12.3 Operator methods in quantum mechanics

Quantum mechanics uses linear operators on Hilbert spaces to describe physical systems. State vectors represent physical states, while observables are represented by self-adjoint operators. Commutation relations between operators determine the compatibility of observables.

The spectral theorem is crucial for understanding quantum operators. It states that self-adjoint operators can be decomposed into projection operators, allowing for calculation of functions of operators. This theorem is essential for analyzing observables and their measurement outcomes.

Mathematical Formulation and Operator Methods in Quantum Mechanics

Mathematical formulation of quantum mechanics

• Quantum mechanics formulated using linear operators acting on a Hilbert space (complex vector space with inner product)
  • State vectors represent physical states are elements of the Hilbert space (wavefunctions)
  • Observables represented by self-adjoint (Hermitian) operators (position, momentum, energy)
• Commutation relations between operators play a crucial role determine compatibility of observables
  • Canonical commutation relations: $[\hat{x}, \hat{p}] = i\hbar$ ($\hat{x}$ position operator, $\hat{p}$ momentum operator)
  • Operators that commute have simultaneous eigenstates can be measured simultaneously
• Expectation values of observables calculated using the inner product gives average value of observable
  • $\langle A \rangle = \langle \psi | \hat{A} | \psi \rangle$ ($|\psi\rangle$ state vector, $\hat{A}$ observable operator)
• Time evolution governed by the time-dependent Schrödinger equation describes dynamics of quantum system
  • $i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle$ ($\hbar$ reduced Planck's constant, $t$ time)
  • The Hamiltonian operator $\hat{H}$ determines the time evolution of the system encodes energy and interactions

Spectral theorem for self-adjoint operators

• Spectral theorem states any self-adjoint operator can be decomposed into a sum of projection operators
  • $\hat{A} = \sum_n a_n |\phi_n\rangle\langle\phi_n|$ ($a_n$ eigenvalues, $|\phi_n\rangle$ corresponding eigenstates)
  • Eigenvalues real, eigenstates orthonormal and complete
• Spectral decomposition allows for calculation of functions of operators enables algebraic manipulation
  • $f(\hat{A}) = \sum_n f(a_n) |\phi_n\rangle\langle\phi_n|$ ($f$ arbitrary function)
• Spectral theorem essential for understanding properties of observables relates spectrum to measurement outcomes
  • Eigenvalues represent possible measurement outcomes discrete or continuous
  • Eigenstates form a complete orthonormal basis for the Hilbert space span the space of states

Solving time-independent Schrödinger equation

• Time-independent Schrödinger equation an eigenvalue problem determines stationary states and energy levels
  • $\hat{H} |\psi_n\rangle = E_n |\psi_n\rangle$ ($E_n$ energy eigenvalues, $|\psi_n\rangle$ corresponding eigenstates)
  • Stationary states have well-defined energy do not evolve in time
• Solving eigenvalue problem involves finding eigenvalues and eigenstates of the Hamiltonian
  • Analytical solutions possible for simple systems (harmonic oscillator, hydrogen atom)
  • Numerical methods used for more complex systems (variational methods, perturbation theory)
• Eigenstates of the Hamiltonian form a complete orthonormal basis allows expansion of arbitrary states
  • Any state can be expanded in terms of the energy eigenstates superposition principle
  • $|\psi\rangle = \sum_n c_n |\psi_n\rangle$, where $c_n = \langle\psi_n|\psi\rangle$ (expansion coefficients)

Spectrum analysis of quantum operators

• Spectrum of an operator set of its eigenvalues characterizes the possible measurement outcomes
  • Discrete spectrum: isolated eigenvalues (bound states in potential wells)
  • Continuous spectrum: continuous range of eigenvalues (scattering states, free particles)
• Eigenfunctions corresponding to discrete eigenvalues square-integrable normalizable
  • Represent bound states typically localized in space (atomic orbitals)
• Eigenfunctions corresponding to continuous eigenvalues not square-integrable non-normalizable
  • Represent scattering states typically delocalized (plane waves)
• Properties of spectrum and eigenfunctions provide insights into physical behavior of the system
  • Energy levels, transition probabilities, selection rules determined by spectrum and symmetries
  • Symmetries of the Hamiltonian reflected in degeneracy of the spectrum (rotational, translational invariance)

Spectral Theory and Applications in Quantum Mechanics