For one-dimensional power-like potentials [Formula: see text], [Formula: see text], the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for [Formula: see text] but below them for [Formula: see text]. The ground state BSE as function of [Formula: see text] are of the same order of magnitude as the exact energies for linear [Formula: see text], quartic [Formula: see text] and sextic [Formula: see text] oscillators but their relative deviation grows with [Formula: see text], reaching the value 4 at [Formula: see text]. For physically important cases [Formula: see text], for the 100th excited state BSE coincide with exact ones in 5–6 figures. It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction [Formula: see text] (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function [Formula: see text] for all [Formula: see text]. It grows slowly with increasing [Formula: see text] for fixed quantum number [Formula: see text], while it decays with quantum number growth at fixed [Formula: see text]. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of [Formula: see text] (and [Formula: see text]).
For one-dimensional power-like potentials [Formula: see text], [Formula: see text], the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for [Formula: see text] but below them for [Formula: see text]. The ground state BSE as function of [Formula: see text] are of the same order of magnitude as the exact energies for linear [Formula: see text], quartic [Formula: see text] and sextic [Formula: see text] oscillators but their relative deviation grows with [Formula: see text], reaching the value 4 at [Formula: see text]. For physically important cases [Formula: see text], for the 100th excited state BSE coincide with exact ones in 5–6 figures. It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction [Formula: see text] (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function [Formula: see text] for all [Formula: see text]. It grows slowly with increasing [Formula: see text] for fixed quantum number [Formula: see text], while it decays with quantum number growth at fixed [Formula: see text]. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of [Formula: see text] (and [Formula: see text]).
Radial power-like potentials: from the Bohr-Sommerfeld ℓ=0 state energies to the exact ones
Two new Bohr-Sommerfeld (B-S) type quantization conditions for d-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{V=sign(m)rm} for m=−3/2,−1,−1/2,1,3,4,6 and V=logr (that corresponds to m=0) are studied in detail for radial quantum numbers nr∈[0,100] and for angular momenta ℓ=1,2,3,4,5 in dimensions d=2,3. It is shown that for large nr the B-S energies EBS always tend to the exact ones. For m=2 (the radial harmonic oscillator) and m=−1 (the d-dimensional analogue of the 3D Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of EBS with the energies EBS(L), obtained for the radial potentials modified by the Langer correction for d=3, are in good agreement in general: in detail, it was checked for the quartic oscillator V=r4. It is demonstrated that introducing a dimensionless {\it WKB correction} γ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with γ being very small for positive degrees m≥0 and sufficiently small for non-positive ones m<0 for all nr,ℓ studied.
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.
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.
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.
An algebraic reformulation of the Bohr - Sommerfeld (BS) quantization rule is suggested and applied to the study of bound states in one-dimensional quantum wells. The energies obtained with the present quantization rule are compared to those obtained with the usual BS and WKB quantization rules and with the exact solution of the Schrödinger equation. We find that, in diverse cases of physical interest in molecular physics, the present quantization rule not only yields a good approximation to the exact solution of the Schrödinger equation, but yields more precise energies than those obtained with the usual BS and/or WKB quantization rules. Among the examples considered numerically are the Poeschl - Teller potential and several anharmonic oscillator potentials, which simulate molecular vibrational spectra and the problem of an isolated quantum well structure subject to an external electric field.
The generalized Bohr-Sommerfeld quantization condition up to the ninth-order approximation is given explicitly, in terms of complete elliptic integrals, for two kinds of quartic oscillators V(z)=v2z2+v4z4: (i) for the quartic double barrier, where v2>0 and ${\mathit{v}}_{4}$0, and the energy is below the potential maximum; and (ii) for the quartic double well, where ${\mathit{v}}_{2}$0 and v4>0. Resonance energies for some quartic double barriers and energy levels for some quartic double wells were evaluated by solving the quantization condition numerically. In some cases the results are very good.