Random Phases and Energy Dispersion

Using 2N+1 successive stationary states centered at nth, we construct a rectangular wave packet in which the stationary states are superimposed with the equal weight √2N+1. With the requirement of the wave packet to be a quasi-classical state, the number N is determined by minimizing the uncertainty ΔxΔp. 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 ≈ 6^1/3n^2/3 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 1/2^N when N = 1 to 7, and does not exceed 0.01 when N ≥ 8.