HOMEWORK ASSIGNMENT 5PHYS852 Quantum Mechanics I, Spring 2008Topics covered:Fine structure, hyperfine structure, time-dependent perturbation theory1. Consider a hydrogen atom in the ground staten= 1,ℓ= 0.Using the classical kinetic energyformulaT=Mv22, at what radius does the electrons velocity begin to exceed 10% of the speed oflight? Using the ground state orbitalR10(r), compute the probability of finding the electron insidethis radius. Do the same for the 2S and 2P levels. If you were to generalize these results, would youexpect relativistic corrections increase or decrease withn? What aboutℓ?2. Compute the complete first-order fine structure shift of the 1S state. One can separate the spin-orbitterm into radial and angular parts giving(vectorL·vectorSR3)=(nℓ||R-3|nℓ)(ℓmℓm2|vectorL·vectorS|ℓmℓms)The angular part vanishes due to an exact symmetry, while the radial part is infinite due to thedivergence of the contribution from very small radius. At such small radius, particularly for S orbitals,
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