Scattering in a three-dimensional fuzzy space

We develop scattering theory in a noncommutative space defined by an $su(2)$ coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spatial noncommutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of noncommutativity on the well’s spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well depths the potential effectively becomes repulsive and that the cross section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle’s wavelength inside the well becomes comparable to the noncommutative length scale.

Figure 1

Illustrations of the various energy regions and the corresponding behavior of the wave function as discussed in the text. The gray regions do not support bound or scattering states. The second, dimensionless energy scale appearing on the right will be used for the scattering analysis. We use the shorthand $M^{\prime }\equiv M+1$. (a) and (b) correspond to the shallow and deep well scenarios respectively.

Figure 2

See discussion in text. (a) The total cross section as a function of incident energy with $U{R}^2=15$. (b) The phase shift of the $l=1$ channel as a function of the well depth at $kR=0.1$. (c) The total cross section as a function of the incident energy for a finite attractive well and a repulsive hardcore potential. Here $M=100$ and $U{R}^2=40768$. (d) The total cross section as a function of the incident energy for a repulsive hardcore potential.