⚛️Quantum Mechanics Unit 7 – Perturbation Theory

Introduction to Perturbation Theory

• Perturbation theory is a powerful mathematical tool used in quantum mechanics to find approximate solutions to complex quantum systems
• Applies when a quantum system can be divided into a solvable unperturbed part and a small perturbation
• Allows for the calculation of corrections to the energy levels and wavefunctions of the unperturbed system
• Perturbative approach becomes necessary when exact solutions to the Schrödinger equation are not available
• Perturbation theory has wide-ranging applications in various fields of physics, including atomic, molecular, and condensed matter physics

Fundamental Concepts

• Unperturbed system refers to a simplified quantum system for which exact solutions to the Schrödinger equation are known
• Perturbation is a small additional term in the Hamiltonian that represents a deviation from the unperturbed system
  • Can be due to external fields, interactions, or any other factors not included in the unperturbed Hamiltonian
• Perturbation parameter ($\lambda$) quantifies the strength of the perturbation relative to the unperturbed system
• Perturbative expansion expresses the perturbed energy levels and wavefunctions as power series in the perturbation parameter
  • $E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ...$
  • $|\psi_n\rangle = |\psi_n^{(0)}\rangle + \lambda |\psi_n^{(1)}\rangle + \lambda^2 |\psi_n^{(2)}\rangle + ...$
• Validity of perturbation theory relies on the perturbation being sufficiently small compared to the energy level spacing of the unperturbed system

Time-Independent Perturbation Theory

• Deals with quantum systems where the perturbation is time-independent
• Rayleigh-Schrödinger perturbation theory is the most commonly used formulation
• Zeroth-order approximation corresponds to the unperturbed system
• First-order correction to the energy is given by the expectation value of the perturbation in the unperturbed state
  • $E_n^{(1)} = \langle\psi_n^{(0)}|H'|\psi_n^{(0)}\rangle$
• Second-order correction involves a sum over all unperturbed states, excluding the state of interest
  • $E_n^{(2)} = \sum_{m \neq n} \frac{|\langle\psi_m^{(0)}|H'|\psi_n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}$
• Higher-order corrections can be calculated using recursive formulas
• Non-degenerate perturbation theory assumes that the unperturbed energy levels are non-degenerate
• Degenerate perturbation theory is used when the unperturbed system has degenerate energy levels
  • Requires diagonalization of the perturbation matrix within the degenerate subspace

Time-Dependent Perturbation Theory

• Addresses quantum systems subjected to time-dependent perturbations
• Interaction picture is often used to simplify the time-dependent Schrödinger equation
• Dyson series expresses the time-evolution operator as a perturbative expansion
  • $U(t,t_0) = 1 - \frac{i}{\hbar} \int_{t_0}^t dt_1 H'_I(t_1) + \left(-\frac{i}{\hbar}\right)^2 \int_{t_0}^t dt_1 \int_{t_0}^{t_1} dt_2 H'_I(t_1) H'_I(t_2) + ...$
• First-order perturbation theory leads to Fermi's golden rule, which gives the transition rate between initial and final states
  • $\Gamma_{i\rightarrow f} = \frac{2\pi}{\hbar} |\langle\psi_f|H'|\psi_i\rangle|^2 \delta(E_f - E_i - \hbar\omega)$
• Higher-order perturbation theory accounts for multi-photon processes and virtual intermediate states
• Resonance occurs when the perturbation frequency matches the energy difference between two states, leading to enhanced transition probabilities

Applications in Quantum Systems

• Stark effect describes the shifting and splitting of atomic energy levels in the presence of an external electric field
  • Linear Stark effect occurs in hydrogen-like atoms and is treated using degenerate perturbation theory
  • Quadratic Stark effect is observed in non-hydrogenic atoms and is a second-order perturbative effect
• Zeeman effect refers to the splitting of atomic energy levels in the presence of an external magnetic field
  • Weak-field Zeeman effect is treated using degenerate perturbation theory
  • Strong-field Zeeman effect requires a non-perturbative approach
• Fine structure of atomic spectra arises from the relativistic correction to the electron's kinetic energy and the spin-orbit coupling
  • Relativistic correction is treated as a perturbation to the non-relativistic Hamiltonian
  • Spin-orbit coupling is a perturbative effect that leads to the splitting of energy levels
• Hyperfine structure results from the interaction between the electron's magnetic moment and the nuclear magnetic moment
  • Fermi contact interaction is the dominant contribution and is treated using perturbation theory
• Van der Waals forces between neutral atoms and molecules can be derived using second-order perturbation theory
  • Dispersion forces arise from the mutual polarization of the electron clouds

Limitations and Approximations

• Perturbation theory is an approximate method and has limitations in its applicability
• Convergence of the perturbative expansion is not always guaranteed
  • Divergent series can occur when the perturbation is too strong or the unperturbed system is not well-behaved
• Breakdown of perturbation theory occurs when the perturbation parameter becomes comparable to or larger than the energy level spacing
• Non-perturbative methods, such as variational techniques or numerical simulations, may be necessary for strongly perturbed systems
• Degenerate perturbation theory can fail if the degeneracy is not completely lifted by the perturbation
  • Quasi-degenerate perturbation theory or multi-level perturbation theory may be required
• Time-dependent perturbation theory assumes that the perturbation is turned on adiabatically
  • Sudden perturbations or non-adiabatic processes may require alternative approaches
• Higher-order corrections become increasingly complex and computationally demanding
  • Truncation of the perturbative expansion at a certain order introduces approximations

Problem-Solving Techniques

• Identify the unperturbed system and the perturbation
  • Choose a suitable basis set for the unperturbed system (e.g., energy eigenstates)
• Determine the order of perturbation theory required based on the desired accuracy and the strength of the perturbation
• Calculate the matrix elements of the perturbation in the chosen basis
  • Use symmetry arguments and selection rules to simplify the calculations
• Apply the appropriate perturbative formulas to obtain the corrections to the energy levels and wavefunctions
  • For degenerate systems, diagonalize the perturbation matrix within the degenerate subspace
• Normalize the perturbed wavefunctions to ensure proper normalization
• Interpret the results in terms of the physical properties of the system
  • Compare the perturbative corrections to experimental observations or numerical simulations
• Assess the validity of the perturbative approach by estimating the convergence of the perturbative expansion
  • Consider higher-order corrections or alternative methods if necessary

Advanced Topics and Extensions

• Brillouin-Wigner perturbation theory is an alternative formulation that avoids the problem of small energy denominators
  • Particularly useful for systems with near-degeneracies or strong perturbations
• Rayleigh-Schrödinger perturbation theory can be generalized to non-Hermitian Hamiltonians
  • Relevant for open quantum systems or systems with complex potentials
• Perturbation theory can be combined with other approximation methods, such as the WKB approximation or the variational method
  • Provides a systematic way to improve the accuracy of the approximations
• Many-body perturbation theory is used to study interacting many-particle systems, such as electrons in solids or nuclei
  • Feynman diagrams are a powerful tool for organizing the perturbative expansion in terms of particle interactions
• Relativistic perturbation theory incorporates relativistic effects into the quantum mechanical description
  • Dirac equation is used as the starting point, and perturbations are added to account for QED corrections or other relativistic effects
• Perturbation theory can be formulated in different representations, such as the position, momentum, or energy representation
  • Choice of representation depends on the symmetries and properties of the system under consideration