ps5 - d~n dt =-i h h ( t ) | [ ~,H ] | ( t ) i (4) and then...

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Harvard University Physics 143b: Quantum Mechanics II Problem Set 5 due Friday October 22 Problem numbers refer to Griffiths, second edition 1. Consider a S = 1 / 2 spin in a time-dependent magnetic field ~ B ( t ) with the Hamiltonian H = - 1 2 ~ B ( t ) · (1) where are the Pauli matrices. For an arbitrary quantum state | Ψ( t ) i , we define the Bloch vector ~n ( t ) by ~n ( t ) = h Ψ( t ) | | Ψ( t ) i . (2) The state | Ψ( t ) i obeys the Schr¨ odinger equation i ¯ h d dt | Ψ( t ) i = H | Ψ( t ) i . (3) Use this to derive the equation of motion obeyed by ~n ( t ). First show that
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Unformatted text preview: d~n dt =-i h h ( t ) | [ ~,H ] | ( t ) i (4) and then express the right-hand-side only in terms of ~ B ( t ) and ~n ( t ). The result is sim-plied version of the Bloch equations (the full Bloch equations also contain dissipative terms generated by couplings to the environment). 2. Griths 9.1 3. Griths 9.3 4. Griths 9.20 (a, b, c were done in class in a slightly dierent notation). 1...
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This note was uploaded on 02/15/2011 for the course PHYS 143B taught by Professor Unknow during the Spring '10 term at Harvard.

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