7.1 Time-independent perturbation theory for non-degenerate states

Perturbation theory helps solve quantum systems that can't be solved exactly. It splits the Hamiltonian into a solvable part and a small perturbation, then calculates energy and wavefunction corrections as power series expansions.

For non-degenerate states, the method gives increasingly accurate corrections to energy levels. The first-order correction is the average of the perturbation in the unperturbed state, while higher orders account for mixing between states.

Energy Corrections for Non-degenerate States

Fundamentals of Time-Independent Perturbation Theory

• Time-independent perturbation theory calculates approximate solutions for quantum mechanical systems lacking exact solutions
• Hamiltonian split into two parts $H = H₀ + λV$
  • $H₀$ unperturbed Hamiltonian with known solutions
  • $V$ perturbation
  • $λ$ small parameter
• Energy and wavefunction corrections expressed as power series expansions in terms of λ
• Zeroth-order approximation corresponds to unperturbed system
• Higher-order terms represent increasingly accurate corrections
• Applies to non-degenerate states with distinct, well-separated unperturbed energy levels

First-Order Energy Correction

• First-order energy correction given by $E₁ = ⟨ψ₀|V|ψ₀⟩$
  • $ψ₀$ unperturbed wavefunction
• Represents average value of perturbation in unperturbed state
• Provides initial estimate of perturbation's effect on energy levels
• Calculated using matrix elements of perturbation operator $V$

Higher-Order Energy Corrections

• Involve summations over intermediate states
• Depend on matrix elements of perturbation operator $V$
• Second-order energy correction given by $E₂ = Σₙ₌₁ |⟨ψₙ|V|ψ₀⟩|² / (E₀ - Eₙ)$
  • Sum excludes n = 0
• Account for mixing of unperturbed state with other states due to perturbation
• Denominator $(E₀ - Eₙ)$ shows states closer in energy contribute more significantly
• Sign of second-order correction always negative, indicating energy lowering due to level repulsion

Perturbation Series Limitations

Convergence Factors

• Perturbation theory may not converge for all systems or large perturbations
• Convergence depends on relative magnitude of perturbation compared to unperturbed energy level spacings
• Rayleigh-Schrödinger perturbation theory assumes perturbed states expressed as linear combinations of unperturbed states
• Series may diverge with nearby energy levels or strong perturbations
• Radius of convergence estimated using complex analysis techniques (residue theorem)

Addressing Convergence Issues

• Resummation techniques improve convergence in some cases
• Alternative perturbation methods (Brillouin-Wigner perturbation theory)
• Padé approximants used to extrapolate series behavior
• Renormalization group methods applied to perturbation theory
• Assess applicability by comparing results with exact solutions or experimental data
• Numerical methods (variational approach) complement perturbative calculations

First- and Second-Order Energy Corrections

Derivation of Energy Corrections

• First-order energy correction derived from time-independent Schrödinger equation
• Expand perturbed wavefunction and energy in powers of λ
• Collect terms of same order in λ to obtain correction equations
• First-order correction $E₁ = ⟨ψ₀|V|ψ₀⟩$ obtained from orthogonality conditions
• Second-order correction derived using first-order wavefunction
• Higher-order corrections systematically derived using perturbation expansion of Schrödinger equation

Interpretation of Energy Corrections

• First-order correction represents direct effect of perturbation on energy levels
• Second-order correction accounts for indirect effects through state mixing
• Negative sign of second-order correction explained by level repulsion principle
• Magnitude of corrections indicates strength of perturbation's influence
• Energy corrections provide insight into symmetry breaking and degeneracy lifting
• Perturbative approach reveals underlying physical mechanisms (coupling between states)
• Corrections used to predict spectroscopic transitions and energy level shifts

Anharmonic Oscillator vs Stark Effect

Anharmonic Oscillator Analysis

• Potential energy includes higher-order terms beyond quadratic term of harmonic oscillator
• Typical anharmonic potential $V(x) = ½kx² + λx³ + μx⁴$
• Perturbation theory calculates corrections due to cubic and quartic terms
• First-order correction vanishes for odd-power terms due to parity
• Second-order correction leads to energy level shifts and non-uniform spacing
• Anharmonicity affects vibrational spectra of molecules (CO₂, H₂O)
• Perturbative results compared with numerical solutions to validate approach

Stark Effect Calculations

• Describes splitting and shifting of spectral lines in external electric field
• Perturbation Hamiltonian $V = -μ·E$ (μ electric dipole moment, E electric field)
• First-order correction vanishes for hydrogen ground state due to parity
• Second-order correction dominates for hydrogen ground state (quadratic Stark effect)
• Linear Stark effect occurs for hydrogen states with different principal quantum numbers
• Matrix elements of electric dipole operator evaluated using spherical harmonics
• Stark effect applications include precision spectroscopy and quantum control (Rydberg atoms)