Quantum gravity and the Coulomb potential

We apply a singularity-resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on $\mathbb{R}$ with the singular $-1/|x|$ potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite-difference approximation of the Schrödinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at $x=0$, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized.

Figure 1

The solid line is the Coulomb potential and the dashed line is the lattice regularized potential (17) for $\lambda =0.1$.

Figure 2

The coefficient $c_0$ as a function of $k=1/\sqrt{-4ϵ}$ for $0.98\le k\le 2.2$, with $\lambda =0.01$. The zeroes are near $k=1$ and $k=2$.

Figure 3

The coefficient $c_0$ as a function of $k=1/\sqrt{-4ϵ}$ for $1.98\le k\le 3.2$, with $\lambda =0.01$. The zeroes are near $k=2$ and $k=3$.

Figure 4

The lowest eigenenergy as a function of $\lambda$ for $0.005\le \lambda \le 0.3$. The vertical error bar of each point is ${10}^{-4}$. Convergence to $ϵ=-0.25$ is apparent as $\lambda \to 0$.

Figure 5

The second-lowest eigenenergy as a function of $\lambda$ for $0.005\le \lambda \le 0.3$. The vertical error bar of each point is $5\times {10}^{-5}$. Convergence to $ϵ=-0.0625$ is apparent as $\lambda \to 0$.