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Physics 452

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. Phys 452. Perturbation theory. Example of non-degenerate system Spring constant expansion in the harmonic oscillator. V. x. Phys 452. 3d. 2p. 2s.

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Physics 452

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  1. Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

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

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

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

  5. Phys 452 Quiz 5 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

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

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

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

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

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

  11. Phys 452 Quiz 5b 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)

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

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

  14. 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

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