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Chapter4_28 - z B B ˆ = r The Hamiltonian in matrix form...

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PHY4604 R. D. Field Department of Physics Chapter4_28.doc University of Florida Electron in a Magnetic Field "Spin" Magnetic Moment: Certain elementary particles (such as electrons) carry intrinsic angular momentum (called "spin" angular momentum ) and an intrinsic magnetic moment (called "spin" magnetic moment ), S S g m e e spin r r r γ µ = = 2 , ( electron ) where S r is the spin angular momentum of the electron and g = 2 is the gyromagnetic ratio and e e m e m eg = = 2 γ . When a magnetic dipole is placed in a magnetic field B r , it experiences a torque, B r r × µ , which tends to line it up parallel to the magnetic field. The energy associated with this torque is B E r r = µ so the Hamiltonian of a spinning charged particle at rest in a magnetic field is B S B H r r r r = = λ µ . Larmour Precession: Consider an electron (spin ½) at rest in a uniform magnetic field which points in the z-direction,
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Unformatted text preview: z B B ˆ = r . The Hamiltonian, in matrix form, is − − = ⋅ − = ⋅ − = 1 1 2 B B S B H h r r r r . The eigenstates of H are the same as S z . Namely >= + 1 | χ with energy ) ( 2 1 B E h − = + and >= − 1 | with energy ) ( 2 1 B E h + = − . The general solution of the time-dependent equation t i H ∂ > ∂ >= | | h Can be expressed in terms of the stationary states as follows: = > + > >= − + − − − + − + 2 / 2 / / / | | ) ( | t B i t B i t iE t iE be ae e b e a t h h where a and b are arbitrary complex constants with 1 | | | | 2 2 = + b a . B S...
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