# Time-Independent Perturbation Theory 1 Time-Independent Perturbation Theory 1 Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Intl Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974) Perturbation Theory Perturbation Theory: A systematic procedure for obtaining approximate solutions to the unperturbed problem, by building on the known exact solutions to the unperturbed case. Time-Independent Perturbation Theory

Schroedinger Equation for 1-D Infinite Square Well Obtain a complete set of orthonormal eigenfunctions If potential is perturbed slightly Find the new eigenfunctions and eigenvalues of H. Derivation of Corrections New Hamiltonian: H = perturbation H0 = unperturbed quantity Write n and En as power series of :

Ist order correction to the nth value Insert into 2nd order correction to the nth value Derivation of Corrections After insertion: Collecting like powers of , First Order Correction to Energy Taking the inner product of: This means: Multiplying by

and integrating. Replace But H0 is hermitean, so and Therefore: First order correction to energy: Expectation value of perturbation, in the unperturbed state. First Order Correction to Wavefunction Rewrite Known function

Becomes inhomogeneous DE Therefore: satisfies First Order Correction to Wavefunction If l = n, m = n Equals Zero !st order energy correction First order correction to wavefunction If n = m, degenerate perturbation theory need to be used.

Example: V(x) V(x) -d/3 Unperturbed State d/3 Perturbed State x d / 3 V ( x ) 0 d / 3 x d

V ( x ) x d / 3 V ( x ) Unperturbed Wave function of Infinitely Deep Square Well x N cos nx / 2 d x N sin nx / 2 d 1 N d for odd n for even n H ( x ) x d / 3 Perturbed Energy Levels are obtained from:

En1 En1 / d

n* H n dx 1* v( x )1dx d/3 d / 3 l*l dx x cos 2 dx sin / 3 1 / 1 / 3 0.61

d/3 2 d d/3 E1 E1 0.61 Energy is increased by 0.61 times the amount of additional potential energy at d / 3 x d To find the perturbed wave function: and

n1 c1n 10 c2n 20 c3n 30 ...... n cm 0 0 m H n 0 En0 Em * 2 v ( x ) 1dx

1 c 0 x N cos nx / 2 d x N sin nx / 2d 1 N d for odd n for even n

2 E10 E 20 ma 2 1 c3 3 2 n1 10 md 2 3 2 cos

3x .......... . 2d Unperturbed levels are degenerate. Perturbation remove degeneracy. Example Suppose we put a delta-function bump in the centre of the infinite square well. where is a constant. Find the second-order correction to the energies for the above potential. Example: Continue Problem 1

Problem 2