The first two period behaviors of a quantum wave packet in an infinite square well potential is studied. First, the short term behavior of expectation value of a quantity on an equally weighted wave packet (EWWP) is in classical limit proved to reproduce the Fej'{e}r average of the Fourier series decomposition of the corresponding classical quantity. Second, in order to best mimic the classical behavior, a nice relation between number N of stationary states in the EWWP with the average quantum number n as N≈n is revealed. Third, since the Fejér average can only approximate the classical quantity, it carries an uncertainty which in large quantum number case is almost the same as the quantum uncertainty.
The mean value of a quantity in an equally weighted wave packet was recently found in the classical limit to be the Fejér average of partial sums of Fourier series expansion of the classical quantity, and the number of stationary states in it is equal to that of partial sums. The incompleteness of the Fejér average in representing a classical quantity enables us to define a classical uncertainty relation which turns out to be the counterpart of the quantum one. In this paper, two typical quantum systems, a harmonic oscillator and a particle in an infinite square well, are used to illustrate the above-mentioned points.
The time-of-flight measurement approach of Peres based on the Salecker-Wigner quantum clock is applied to the one-dimensional scattering of a wave packet from a rectangular barrier. By directly evaluating the expectation value of the clock-time operator in the asymptotic states of wave packet long after the scattering process, we derive an average wave-packet clock time for the barrier interaction, which is expressed as an average of the stationary-state clock time over all possible initial scattering states of the wave packet. We show that the average wave-packet clock time is identical to the average dwell time of a wave packet.