Quantum mechanics II Winter 2012 Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

Phys 452 Homework Friday Jan 20: Homework # 4 Pb 6.6, 6.8, 6.9, 6.10

Example of non-degenerate system Phys 452 Perturbation theory Example of non-degenerate system Spring constant expansion in the harmonic oscillator V x

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Example of degenerate system 1s 2s 2p 3d Atomic energy levels perturbation “lift the degeneracy”

Quiz 5 Phys 452 Taking the electron spin into account, in how many energy levels, the (3d) band of Fe atom is susceptible to split, under the application of a magnetic field? A. 2 B. 3 C. 5 D. 6 E. 10

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Two-fold degeneracy are the eigenvalues of In fact, we are solving for

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Two-fold degeneracy E d=2 E0

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Two-fold degeneracy In fact, we are solving for If Then, using Choose a “good” clever basis (to make W diagonal) Pb 6.6

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Higher –order degeneracy Find eigenvalues of d = 3 3 energies Find eigenvalues of d = N N energies Pb 6.10 Try to find a “good” basis to make W diagonal

Degenerate perturbation theory Phys 452 Degenerate perturbation theory General method Start with an ortho-normal basis of the unperturbed states If the state is non-degenerate: If the state is degenerate: build l Diagonalize W : the eigenvalues are Pb 6.8 & 6.9

Quiz 5b Phys 452 Here is a list of energy levels for a given system (V,5V,3V,3V,2V, 3V,5V,5V,V,7V) How many levels are degenerate and non-degenerate? A. (3,1) B. (2,3) C. (3,2) D. (1,4) E. (4,1)

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Example: cubic infinite square well z Perturbation: in a sub-cell y l x Build W for the first excited state Diagonalize W

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Example: problem 6.8 z Perturbation: y l ? x Build W for the first excited state Diagonalize W

Degenerate perturbation theory Phys 452 Degenerate perturbation theory Problem 6.9 e is the pertubation Identify the unperturbed states determine which one are degenerate or non degenerate Find the exact solution of the perturbed Hamiltonian Compare with the first order correction Use W for degenerate state Compare with the second order correction for non- degenerate states