Perturbation Theory Lecture 1 Books Recommended: Quantum Mechanics, concept and applications by Nouredine Zetili Introduction to Quantum Mechanics by D.J. Griffiths Cohen Tanudouji, Quantum Mechanics II Introductory Quantum Mechanics, Rechard L. Liboff

Approximate Methods: Approximate methods for stationary states corresponds to time independent Hamiltonian. Examples: perturbation theory, variational method, and the WKB method. Perturbation theory: Exactly solvable part + small correction In case we are not able to reduce the Hamiltonian of the problem to exactly solvable part+ small correction then we use variation method or WKB method

Time independent perturbation theory is splited into two parts, exactly solvable part and small perturbative part ---------(1) Small perturbative part can be written as ---------(2) From (1) -----(3)

Non-degenerate perturbation theory We have ---------(4) Perturbed Eigen value and Eigen state is expanded as ------(5) ------(6) For λ = 0 we get unperturbed part.

Use (5) and (6) in (3), we get ----(7) Now we compare powers of λ on both sides

Zero power of λ ----(8) 1st power of λ ----(9) 2nd power of λ ----(10)

Now we have to find eigen values and eigen functions not much different from ----(11) Using (6) in (11), we get ----(12) Coefficients of various powers of λ vanishes ----(13)

First order correction to : Multiplying both sides of (9) by -------(14) Where we used = 0 and Thus, from (5), using (14) --------(15)

Expand in terms of basis: ------(16) From (9), multiplying by --------(17)

Using (17) in (16), we get ----(18) We have ---------(19)

Second order correction: Multiply both sides by -------(20) Where we used Using (18) in (20) ----(21)

Thus En upto 2nd order Validity :