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Unformatted text preview: Physics 6210/Spring 2007/Lecture 13 Lecture 13 Relevant sections in text: § 2.1 Example: Spin 1/2 in a uniform magnetic field. Let us consider the dynamical evolution of an electronic spin in a (uniform) magnetic field. We ignore the translational degrees of freedom of the electron. The electron magnetic moment is an observable which we represent as μ = e mc S , (Here e > 0 is the magnitude of the electron electric charge.) The Hamiltonian for a magnetic moment in a (uniform, static) magnetic field B is taken to be H = μ · B = e mc S · B . Let us choose the z axis along B so that H = eB mc S z . Evidently, S z eigenvectors are energy eigenvectors. Thus the S z eigenvectors are stationary states. Let us consider the time evolution of a general state,  ψ ( t ) i = a  + i + b i ,  a  2 +  b  2 = 1 . We have  ψ ( t ) i = e i ¯ h eB mc S z  ψ ( t ) i = ae iωt/ 2  + i + be iωt/ 2 i , where ω = eB mc . From this formula you can easily see that if the initial state is an S z eigenvector, then it remains so for all time. To see dynamical evolution, we pick an initial state that is not an energy eigenvector. For example, suppose that at t = 0, a = b = 1 √ 2 , i.e.,  ψ (0) i =  S x , + i . What is the probability for getting S x = ± ¯ h 2 at time t ? We have (exercise) P rob ( S x = ¯ h 2 ) = h S x , +  ψ ( t ) i 2 = cos 2 ( ωt 2 ) , P rob ( S x = ¯ h 2 ) = h S x , ψ ( t ) i 2 = sin 2 ( ωt 2 )...
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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.
 Spring '07
 M
 Physics, mechanics

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