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arXiv:2006.01434

Exact Quantization via Summation of Wentzel-Kramers-Brillouin Series to All Orders

B. M. Tripathi

Published Jun 2, 2020• 0 citations
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Abstract

We present, for the first time ever, with the help of two conjectures, an \textit{exact} quantization condition for two turning-point eigenvalue problems (that recurrently appear in several branches of science) by explicitly summing up the Wentzel-Kramers-Brillouin series to all orders. Surprisingly, the quantization condition looks elegant and resembles exactly the Bohr-Sommerfeld quantization relation with just an extra term having a square of Planck's constant (appearing as a form of repulsive quantum potential). We also find that our exact quantization relation produces Langer-modification for spherically symmetric $3-$dimensional potentials.

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We present, for the first time ever, with the help of two conjectures, an \textit{exact} quantization condition for two turning-point eigenvalue problems (that recurrently appear in several branches of science) by explicitly summing up the Wentzel-Kramers-Brillouin series to all orders. Surprisingly, the quantization condition looks elegant and resembles exactly the Bohr-Sommerfeld quantization relation with just an extra term having a square of Planck's constant (appearing as a form of repulsive quantum potential). We also find that our exact quantization relation produces Langer-modification for spherically symmetric 3−3-3−dimensional potentials.
First, two conditions are specified for the lowest order Wentzel–Kramers–Brillouin quantization rule to yield exact results. These rules are related to the periodic orbit decomposition of the quantum density of states. This approach is then applied to supersymmetric quantum mechanics. It leads to a new derivation of the result that shape-invariant potentials give exact results when the classical action is calculated with the square of the super potential, but without the Maslov index or the Langer correction.PACS Nos.: 03.65.Sq, 12.60.Jv
We extend the Wentzel–Kramers–Brillouin (WKB) approximation to the semi-relativistic spinless Salpeter equation. The Bohr–Sommerfeld quantization rule for a one-dimensional potential is derived by properly connecting the WKB wave functions. Since the latter diverges at the classical turning points (TPs), we develop a uniform WKB approach that yields an approximate wave function valid everywhere including the TPs, along with a modified quantization condition. The method is illustrated for the symmetric linear potential, and comparison with exact numerical results shows excellent agreement, significantly improving upon the standard WKB method.
We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schrödinger equation. Based on an ansatz for the wave function, subject to an auxiliary condition on its first derivative, we obtain a set of coupled differential equations that decouple, via an alternating perturbation method, into the well-known Bremmer series. The perturbation improves the wave function amplitudes alternately, and its phase is refined through recursive diagonalization. From this construction, we derive a general quantization formula that encodes the geometric-optics-like physics of the system. When the ratio of the differential reflection coefficient to the classical momentum remains constant, the formula reduces to a closed-form quantization condition consistent with that obtained by re-summing the WKB series to all orders.
Extensions of the Bohr–Sommerfeld quantization formula to smooth double wells are established which preserve the formula’s generality and simplicity as well as its accuracy and the ease of use. Moreover, the proposed quantization equations for double wells, while being free from the limitations of the Wentzel–Kramers–Brillouin approximation which is restricted to deep energy levels alone, are equally well suited for determining higher energy levels lying in the vicinity of the barrier’s top as well as above the latter, and they have the correct connection to the conventional Bohr–Sommerfeld formula as the particle’s energy becomes large enough. As a particular application, a simple quantization equation for the pure quartic oscillator is derived which yields correct energy eigenvalues for all eigenstates of the oscillator including the ground one, with relative errors being all less than 1%. Conditions for the validity of the extended quantization equations are discussed and their numerical verification is made.
A discussion of the Wentzel-Brillouin-Kramers method of obtaining an approximate solution of the wave equation is given from the point of view that it forms a link between the quantum theory of Bohr and the new quantum mechanics. This becomes clear when one compares the probability distributions (see Figs. 2 and 3) and calculates the mean values rk−\stackrel{-}{{r}^{k}}rk− (see Table I). It is shown that for high quantum numbers these become the classical values. The method leads also, as shown by Zwaan, to quantization by the classical phase integrals with the use of half-integer quantum numbers. In the last section the method is applied to the charged shell atom model. It is shown that the condition of smooth joining of the wave function is practically equivalent to half-integer quantization of the sum of the inner and outer phase integrals. Of course there is no longer a sharp distinction between penetrating and non-penetrating orbits. The Land\'e formula for the doublet separations is derived. The value of ${\ensuremath{\psi}}^{2}(0)$ (0), which occurs in the calculation of the hyperfine separations in 2S^{2}S2S states, is also given.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.