precess - -i ) sin 2 cos 2 e iE t/ h-e-iE t/ h / 2 i = =-i...

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Physics 225/315 February 8, 2008 Precession Example– Magnetic Field in the x, z plane. Continued θ z B x ± B is in the x, z plane and makes an angle θ with the z axis. ± B = B e cos θ ˆ z + B e sin θ ˆ x Write this in the matrix representation using S z and S x . H = - γ ± S · ± B = - γB e ¯ h 2 ˆ cos θ sin θ sin θ - cos θ ! The eigenstates are cos θ 2 sin θ 2 and sin θ 2 - cos θ 2 ( convenient sign convention ) The time evolution of the state is | ψ ( t ) ± = a 1 cos θ 2 sin θ 2 e - iE 0 t/ ¯ h + a 2 sin θ 2 - cos θ 2 e iE 0 t/ ¯ h The initial condition At t = 0 is spin in the + z direction | ψ ( t = 0) ± = ˆ 1 0 ! find a 1 and a 2 1 = a 1 cos θ 2 + a 2 sin θ 2 0 = a 1 sin θ 2 - a 2 cos θ 2 a 1 = cos θ 2 a 2 = sin θ 2 satisfies these equations. | ψ ( t ) ± = cos 2 θ 2 sin θ 2 cos θ 2 e - iE 0 t/ ¯ h + sin 2 θ 2 - sin θ 2 cos θ 2 e iE 0 t/ ¯ h Check that this becomes the initial state | + ± at t = 0 1
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Probability for observing spin down |-± at time t The amplitide is ²-| ψ ( t ) ± or 0 1 · ˆ ψ + ψ - ! = ψ - Then ψ - = sin θ 2 cos θ 2 e - iE 0 t/ ¯ h - e iE 0 t/ ¯ h · = 2(
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Unformatted text preview: -i ) sin 2 cos 2 e iE t/ h-e-iE t/ h / 2 i = =-i sin sin ( E t/ h ) use E h = 2 . The probability is P-= |-| ( t ) | 2 P-= sin 2 sin 2 t 2 At t = 0 P-= 0 Is this expected? Notice that P-ranges from 0 to 1. At later times what is the maximum probability for nding spin down? The maximum occurs when sin 2 t 2 = 1 the maximum value is P max-= sin 2 What is P max-when = 0 , / 4 , / 2 ? Maximum spin direction B Field P-at Max P-/ 4 1/2 / 2 1 0.01 0.02 0.03 0.04 0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 time (microsec) Prob down Probability for spin down 2...
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precess - -i ) sin 2 cos 2 e iE t/ h-e-iE t/ h / 2 i = =-i...

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