Stationary states are crucial in quantum mechanics, representing systems with constant probability densities over time. They're solutions to the time-independent Schrödinger equation, forming a basis for understanding quantum systems' behavior and energy levels.

Energy eigenvalues, associated with stationary states, reveal the allowed energies of quantum systems. From simple setups like particles in boxes to complex atoms, these eigenvalues help us grasp how energy is quantized and how quantum systems evolve.

Stationary States and the Schrödinger Equation

Defining Stationary States

• Quantum states with time-independent probability densities represented by wavefunctions as eigenfunctions of the Hamiltonian operator
• Solutions to the time-independent Schrödinger equation (TISE) take the general form $\psi(x,t) = \psi(x)e^{-iEt/\hbar}$
  • $\psi(x)$ denotes the spatial part
  • $e^{-iEt/\hbar}$ represents the time-dependent phase factor
• Probability density $|\psi(x,t)|^2$ remains constant over time as the time-dependent phase factor cancels out when taking the absolute square
• Form a complete orthonormal basis for the Hilbert space of the quantum system
  • Allows expression of any state as a linear combination of stationary states
• Examples of stationary states
  • Ground state of a hydrogen atom
  • Specific energy level of a quantum harmonic oscillator

Time-Independent Schrödinger Equation

• Eigenvalue equation for the Hamiltonian operator
• Energy eigenvalues correspond to allowed energies of the system
• TISE takes the form $\hat{H}\psi = E\psi$
  • $\hat{H}$ represents the Hamiltonian operator
  • $E$ denotes the energy eigenvalue
  • $\psi$ signifies the wavefunction (eigenfunction)
• Solving TISE involves finding solutions satisfying both the differential equation and boundary conditions
• For bound states, normalizability of the wavefunction leads to energy level quantization
• Examples of systems described by TISE
  • Particle in a box (infinite square well)
  • Quantum harmonic oscillator
  • Hydrogen atom

Energy Eigenvalues and Eigenfunctions

Simple Quantum Systems

• Particle in a box (infinite square well)
  • Energy eigenvalues: $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$
  • Eigenfunctions: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$
  • $n$ represents the quantum number, $L$ denotes the well width, $m$ signifies particle mass
• Quantum harmonic oscillator
  • Energy eigenvalues: $E_n = (n + \frac{1}{2})\hbar\omega$
  • Eigenfunctions expressed using Hermite polynomials
  • $\omega$ represents the angular frequency of the oscillator
• Hydrogen atom
  • Energy eigenvalues: $E_n = -\frac{13.6 \text{ eV}}{n^2}$
  • Eigenfunctions involve spherical harmonics and Laguerre polynomials
  • $n$ denotes the principal quantum number

Calculation Methods

• Analytical solutions for simple potentials involve solving differential equations
• Numerical methods for complex potentials
  • Shooting method iteratively adjusts initial conditions to match boundary conditions
  • Matrix diagonalization converts TISE into a matrix eigenvalue problem
• Perturbation theory for systems close to exactly solvable ones
  • Treats complex potentials as perturbations to simpler, known systems
  • Provides approximate solutions for energy levels and wavefunctions
• Variational method for estimating ground state energy
  • Uses trial wavefunctions to find upper bounds on the ground state energy
  • Improves accuracy by optimizing parameters in the trial wavefunction

Physical Interpretation of Stationary States

Properties of Stationary States

• Represent states of definite energy in quantum systems
  • Energy as a conserved quantity precisely defined
• Exhibit no probability current due to constant probability density over time
• Uncertainty principle applies
  • Energy uncertainty $\Delta E = 0$ implies complete time uncertainty $\Delta t \rightarrow \infty$
• Time-independent expectation values for observables reflect stable nature
• Often associated with standing waves in bound systems
  • Analogous to standing waves on a string with fixed endpoints (guitar string)

Dynamics and Superposition

• Superpositions of stationary states with different energies result in time-dependent states
  • Leads to dynamic behavior and probability currents
  • Example: electron in a superposition of ground and excited states in an atom
• Coherent states in quantum harmonic oscillator
  • Superposition of multiple energy eigenstates
  • Exhibit classical-like oscillatory behavior
• Quantum beats observed in atomic systems
  • Result from superposition of closely spaced energy levels
  • Manifest as periodic intensity fluctuations in spectroscopic measurements

Energy Spectrum of Quantum Systems

Characteristics of Energy Spectra

• Consists of all possible energy eigenvalues
  • Can be discrete, continuous, or a combination
• Bound states in finite potential wells have discrete energy levels
  • Example: electron energy levels in an atom
• Unbound states form a continuous spectrum above the well
  • Example: free electron states in a metal
• Ground state represents the lowest energy stationary state
• Excited states correspond to higher energy levels
• Energy level spacing determines transition frequencies between states
  • Observable in spectroscopic measurements (atomic emission spectra)

Advanced Concepts

• Degeneracy occurs when multiple distinct stationary states share the same energy eigenvalue
  • Often due to symmetries in the system (spherical symmetry in hydrogen atom)
• Density of states describes available energy states per unit energy interval
  • Crucial for understanding thermal and statistical properties (heat capacity of solids)
• Selection rules determine allowed or forbidden transitions between energy levels
  • Derived from symmetry considerations and conservation laws
  • Example: electric dipole transitions in atoms ($\Delta l = \pm 1$)
• Band structure in solid-state physics
  • Energy spectra of electrons in periodic potentials of crystalline solids
  • Explains electrical and optical properties of materials