Some consequences of the generalized uncertainty principle induced ultraviolet wave-vector cutoff in one-dimensional quantum mechanics

A projection method is proposed to treat the one-dimensional Schrödinger equation for a single particle when the generalized uncertainty principle (GUP) generates an ultraviolet (UV) wave-vector cutoff. The existence of a unique coordinate representation called the naive one is derived from the one-parameter family of discrete coordinate representations. In this bandlimited quantum mechanics a continuous potential is reconstructed from discrete sampled values observed by means of a particle in maximally localized states. It is shown that bandlimitation modifies the speed of the center and the spreading time of a Gaussian wave packet moving in free space. Indication is found that GUP accompanied by bandlimitation may cause departures of the low-lying energy levels of a particle in a box from those in ordinary quantum mechanics to be much less suppressed than commonly thought when GUP without bandlimitation is at work.

Figure 1

Sampled values (boxes) and the reconstructed bandlimited continuous potential (solid line) for $V(x)=V_{0}a\delta (x)$. The numerical result obtained by terminating the sum in Eq. (20) at $n=\pm 50$ and the analytic one given by Eq. (g4) cannot be distinguished in the figure.

Figure 2

Sampled values (boxes), the reconstructed bandlimited continuous potential (solid line), and its analytic approximation (dashed line) for the potential step $V(x)=V_0\Theta (-x)$. The numerical result (solid line) has been obtained by terminating the sum in Eq. (20) at $n=\pm 50$.