Time-independent and time-dependent perturbation theory are two approaches used in quantum mechanics to analyze the effects of perturbations on the behavior of a quantum system. The main difference between the two lies in the nature of the perturbation and the way it is treated mathematically.
- Time-Independent Perturbation Theory: Time-independent perturbation theory is employed when the perturbation acting on a system does not explicitly depend on time or varies very slowly compared to the characteristic timescale of the system. It is typically used for studying the behavior of a stationary quantum system under the influence of a small external perturbation. The theory provides a way to calculate corrections to the energy levels and wavefunctions of the unperturbed system.
In time-independent perturbation theory, the Hamiltonian of the system is divided into two parts: the unperturbed Hamiltonian (H₀), which describes the system in the absence of any perturbation, and the perturbation term (V), which represents the small additional influence on the system. The key steps in this approach involve calculating the corrections to the energy eigenvalues (E₀) and the corresponding wavefunctions (ψ₀) of the unperturbed system up to a certain order in the perturbation. This is achieved through iterative techniques, such as the Rayleigh-Schrödinger perturbation theory or the method of degenerate perturbation theory.
- Time-Dependent Perturbation Theory: Time-dependent perturbation theory is employed when the perturbation acting on a system explicitly depends on time and may vary significantly over time. It is used to analyze the time evolution of a quantum system under the influence of a time-varying perturbation. This theory is particularly useful for studying processes such as absorption and emission of photons, scattering, and transitions between energy levels.
In time-dependent perturbation theory, the time-dependent Schrödinger equation is solved to find the time-dependent wavefunction of the system. The perturbation term is treated as a time-dependent function that is introduced into the Hamiltonian of the system. The theory involves expanding the time-dependent wavefunction in terms of a series of terms that incorporate the effects of the perturbation at different orders. The coefficients of these terms are determined by solving a set of coupled differential equations, typically through iterative methods or using techniques like the Dyson series.
In summary, time-independent perturbation theory is used for analyzing stationary quantum systems under the influence of small perturbations, while time-dependent perturbation theory is employed for studying the time evolution of quantum systems under time-varying perturbations.