,Random Phases and Energy Dispersion

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Using 2N + 1 successive stationary states centred at nth, we construct a rectangular wavepacket in which the stationary states are superimposed with the equal weight √2N + 1. With the requirement of the wavepacket to be a quasi-classical state, the number N is determined by minimizing the uncertainty △x△p. Since the stationary state can only be determined to within an arbitrary multiplicative complex phase factor of unit magnitude, a number of N is obtained as a set of the phases are given. For a harmonic oscillator, when all of the phase factors are essentially the same, we have N ≈ [61/3n2/3] with [x] signifying the integral part of positive number x. When every phase in the phase factors is given by a random number generated in a closed interval [0, 2π] and when n ≥ 10, the probability of appearance of N is roughly 1/2N when N = 1 to 7, and does not exceed 0.01 whenN ≥ 8.
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