Perturbation Theory
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Perturbation Theory: A quantum-mechanical approximation method used to find an approximate solution to a problem by starting from the exact solution of a related, simpler problem 1. Hamiltonian Operator: An operator corresponding to the total energy of the system, including both kinetic and potential energies 1. Schrödinger Equation: A fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time 1. Eigenfunctions and Eigenvalues: Solutions to the Schrödinger equation where eigenfunctions represent the possible states of the system and eigenvalues represent the corresponding energy levels 1. Unperturbed System: A system whose Hamiltonian is exactly solvable and serves as the starting point for perturbation theory 1. Perturbed System: A system whose Hamiltonian is slightly different from the unperturbed system, making it more complex to solve 1. Perturbation: The difference between the Hamiltonians of the perturbed and unperturbed systems 1. Nondegenerate Perturbation Theory: Perturbation theory applied to energy levels that are not degenerate (i.e., each energy level corresponds to a unique state) 1. Degenerate Perturbation Theory: Perturbation theory applied to energy levels that are degenerate (i.e., multiple states share the same energy level) 1. First-Order Energy Correction: The initial correction to the energy of a system due to perturbation 1. Second-Order Energy Correction: The subsequent correction to the energy of a system, taking into account the first-order correction 1. Intermediate Normalization: A simplification method where the perturbed wave function is required to satisfy a specific normalization condition 1. Configuration Interaction: The mixing of different configurations in the wave function due to perturbation 1. Variation-Perturbation Method: A method that combines variational principles and perturbation theory to estimate higher-order energy corrections 1. Coulomb Integral: An integral representing the electrostatic energy of repulsion between two charge distributions 1. Exchange Integral: An integral representing the interaction between two electrons when their positions are exchanged 1. Transition Dipole Moment: A measure of the probability of a transition between two states due to interaction with electromagnetic radiation 1. Selection Rules: Rules that determine the allowed transitions between quantum states based on the change in quantum numbers 1. Time-Dependent Perturbation Theory: Perturbation theory applied to systems exposed to time-dependent perturbations, such as electromagnetic radiation 1. Stimulated Emission: The process by which an atom or molecule emits a photon when exposed to radiation, causing a transition to a lower energy state 1. Spontaneous Emission: The process by which an atom or molecule emits a photon without external stimulation, causing a transition to a lower energy state 1. Absorption: The process by which an atom or molecule absorbs a photon, causing a transition to a higher energy state 1. Secular Equation: An algebraic equation used to find the energy corrections in degenerate perturbation theory 1. Hermitian Operator: An operator whose eigenvalues are real and whose eigenfunctions form a complete orthonormal set 1. Orthonormality: The property of eigenfunctions being orthogonal and normalized 1.