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PERTURBATION THEORY Time Independent Theory

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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.

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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 as the co-efficient of the perturbative term in the Hamiltonian, to observe the effect of the Hamiltonian.

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Eigen Value changes with perturbation intensity E 4 0 E 3 0 E 2 0 E 1 0 Observation: Start with a negligible value of 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 smoothly varies the new solution as well.

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

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Lecture 35 Numerical Analysis. Chapter 7 Ordinary Differential Equations.

Lecture 35 Numerical Analysis. Chapter 7 Ordinary Differential Equations.

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