Exact quantization of nonsolvable potentials: The role of the quantum phase

Semiclassical quantization is exact only for the so-called solvable potentials, such as the harmonic oscillator. In the nonsolvable case, the semiclassical phase, given by a series in $\hslash$, yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows us to obtain the quantum wave function while locally approaching the best predivergent semiclassical expansion. An iterative procedure allowing us to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.

Figure 1

(Color online) The total quantum phase $\sigma (s_2,E)∕\pi$ (solid curves) is plotted along with the semiclassical quantization curve $[S(t_2,E)-S(t_1,E)]∕\pi +1∕2$ (dashed lines) for different values of $\lambda$: from left to right $\lambda =0.01$ (red), 1 (pink), 5 (purple), 20 (blue), 50 (light blue), and 1000 (green). The exact ground state energy for $\lambda =1000$ obtained by solving $\sigma (s_2,E)∕\pi =1$ is shown by a box and the triangle shows the WKB quantized level off by 37%. Second-order quantization reduces the error to about 7% (depending on the terminant) and the $\hslash$ expansion diverges beyond this order.

Figure 2

(Color online) Derivative of the quantum phase ${\partial }_x\sigma$ (solid black curve) and of the classical action ${\partial }_{x}S=p$ (thick curve) in the 12-6 LJ potential characteristic of cold atom collisions (scaled a.u.). These quantities, plotted for the most excited bound state, are shown in the zone near the outer turning point $t_2\approx 9.5$. The inset shows the zone near the inner turning point $t_1\approx 0.9$ (note the very different scales).

Figure 3

(Color online) Exact energy levels (boxes) of the 12-6 LJ potential obtained by quantizing $\sigma (s_2,E)$ (solid curve), in scaled a.u. The inset zooms on the last state below threshold $(E=0)$ and also shows the WKB prediction, about 40% too low (triangle).