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University of California at San Diego – Department of Physics – Prof. John McGreevy Quantum Mechanics C (130C) Winter 2015 Final exam cover sheet Please remember to put your name on your exam booklet. This is a closed-book exam. 6 There are problems, each with several parts, of varying levels of difficulty; make sure you try all of the parts. None of the problems require very extensive calculation; if you find yourself involved in a morass of calculation, step back and think. Good luck. Possibly useful information: U(t) = e iHt/~ satisfies i~@t U = [H, U]. ✓ ◆ ✓ ◆ ✓ ◆ 01 0 i 1 0 x y z = , = , = 10 i 0 0 1 | "n̂ i = e i'/2 ✓ ✓ cos | "ẑ i + e+i'/2 sin | #ẑ i 2 2 ~ · n̂| "n̂ i = | "n̂ i i↵n̂·~ = 1 cos ↵ in̂ · ~ sin ↵. ◆ 1 ✓ ◆ cos ✓ sin ✓ cos ✓ sin ✓ = sin ✓ cos ✓ sin ✓ cos ✓ e ✓ satisfies ✓ ◆ 1 p2 1 HSHO = ~! a a + = + m! 2 q2 2 2m 2 r r ~ 1 ~m! q= a + a† , p = a a† ; [q, p] = i~ =) [a, a† ] = 1. 2m! i 2 † 1