r - qc / 9 80 90 07 v 1 1 S ep 1 99 8 Spreading of wave packets , Uncertainty Relations and the de Broglie frequency

The spreading of quantum mechanical wave packets are studied in two cases. Firstly we look at the time behavior of the packet width of a free particle confined in the observable Universe. Secondly, by imposing the conservation of the time average of the packet width of a particle driven by a harmonic oscillator potential, we find a zero-point energy which frequency is the de Broglie frequency. The quantum mechanical wave-packet spreading is a subject of current interest as can be verified in some recently published papers [1] and [2]. As pointed out by Grobe and Fedorov [1] the ionization of atoms can be supressed in superstrong fields. This phenomenon has been called stabilization and is characterized by decreasing ionization probability with increasing laser intensity. The wave-packet spreading plays a key role in the final degree of stabilization. On the other hand, Dodonov and Mizrahi [2] have adressed to the " Strict lower bound for the spatial spreading of a relativistic particle " where they provide a strict inequality for the minimal possible extension of a wave packet corresponding to the physical state of a relativistic particle. In this letter we intend to study the spreading of wave-packets in two particular situations. In the first case we explore the consequences of the finiteness of the Universe in the spreading of a free particle wave-packet. In the second one, we want to study the wave-packet of a particle described by an one-dimensional harmonic oscillator. As we will see this can lead to interesting consequences related to the interpretation of the de Broglie frequency of a particle. For a one-dimensional wave-packet let us define [3] (∆q) 2 = q 2 − q 2 , (1) (∆p) 2 = p 2 − p 2 , (2) Where (∆q) 2 and (∆p) 2 in the above relations are, respectively the variancies of the quantities q and p, representing the position and the momentum of a particle. An interesting interpretation of wave-packet spreading can be found in Gasiorowicz [4].