Hydrogen-atom spectrum under a minimal-length hypothesis

The energy spectrum of the Coulomb potential with minimal length commutation relations $[X_i,P_j]=i\hslash \{{\delta }_{ij}(1+\beta P^2)+{\beta }^{\prime }{P}_i{P}_j\}$ is determined both numerically and perturbatively for arbitrary values of ${\beta }^{\prime }∕\beta$ and angular momenta $\ell$. The constraint on the minimal length scale from precision hydrogen spectroscopy data is of the order of a few ${\mathrm{GeV}}^{-1}$, weaker than previously claimed.

Figure 1

Plot of energy eigenvalues of the Coulomb potential in units of the regular ground-state energy as a function of the minimal length in units of the Bohr radius $\xi =\hslash \sqrt{\beta +{\beta }^{\prime }}∕a_0$. For principal quantum numbers $n=1$, 2, 3, two extreme cases are represented: ${\beta }^{\prime }=0$ (thick lines) and $\beta =0$ (thin lines). Continuous line, $\ell =0$; dotted line, $\ell =1$; dash-dotted line, $\ell =2$.

Figure 2

Comparison of different results for (a) $2P$ states with $\eta =0$, 1 and (b),(c) $1S$ states with $\eta =0,1∕3,1$. Solid lines, numeric result; dotted lines, perturbative result; dash-dotted lines, Ref. 6; double-dashed lines, Ref. 7. The perturbative expression is given for (a) by (16) and for (b),(c) by formula (17), with the coefficient $C$ chosen such that it agrees with the numerics at ${\xi }^2={10}^{-6}$.

Figure 3

Constraint on the minimal length obtained as a function of $\eta$, including the two highest-order terms (continuous line) and just the leading-order term (dash-dotted line).