin operators ($\hat{S}_x$, $\hat{S}_y$, $\hat{S}_z$) for intrinsic angular momentum of particles
• Pauli matrices as Hermitian operators representing spin-1/2 systems
• Density operator $\hat{\rho}$ for describing mixed quantum states
• Ladder operators (creation and annihilation operators) in quantum harmonic oscillator

Eigenvalues of Hermitian Operators

Proof of Real Eigenvalues

• Start with eigenvalue equation for Hermitian operator A: $A|\psi\rangle = \lambda|\psi\rangle$
• Take inner product with $\langle\psi|$: $\langle\psi|A|\psi\rangle = \lambda\langle\psi|\psi\rangle$
• Use Hermitian property A† = A: $\langle\psi|A^{\dagger}|\psi\rangle = \lambda^\langle\psi|\psi\rangle$
• Equate right-hand sides: $\lambda\langle\psi|\psi\rangle = \lambda^\langle\psi|\psi\rangle$
• Conclude \lambda = \lambda^ since $\langle\psi|\psi\rangle \neq 0$ for normalized states
• Result demonstrates all eigenvalues of Hermitian operators are real numbers
• Consistency with physical interpretation of eigenvalues as measurement outcomes

Properties and Implications

• Discrete spectrum for bounded operators in finite-dimensional Hilbert spaces
• Continuous spectrum possible for unbounded operators in infinite-dimensional spaces
• Degeneracy occurs when multiple eigenstates share same eigenvalue
• Spectral theorem guarantees existence of complete set of eigenfunctions
• Eigenvalues determine allowed energy levels in quantum systems (hydrogen atom, particle in a box)
• Measurement process collapses wavefunction to eigenstate with probability given by $|\langle\psi|n\rangle|^2$
• Examples: energy eigenvalues of hydrogen atom, angular momentum eigenvalues of orbital states

Eigenfunctions of Hermitian Operators

Orthogonality and Normalization

• Consider two distinct eigenstates $|\psi_i\rangle$ and $|\psi_j\rangle$ with eigenvalues $\lambda_i$ and $\lambda_j$
• Prove orthogonality for $\lambda_i \neq \lambda_j$: $\langle\psi_i|\psi_j\rangle = 0$
• Use Gram-Schmidt process for degenerate eigenvalues to construct orthogonal set
• Normalize eigenfunctions to obtain orthonormal set: $\langle\psi_i|\psi_j\rangle = \delta_{ij}$ (Kronecker delta)
• Significance of orthonormality in quantum mechanics (probability interpretation, superposition principle)
• Examples: spherical harmonics as eigenfunctions of angular momentum operators

Completeness and Expansion

• Prove completeness by showing sum of projection operators equals identity: $\sum_i |\psi_i\rangle\langle\psi_i| = I$
• Demonstrate any state in Hilbert space expressible as linear combination of eigenfunctions
• Expansion of arbitrary wavefunction: $|\psi\rangle = \sum_i c_i|\psi_i\rangle$ with $c_i = \langle\psi_i|\psi\rangle$
• Significance of completeness in quantum mechanics (resolution of identity, spectral decomposition)
• Role in solving time-independent Schrödinger equation
• Applications in perturbation theory and variational methods
• Examples: expansion of wavefunction in terms of energy eigenstates, Fourier series as expansion in momentum eigenstates