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Transcript

5.6 Calibration of the Optical Potential 195 Because we have scaled the units so that = 1, the scaled well depth Î±p represents the âdegree of quantumnessâ of the pendulum, with larger values representing more classical behavior (and thus a smaller wave-packet area in phase space), as reï¬ected in this quantum scaling factor. Hence, we should expect that the quantum wave packet moves with a longer period due to the reduced effective potential amplitude, and also that the wave packet motion will be further retarded by âclassicalâ anharmonic effects in the effective potential. 5.6.2.1 Wigner-Function Derivation Hug and Milburn [Hug01] have recently produced a more formal derivation of a quantum scaling factor based on the Wigner-function dynamics, in the context of the amplitude-modulated pendulum. Here we adapt this calculation to the ordinary pendulum, since the derivation does not depend on the temporal modulation of the potential. We begin with the general equation of motion for the Wigner function (which we introduced in Chapter 1 as the Moyal bracket), â s i 1 kÌ ât W (x, p) = âpâx W (x, p) + âxs V (x, t)âps W (x, p) kÌ s=0 s! 2i s â 1 kÌ s s â âx V (x, t)âp W (x, p) , â s! 2i s=0 (5.39) where we will keep the scaled Planck constant kÌ explicit for the time being. We can then insert the pendulum potential, V (x) = âÎ±p cos(x) , (5.40) with the result â ât W = âpâx W + 1 Î±p sin(x) kÌ s! s=0 s kÌ [1 â (â1)s ]âps W . 2 (5.41) If make use of the Taylor expansion s â 1 s kÌ W (x, p + kÌ/2) = , [âp W (x, p)] s! 2 s=0 (5.42) then Eq. (5.41) becomes ât W (x, p) = âpâx W (x, p) + Î±p sin(x) [W (x, p + kÌ/2) â W (x, p â kÌ/2)] . kÌ (5.43)