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
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 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.
Abstract Using both WKB approximation methods and complete eigenstate solutions, we examine four standard, textbook level, bound state problems in quantum mechanics. We show how the action integral from the Bohr–Sommerfeld quantization condition, extended to the WKB energy quantization condition, can be expressed in the language of semi-classical and exact quantum results. Relating classical time averaging and quantum expectation value methods, the action integral can be used to connect the average value of speed squared times period across such systems giving ⟨ v 2 ⟩ n τ n = ( n + C ~ ) ( h / m ) , where n is the bound state quantum number ( n = 0 , 1 , 2 , . . . ) supplemented by the Maslov correction ( C ~ ) . We then extend this analysis using exact results for an infinite hierarchy of potentials, one derived from the repeated supersymmetrization of the familiar infinite well, as a more mathematically sophisticated example. These discussions provide a novel pedagogical twist on h as the ‘quantum of action’, accessible at several levels of the undergraduate quantum mechanics curriculum.
Abstract. – An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to Nπ, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom. In the development of quantum mechanics, the Bohr-Sommerfeld quantization rules of the old quantum theory [1] occupy a position intermediate between classical and quantum mechanics. The WKB approximation [2–4] is a method for the approximate treatment of the Schrödinger wave function with another quantization rule [1]. Both quantization rules are approximate. We are going to derive an exact quantization rule for the one-dimensional Schrödinger equation: d2 ψ(x) = −2M dx2 2 [E − V (x)] ψ(x), (1) ¯h where M is the mass of the particle, and the potential V (x) is a piecewise continuous real function of x satisfying for definiteness V (x)> E, − ∞ < x < xA or xB < x < ∞, V (x) = E, x = xA or x = xB, V (x) < E, xA < x < xB, (2) where xA and xB are two turning points. Between two turning points, the momentum is k(x) = √ 2M [E − V (x)]/¯h.
By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that a necessary condition for the first-order integral to give an exact quantization rule is the vanishing of the higher-order integrals. When this condition is satisfied, the resulting quantization condition is of the form of all previously known exact-quantization rules. The higher-order integrals are shown to vanish for certain cases of interest. An error in Paper I [C. Rosenzweig and J. B. Krieger, J. Math. Phys. 9, 849 (1968)] is noted and an examination of the higher-order integrals shows that a proposed quantization rule given there is not exact.
WKB theory provides a plausible link between classical mechanics and quantum mechanics in its semi-classical limit. Connecting the WKB wave function across the turning points in a quantum well, leads to the WKB quantization condition. In this work, I focus on some improvements and recent developments related to the WKB quantization condition. First I discuss how the combination of super-symmetric quantum mechanics and WKB, gives the SWKB quantization condition which is exact for a large class of potentials called shape invariant potentials. Next I turn to the fact that there is always a probability of refection when the potential is not constant and the phase of the wave function should account for this refection. WKB theory ignores refection except at turning points. I explain the work of Friedrich and Trost who showed that by including the correct "refection phase" at a turning point, the WKB quantization condition can be made to give exact bound state energies. Next I discuss the work of Cao and collaborators which takes refection of the wave function into account everywhere. We show that Cao's method provides a way to compute the F-T refection phase. Finally I discuss a paper of Fabre and Guery-Odelin who used the exponential potential to study the accuracy of WKB. In their results the accuracy deteriorates as the energy increases, which is inconsistent with Bohr's correspondence principle. Using the Friedrich and Trost method, we resolved this problem.
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− (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 states, is also given.