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Unformatted text preview: HOMEWORK ASSIGNMENT 5 PHYS852 Quantum Mechanics I, Spring 2008 Topics covered: Fine structure, hyperfine structure, time-dependent perturbation theory 1. Consider a hydrogen atom in the ground state n = 1, ℓ = 0. Using the classical kinetic energy formula T = Mv 2 2 , at what radius does the electrons velocity begin to exceed 10% of the speed of light? Using the ground state orbital R 10 ( r ), compute the probability of finding the electron inside this radius. Do the same for the 2S and 2P levels. If you were to generalize these results, would you expect relativistic corrections increase or decrease with n ? What about ℓ ? 2. Compute the complete first-order fine structure shift of the 1S state. One can separate the spin-orbit term into radial and angular parts giving ( vector L · vector S R 3 ) = ( nℓ || R- 3 | nℓ )( ℓm ℓ m 2 | vector L · vector S | ℓm ℓ m s ) The angular part vanishes due to an exact symmetry, while the radial part is infinite due to the divergence of the contribution from very small radius. At such small radius, particularly for S orbitals,divergence of the contribution from very small radius....
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