852quiz3_Solution - Answer: The ±rst-order ±ne-structure...

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NAME: QUIZ 3 PHYS852 Quantum Mechanics II, Spring 2009 Hydrogen Fine Structure Quiz: a.) List the three efects which combine to give the ±ne structure, and give a brie² description o² each physical process. Answer : The three efects are: Relativistic Mass Efect : ±rst relativistic correction to kinetic energy. Spin-Orbit interaction : magnetic interaction between electron spin and the magnetic ±eld it sees due to its motion around the nucleus. Darwin term : Relativistic electron see’s proton smeared out over the compton length. b.) Which o² the three terms is responsible ²or selecting a speci±c basis in the n , , s , subspace? What are the good eigenstates ²or computing the ±ne structure via perturbation theory? Answer: The only term which is not diagonal in the | nℓsm m s a basis is the spin-orbit term: v L · v S . The good eigenstates which diagonalize this term are | nℓsjm j a states. c.) On what quantum numbers should the ±rst-order ±ne-structure energy shi²t depend?
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Unformatted text preview: Answer: The ±rst-order ±ne-structure should energy-shi²t should in principle depend only on n , ℓ , s , and j . It cannot depend on m j because o² overall spherical symmetry. d.) Write a ²ormula which gives a crude order-o²-magnitude estimate o² the ±ne-structure shi²t o² the hydrogen 1s level (you probably want to make use o² the ±ne-structure constant). Anwer: E (1) fs ∼ | E (0) 1 | α 2 e.) Use your ²ormula to estimate the 1s ±ne structure shi²t in Joules and/or electron Volts. Answer: E (1) fs ∼ 2 . 18 × 10-18 J 137 2 now 137 2 ≈ (100 + 40)(100 + 40) = 10000 + 8000 + 1600 = 18600 ≈ 2 × 10 4 which gives E (1) fs ∼ 2 × 10-18 J 2 × 10 4 ≈ 10-22 J ≈ 6 × 10-4 eV This is a ²actor 3 larger than the ±ne-structure shi²t o² the 1s level E (1) fs = 3 . 5 × 10-23 J....
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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