23 - Physics 6210/Spring 2007/Lecture 23 Lecture 23...

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Unformatted text preview: Physics 6210/Spring 2007/Lecture 23 Lecture 23 Relevant sections in text: § 3.2, 3.5 Spin precession as a rotation It is enlightening to return to the dynamical process of spin precession in light of our new results on rotations. You will recall that a spin system with magnetic moment ~μ when placed in a uniform magnetic field ~ B can be described by the Hamiltonian H =- ~μ · ~ B, where ~μ = μ ~ S. You will recall that the behavior of the spin observables could be viewed as precession about ~ B , i.e., a continuously developing ( in time) rotation about an axis along ~ B . We can now see this result immediately. Let ˆ n be a unit vector along ~ B , so that H =- μB ˆ n · S . This means that the time evolution operator is U ( t, t ) = e i ¯ h ( t- t ) μB ˆ n · ~ S . This operator represents a rotation about ˆ n by an “angle” μB ( t- t ), which is exactly our previous result for the dynamics. Note that, while the physical observables are precessing with frequency μB , the state vector itself has is precessing at half the frequency since, e.g., it takes a 4 π rotation to get the state vector to return to its initial value. It is possible to experimentally “see” this difference in frequencies (thereby confirming the projective representation being used) by a pair of spin 1/2 systems, one of which propagates freely and one of which travels through a region with a magnetic field. The latter spin will precess according to the time it spends in the magnetic field. The two particles can be brought together to form an interferencein the magnetic field....
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.

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23 - Physics 6210/Spring 2007/Lecture 23 Lecture 23...

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