Perturbation theory in quantum mechanics is a pivotal approximation method for studying complex systems. It involves a small perturbation to a known system, enabling approximate solutions. The theory is divided into time-independent and time-dependent categories, each with specific applications in atomic spectra, quantum states evolution, and more. It's also crucial for understanding phenomena like the Zeeman and Stark effects, and for advancements in quantum electrodynamics.
Show More
Perturbation theory is an approximation technique used in quantum mechanics to study complex systems
Time-Independent Perturbation Theory
Time-independent perturbation theory is used for systems with a constant Hamiltonian
Time-Dependent Perturbation Theory
Time-dependent perturbation theory is used for systems with a varying Hamiltonian
The Hamiltonian, representing the total energy of a quantum system, is a key element in perturbation theory
Time-independent perturbation theory is based on the stationary Schrödinger equation and is used to calculate energy level shifts
The first-order energy correction for a non-degenerate state is given by the inner product of the unperturbed wave function and the perturbation term
Time-independent perturbation theory is crucial for understanding phenomena such as the fine structure of atomic spectra
Time-dependent perturbation theory uses the time-dependent Schrödinger equation to predict the evolution of quantum states under external influences
Time-dependent perturbation theory is essential for calculating transition probabilities between states, as given by Fermi's Golden Rule
Time-dependent perturbation theory is used to understand processes such as photon absorption and emission by atoms
Harmonic perturbation theory is a subset of perturbation theory used to study the effects of small anharmonicities on the behavior of the quantum harmonic oscillator
Harmonic perturbation theory is used to calculate corrections to the energy levels of nearly harmonic systems, providing insight into vibrational spectra
Harmonic perturbation theory is used in quantum field theory to study the behavior of particles and their interactions
00
There are two categories of ______ theory: time-independent, for systems with a constant Hamiltonian, and time-dependent, for systems with a ______ Hamiltonian.
perturbation
varying
01
Time-dependent perturbation theory basis equation
Relies on time-dependent Schrödinger equation for evolving quantum states.
02
Fermi's Golden Rule formula components
Transition probabilities P_{i→f}(t), interaction Hamiltonian H', initial and final states Ψ_i^{(0)}, Ψ_f^{(0)}, density of states ρ(E_f).
