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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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