This investigation concerns some model potentials with very steep walls, for which the quantal Bohr–Sommerfeld half-integer quantization condition, if necessary generalized to correspond to modified phase-integral approximations of arbitrary order, and used without or with higher-order corrections included, can be used for obtaining, very accurately, the energy eigenvalues of the bound states, apart possibly from the lowest ones. One of the cases treated is the potential proportional to cot2x, for which a modified Bohr–Sommerfeld half-integer quantization condition yields the energy eigenvalues exactly. That the Bohr–Sommerfeld half-integer quantization condition is applicable to potentials with very steep walls may at first sight seem surprising in view of the well-known fact that the energy eigenvalues of a square-well potential with infinitely high walls are obtained exactly from the Bohr–Sommerfeld integer quantization condition, i.e., the quantization condition obtained by replacing (s+1/2) π by (s+1) π in the right-hand member of the Bohr–Sommerfeld half-integer quantization condition. From the study of a potential with horizontal bottom and linearly rising walls, which goes over into a square well when the inclination of the walls tends to zero, it can easily be understood, why the half-integer quantization condition is appropriate when the steep walls have a finite slope.
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.
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.
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.
We use old quantization to predict features of the energy spectrum for bound states in spherically symmetric potentials. We review the application of this method to simple quantum mechanical systems with known analytic results and show that old quantization, while not generally capable of predicting the exact quantum-mechanical spectrum, does consistently give useful information. We then apply the method to a logarithmic potential and compare the results with a numerically calculated spectrum. Finally, we use these techniques to predict the number of bound states in the Yukawa potential and again compare the results with those obtained numerically. In all cases old quantization provides good predictive power for the main features of the true quantum-mechanical problem, and thus can serve as a “back-of-the-envelope” technique for providing rough, qualitative information about the system.
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]).