Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel

Homework Phys 452 Tuesday Jan 24: Homework # 5 Pb: 6.12, 6.13, 6.14 & 6.33 Thursday Jan 26: Homework # 6

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

Phys 452 Application: The fine structure of hydrogen Motion of the electron Coulomb interaction between e - and nucleus Bohr’s energies where Bohr radius

Phys 452 The fine structure of hydrogen Spherical harmonics

Phys 452 The fine structure of hydrogen Fine structure + Lamb shift + Sources of perturbation: Hyperfine splitting + +… Bohr’s energy E = where fine structure constant

Phys 452 Quiz 6a How small is the fine structure correction compared to the Bohr’s energies ? A. Same order of magnitude B. Factorsmaller C. Factorsmaller D. Factorsmaller E. Factorsmaller

Phys 452 The fine structure of hydrogen Bohr’s energy Fine structure Lamb shift Hyperfine splitting Relativistic correction Spin- orbit coupling

Phys 452 The fine structure of hydrogen Relativistic correction What is the relativistic correction to the Hamiltonian? Momentum Kinetic energy

Phys 452 The fine structure of hydrogen Relativistic correction The first-order correction to the Bohr’s energy will be: Developing H’ in terms of p Pb 6.14 Application to harmonic oscillator

Phys 452 The fine structure of hydrogen Relativistic correction The first order correction to the Bohr’s energy is:

Phys 452 The fine structure of hydrogen Relativistic correction Expectation values Pb 6.12Pb 6.33 Pb 6.13 For example (1s) Introducing the azimuthal quantum number

Phys 452 Quiz 6b Give an estimate of the relativistic correction for the ground level of hydrogen ? A. B. C. D. E.

Phys 452 Quiz 6c Does the relativistic correction lift the degeneracy of the atomic levels ? A. Not at all B.Yes, partially C.Yes, completely D. It depends on the speed of the electron E. It depends on external fields

Phys 452 Quiz 6d What is the relativistic correction for the band (3d)? A. B. C. D. E.