🎵Spectral Theory Unit 7 – Perturbation theory

Key Concepts and Definitions

• Perturbation theory studies the effects of small changes or disturbances on a system
• Unperturbed system represents the original, undisturbed system before perturbations are applied
  • Often has known solutions or is easier to solve compared to the perturbed system
• Perturbation refers to a small change or disturbance introduced to the unperturbed system
  • Can be represented as an additional term in the equations describing the system
• Perturbation parameter ($\epsilon$) quantifies the magnitude of the perturbation
  • Typically assumes values much smaller than 1 ($\epsilon \ll 1$)
• Perturbative expansion expresses the solution as a power series in terms of the perturbation parameter
  • Each term in the series represents a higher-order correction to the unperturbed solution
• Perturbative corrections are the additional terms in the perturbative expansion beyond the unperturbed solution
  • First-order correction corresponds to the term proportional to $\epsilon$, second-order correction to $\epsilon^2$, and so on

Historical Context and Development

• Perturbation theory has roots in celestial mechanics, where it was used to study planetary motions
  • Early work by astronomers like Pierre-Simon Laplace and Joseph-Louis Lagrange in the 18th and 19th centuries
• Quantum mechanics heavily relies on perturbation theory due to the complexity of many-body systems
  • Pioneering contributions by Erwin Schrödinger, Werner Heisenberg, and Paul Dirac in the early 20th century
• Time-independent perturbation theory developed to solve time-independent Schrödinger equation with small perturbations
• Time-dependent perturbation theory introduced to handle time-dependent perturbations and transitions between quantum states
• Degenerate perturbation theory emerged to address systems with degenerate energy levels
  • Involves lifting the degeneracy and determining the correct linear combination of degenerate states
• Relativistic quantum mechanics and quantum field theory extensively employ perturbation methods
  • Feynman diagrams provide a visual representation of perturbative expansions in particle physics

Mathematical Foundations

• Perturbation theory relies on the assumption that the perturbed system can be approximated by the unperturbed system
  • Validity of this assumption depends on the smallness of the perturbation parameter
• Taylor series expansion is a fundamental tool in perturbation theory
  • Expresses functions as power series in terms of a small parameter
• Rayleigh-Schrödinger perturbation theory is a widely used formulation in quantum mechanics
  • Assumes the perturbed Hamiltonian can be written as $H = H_0 + \epsilon V$, where $H_0$ is the unperturbed Hamiltonian and $V$ is the perturbation
• Brillouin-Wigner perturbation theory is an alternative formulation that avoids some of the limitations of Rayleigh-Schrödinger theory
  • Particularly useful for systems with strong coupling or near-degenerate energy levels
• Variational methods, such as the Rayleigh-Ritz method, provide upper bounds on the ground state energy
  • Can be combined with perturbation theory to improve accuracy

Types of Perturbation Methods

• Time-independent perturbation theory deals with time-independent perturbations to the Hamiltonian
  • Calculates corrections to the energy levels and eigenstates of the unperturbed system
• Time-dependent perturbation theory handles time-dependent perturbations and transitions between quantum states
  • Fermi's golden rule gives the transition rate between states under a perturbation
• Degenerate perturbation theory addresses systems with degenerate energy levels
  • Involves constructing a matrix in the degenerate subspace and diagonalizing it to find the perturbed energies and states
• Many-body perturbation theory deals with systems of interacting particles, such as electrons in solids
  • Green's function methods and Feynman diagrams are commonly used tools
• Relativistic perturbation theory extends perturbation methods to relativistic quantum mechanics and quantum field theory
  • Dyson series and renormalization techniques are employed to handle divergences

Applications in Quantum Mechanics

• Perturbation theory is extensively used in atomic and molecular physics to calculate energy levels and transitions
  • Examples include the Stark effect (electric field perturbation) and the Zeeman effect (magnetic field perturbation)
• In solid-state physics, perturbation theory helps understand electronic band structures and properties of materials
  • Electron-phonon interactions and impurity scattering can be treated as perturbations
• Quantum chemistry relies on perturbation methods to calculate molecular properties and reaction rates
  • Møller-Plesset perturbation theory is a widely used post-Hartree-Fock method
• Quantum field theory heavily employs perturbation theory to calculate scattering amplitudes and cross-sections
  • Feynman diagrams provide a systematic way to organize perturbative calculations
• Perturbative methods are also applied in quantum optics, quantum information, and quantum computing
  • Examples include the study of light-matter interactions and the design of quantum algorithms

Convergence and Limitations

• Convergence of the perturbative expansion is a crucial consideration in perturbation theory
  • Series may converge rapidly, slowly, or not at all depending on the magnitude of the perturbation
• Small perturbation parameter ($\epsilon \ll 1$) is necessary for the perturbative approach to be valid
  • Larger perturbations may require higher-order corrections or alternative methods
• Divergent series can arise in perturbation theory, especially in quantum field theory
  • Renormalization techniques are employed to handle divergences and extract meaningful results
• Strongly coupled systems or systems with large perturbations may not be amenable to perturbative treatment
  • Non-perturbative methods, such as variational techniques or numerical simulations, may be necessary
• Degenerate perturbation theory can fail if the degeneracy is not lifted by the perturbation
  • Requires careful analysis of the degenerate subspace and symmetries of the system

Advanced Techniques and Extensions

• Resummation techniques, such as Padé approximants and Borel resummation, can improve the convergence of perturbative series
  • Useful for handling divergent series or extending the range of validity of perturbation theory
• Renormalization group methods provide a systematic way to handle scale-dependent perturbations
  • Widely used in quantum field theory and statistical physics to study critical phenomena and phase transitions
• Effective field theories allow perturbative calculations in the presence of a separation of scales
  • Low-energy effective theories can be derived by integrating out high-energy degrees of freedom
• Symmetry-based perturbation theory exploits the symmetries of the system to simplify calculations
  • Examples include the use of angular momentum selection rules and the Wigner-Eckart theorem
• Adiabatic perturbation theory deals with slowly varying perturbations and adiabatic evolution
  • Relevant for studying quantum systems under slowly changing external fields or parameters

Problem-Solving Strategies

• Identify the unperturbed system and the perturbation
  • Determine the unperturbed Hamiltonian ($H_0$) and the perturbation term ($\epsilon V$)
• Assess the magnitude of the perturbation parameter ($\epsilon$) to ensure the validity of perturbative approach
• Choose the appropriate perturbation method based on the nature of the problem
  • Time-independent, time-dependent, degenerate, or many-body perturbation theory
• Calculate the unperturbed energies and eigenstates
  • Solve the unperturbed Schrödinger equation $H_0 \psi_n^{(0)} = E_n^{(0)} \psi_n^{(0)}$
• Apply the perturbation theory formalism to determine the corrections to the energies and eigenstates
  • Use the appropriate formulas for the chosen perturbation method
• Evaluate the perturbative corrections up to the desired order
  • First-order correction: $E_n^{(1)} = \langle \psi_n^{(0)} | V | \psi_n^{(0)} \rangle$, $|\psi_n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle \psi_m^{(0)} | V | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} |\psi_m^{(0)}\rangle$
  • Higher-order corrections involve more complex expressions
• Assess the convergence of the perturbative expansion and the validity of the results
  • Compare the magnitude of successive corrections and check if they decrease rapidly
• Interpret the physical meaning of the perturbative corrections
  • Relate the corrections to observable quantities such as energy shifts, transition rates, or response functions