1 / 6

PERTURBATION THEORY

PERTURBATION THEORY. Time Independent Theory. A Necessity:. Exactly solvable problems are very few and do not really represent a real physical system completely. In many cases, the primary interaction on the system maybe exactly solvable and is hence the dominant contribution.

chelsi
Download Presentation

PERTURBATION THEORY

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PERTURBATION THEORY Time Independent Theory

  2. A Necessity: • Exactly solvable problems are very few and do not really represent a real physical system completely. • In many cases, the primary interaction on the system maybe exactly solvable and is hence the dominant contribution. • We develop a theory which takes the exact solution as a base to build on, for more accurate description of the system.

  3. Recipe of the Theory • Take the Hamiltonian of an exactly solvable problem. • Add a perturbative term to the original Hamiltonian, satisfying the condition, The terms in the perturbation need to be smaller than the energy differences of the original Hamiltonian. • Introduce a parameter l as the co-efficient of the perturbative term in the Hamiltonian, to observe the effect of the Hamiltonian.

  4. Eigen Value changes with perturbation intensity • Observation: Start with a negligible value of l such that the perturbation is very minuscule to start with. This causes the change in the Eigen-values and functions to vary slightly from the original Hamiltonian. Smoothly varying the parameter l smoothly varies the new solution as well.

  5. Taylor Expansion • The implication is, the existence of a continuous Eigen-functions and Eigen-values: and for an Hamiltonian parameterized by l as • Hence for small values of perturbation, the solutions can be expanded in Taylor series around the known solution with l = 0, as for the eigen-value equation: The original equation:

More Related