Heisenberg uncertainty relation for position and momentum beyond central potentials

Recently, Sánchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\sigma }_r{\sigma }_p\ge 3\hslash /2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\sigma }_r{\sigma }_p$ which involves the mean value and the variance of the square of the angular momentum operator.

Figure 1

The dependence on $L_{\mathrm{inv}}$ of the new bound (6) for some fixed values of ${\mathcal{R}}_{\mathrm{inv}}$: ${\mathcal{R}}_{\mathrm{inv}}={10}^3$, red dashed; ${\mathcal{R}}_{\mathrm{inv}}={10}^4$, green dashed-dotted; ${\mathcal{R}}_{\mathrm{inv}}={10}^5$, blue dotted. The black line represents the reference bound (2). All bounds are above the Heisenberg bound $3/2$.