Eigenvalues and eigenfunctions are key concepts in quantum mechanics. They represent allowed values of observables and describe quantum states. Understanding these helps us solve the Schrödinger equation and predict system behavior.

In the broader context of quantum mechanical postulates, eigenvalues and eigenfunctions are crucial. They form the mathematical foundation for describing quantum states, measurements, and time evolution of systems. This knowledge is essential for grasping quantum mechanics.

Eigenvalues and Eigenfunctions in Quantum Mechanics

Definition and Mathematical Representation

• Eigenvalues represent the allowed values of an observable quantity (energy, angular momentum) that correspond to a specific quantum state of a system
• Eigenfunctions, or eigenstates, are the wave functions that describe the quantum states associated with specific eigenvalues
• The eigenvalue equation in quantum mechanics: $\hat{H}\psi = E\psi$, where $\hat{H}$ is the Hamiltonian operator, $\psi$ is the eigenfunction, and $E$ is the eigenvalue
  • Eigenvalues and eigenfunctions are obtained by solving this eigenvalue equation for a given quantum mechanical system
• The set of eigenvalues and eigenfunctions for a quantum system is unique and forms a complete basis for describing the system's possible states (Hilbert space)

Properties of Eigenvalues and Eigenfunctions

• Eigenfunctions form a complete set of orthonormal functions
  • Orthogonal: Inner product is zero for different eigenfunctions, $\langle\psi_i|\psi_j\rangle = 0$ for $i \neq j$
  • Normalized: Inner product of an eigenfunction with itself is unity, $\langle\psi_i|\psi_i\rangle = 1$
• Eigenvalues are real numbers for Hermitian operators (operators equal to their adjoint)
  • Most physical observables are represented by Hermitian operators in quantum mechanics
• Degenerate eigenvalues occur when multiple eigenfunctions correspond to the same eigenvalue
  • Any linear combination of these eigenfunctions is also a valid eigenfunction for that eigenvalue

Significance of Eigenvalues and Eigenfunctions

Solving the Schrödinger Equation

• The Schrödinger equation describes the behavior of a quantum system (particle in a potential well, electron in an atom)
• To solve the Schrödinger equation for a specific system, one must find the eigenvalues and eigenfunctions of the system's Hamiltonian operator
  • Eigenvalues represent the allowed energy levels of the quantum system
  • Eigenfunctions describe the spatial distribution of the particle's wave function at each energy level
• The general solution to the Schrödinger equation is expressed as a linear combination of eigenfunctions
  • Determines the probability distribution and expectation values of observable quantities for the quantum system

Quantum State Expansion and Time Evolution

• Any quantum state can be expanded as a linear combination of eigenfunctions (basis states)
  • Coefficients represent the probability amplitudes for measuring the corresponding eigenvalues
• The time evolution of a quantum state is described by the time-dependent Schrödinger equation
  • The coefficients of the eigenfunction expansion evolve in time, while the eigenfunctions themselves remain constant
• Unitary time evolution operators can be expressed in terms of the eigenfunctions and eigenvalues of the Hamiltonian

Determining Eigenvalues and Eigenfunctions

Simple Quantum Mechanical Systems

• Particle in a one-dimensional infinite potential well (or "box")
  • Eigenvalues: $E_n = \frac{n^2 h^2}{8mL^2}$, where $n$ is a positive integer, $h$ is Planck's constant, $m$ is the particle's mass, and $L$ is the width of the well
  • Eigenfunctions: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$, sinusoidal standing waves
• Quantum harmonic oscillator
  • Eigenvalues: $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$, where $n$ is a non-negative integer, $\hbar$ is the reduced Planck's constant, and $\omega$ is the angular frequency of the oscillator
  • Eigenfunctions: Hermite polynomials multiplied by a Gaussian function
• Hydrogen atom
  • Eigenvalues: $E_n = -\frac{13.6 \text{ eV}}{n^2}$, where $n$ is the principal quantum number
  • Eigenfunctions (atomic orbitals): Described by spherical harmonics and radial functions involving Laguerre polynomials

Numerical Methods for Complex Systems

• Shooting method
  • Iteratively adjusts initial conditions to satisfy boundary conditions and find eigenvalues
• Matrix diagonalization
  • Represents the Hamiltonian as a matrix in a chosen basis and diagonalizes it to find eigenvalues and eigenvectors (eigenfunctions)
• Variational method
  • Uses trial wave functions with adjustable parameters to minimize the energy and approximate the ground state eigenvalue and eigenfunction

Physical Meaning of Eigenvalues and Eigenfunctions

Measurement and Probability

• Eigenvalues represent the possible outcomes of measurements of observable quantities (energy, angular momentum, position)
• The probability of measuring a specific eigenvalue is determined by the square of the absolute value of the corresponding eigenfunction, $|\psi|^2$ (probability density)
• Expectation value of an observable quantity: $\langle\hat{O}\rangle = \langle\psi|\hat{O}|\psi\rangle$, where $\hat{O}$ is the operator corresponding to the observable
• Eigenfunctions describe the spatial distribution of the particle's probability density
  • Provides information about the likelihood of finding the particle at a given position or with a given momentum

Quantum State Superposition and Interference

• Quantum states can be in a superposition of eigenfunctions
  • The coefficients of the superposition represent the probability amplitudes for each eigenstate
• Interference effects arise from the superposition of eigenfunctions
  • Constructive interference occurs when probability amplitudes add coherently
  • Destructive interference occurs when probability amplitudes cancel out
• Measuring an observable quantity collapses the quantum state to one of the eigenfunctions corresponding to the measured eigenvalue

Connection to Classical Mechanics

• In the classical limit (large quantum numbers), the behavior of quantum systems approaches that of classical systems
  • Bohr correspondence principle: Quantum mechanics must reduce to classical mechanics in the limit of large quantum numbers
• The spacing between adjacent eigenvalues becomes smaller as the quantum numbers increase
  • Allows for the transition from discrete to continuous values of observables
• Ehrenfest's theorem relates the time evolution of quantum expectation values to classical equations of motion