«Quasilinear Approximation and WKB∗ R. Krivec1, V.B. Mandelzweig2,∗∗, F. Tabakin3 1 J. Stefan Institute, PO Box 3000, 1001 Ljubljana, Slovenia ...»
Few-Body Systems 0, 52–57 (2003)
c by Springer-Verlag 2003
Printed in Austria
Quasilinear Approximation and WKB∗
R. Krivec1, V.B. Mandelzweig2,∗∗, F. Tabakin3
J. Stefan Institute, PO Box 3000, 1001 Ljubljana, Slovenia
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260,
Quasilinear solutions of the radial Schr¨dinger equation for diﬀerent po- o tentials are compared with corresponding WKB solutions. For this study, the Schr¨dinger equation is ﬁrst cast into a nonlinear Riccati form. While the o WKB method generates an expansion in powers of, the quasilinearization method (QLM) approaches the solution of the Riccati equation by approxi- mating its nonlinear terms by a sequence of linear iterates. Although iterative, the QLM is not perturbative and does not rely on the existence of any kind of smallness parameter. If the initial QLM guess is properly chosen, the usual QLM solution, unlike the WKB, displays no unphysical turning point sin- gularities. The ﬁrst QLM iteration is given by an analytic expression. This allows one to estimate analytically the role of diﬀerent parameters, and the inﬂuence of their variation on the boundedness or unboundedness of a criti- cally stable quantum system, with much more precision than provided by the WKB approximation, which often fails miserably for systems on the border of stability. It is therefore demonstrated that the QLM method is preferable over the usual WKB method.
1 Introduction and brief history The application of a very powerful approximation technique called the quasi- linearization method (QLM), which was introduced years ago by Bellman and Kalaba  in theory of linear programming, to various physics problems was discussed in a series of recent papers [1, 2]. The QLM approaches the solution of a nonlinear diﬀerential equation by approximating the nonlinear terms by a sequence of linear ones, and is not based on the existence of any smallness pa- rameter. Indeed, although iterative the QLM is not perturbative and it has very ∗ Article based on the presentation by V.B. Mandelzweig at the Workshop on the Dynamics and Structure of Critically Stable Quantum Few-Body Systems, Trento, 2003 ∗∗ E-mail address: firstname.lastname@example.org R. Krivec, V
fast quadratic convergence. It has been shown [1, 2] that the QLM solution sums an inﬁnite number of terms of the expansion in powers of the diﬀerent parameters and therefore well approximates the exact solution in a wide region of the parameter values.
When the structure of a critically stable quantum system is analyzed, understanding the analytic behavior of the solution as a function of diﬀerent physical parameters is often of decisive importance. In one-dimensional two-body problems, or in three-dimensional two-body problems with central potentials, such understanding is usually provided by the application of the WKB method, which supplies an approximate solution accurate up to some low (usually the ﬁrst) power of.
The derivation of the WKB solution starts by casting the radial Schr¨dinger o equation into nonlinear Riccati form and solving that equation by expansion in powers of. It is interesting instead to solve this nonlinear equation with the help of the quasilinearization technique and compare with the WKB results. Such a procedure was performed in the work , where it was shown that the ﬁrst QLM iteration reproduces the structure of the WKB series generating an inﬁnite series of the WKB terms, but with diﬀerent coeﬃcients. Besides being a better approximation, the ﬁrst QLM iteration is also expressible in a closed integral form. Similar conclusions are reached for all the higher QLM approximations and it can be shown  that the n-th QLM iteration yields the correct structure of the inﬁnite WKB series and reproduces 2n terms of the expansion of the solution in powers of exactly, as well as a similar number of terms approximately.
That the ﬁrst QLM iteration already provides a much better approximation to the exact solution than the usual WKB is obvious, not only from comparison of terms of the QLM and WKB series [4, 5], but also from the fact that the quantization condition in the ﬁrst QLM iteration leads to exact energies for many potentials ; namely, for the Coulomb, harmonic oscillator, P¨schl-Teller, o Hulthen, Hylleraas, Morse, Eckart and some other well known physical potentials, which have a simple analytic structure. By comparison, the WKB approximation reproduces exact energies only in the case of the ﬁrst two potentials.
The goal of the present work is to show that also in the general case of arbitrary potentials that do not have a simple analytic structure, both the wave functions and energies are very well reproduced by the ﬁrst QLM iteration and show signiﬁcant improvement over those obtained by the usual WKB approximation. In addition, we show that if the initial QLM guess is properly chosen, the wave function in the ﬁrst QLM iteration, unlike the WKB wave function, is free of unphysical turning point singularities. Since the ﬁrst QLM iteration is given by an analytic expression [1, 2, 4], it allows one to analytically estimate the role of diﬀerent parameters and the inﬂuence of their variation on boundedness or unboundedness of a critically stable quantum system with much more precision than obtained by the WKB approximation, which often fails miserably for systems on the border of stability. In addition, we show that ﬁve QLM iterations are usually enough to obtain both the wave function and energies with the extreme accuracy of ten signiﬁcant ﬁgures.
54 Quasilinear Approximation and WKB 2 Computation and the results
Figure 1. Comparison of the Langer WKB solution χL (dashed curve), the exact solution χexact (dotted curve) and the ﬁrst QLM iteration χ1 (solid curve) for the ground state of the anharmonic oscillator.
The last two are almost indistinguishable on the plot. Here x = κr, κ2 = 2mE/ 2.
2.1 Ground state of the anharmonic oscillator V (r) = 2 (r2 + r4 ) The exact energy of this state is 2.324406352 in atomic units with mass m = 1.
The WKB energy is diﬀerent by 2.14 % and equals 2.27460 in the same units, while the ﬁrst-iteration QLM energy equals 2.32575 and diﬀers from the exact energy only by 0.058 %. The ﬁfth-iteration QLM energy coincides with the exact energy in all ten digits.
The graphs corresponding to the Langer WKB solution, the exact solution and the ﬁrst QLM iteration are displayed in Fig. 1. One can see that while the Langer solution is noticeably diﬀerent from the exact solution, the curve of the ﬁrst QLM iteration is almost indistinguishable from the exact curve.
This could be followed more precisely by looking at Fig. 2 where the logarithm of the diﬀerence between the exact and WKB solutions and between the exact solution and the ﬁrst QLM iteration are shown. One can see that the diﬀerence between the exact solution and the ﬁrst QLM iteration is two orders of magnitude smaller than the diﬀerence between the exact and the WKB solutions, that is just one QLM iteration increases the accuracy of the result by a remarkable two orders of magnitude.
2.2 Second excited state of the linear potential V (r) = 27/2 r The exact energy in this case is 9.352429642 in atomic units. The WKB energy is diﬀerent by 0.49 % and equals 9.39863 in the same units. The ﬁrst-iteration QLM energy equals 9.3582123 and diﬀers from the exact one only by 0.062 %.
The QLM energy, as in the case of the anharmonic potential, coincides with the exact one after the ﬁfth iteration in all ten digits.
56 Quasilinear Approximation and WKB 0
Figure 2. Logarithm of the diﬀerences of the functions uexact and uL corresponding to the exact solution χexact and the WKB solution χL (dashed curve), and of uexact and u1 corresponding to the exact solution χexact and the ﬁrst QLM iteration χ1 (solid curve) for the ground state of the anharmonic oscillator.
u(x) is deﬁned as u(x) = − arctan(κχ(r)/χ (r)) = − arctan(E/y(λr)).
QLM iteration is performed on the monotonic function u(x), not on the singular function y(λr).
Figure 3. As in Fig.
2, but for the second excited state of the linear potential.
The accuracy of the WKB approximation increases for higher excitations.
Therefore in the case of the second excited state in the linear potential both the Langer WKB and QLM curves are indistinguishable from the exact one.
Fig. 3 shows, however, that also in this case the diﬀerence between the exact solution and the ﬁrst QLM iteration is two orders of magnitude smaller than the diﬀerence between the exact and WKB solutions.
R. Krivec, V. Mandelzweig and F. Tabakin 57
The quasilinear solutions of the radial Schr¨dinger equation for diﬀerent poteno tials are compared with the corresponding WKB solutions. It is shown by two typical examples that the use of the Langer WKB solution as initial guess already in the ﬁrst QLM approximation gives energies and wave functions two orders of magnitude more accurate than the WKB results. Such a QLM solution, unlike the usual WKB solution, displays no unphysical turning point singularities. The ﬁrst QLM iteration is given by an analytic expression. It allows one therefore to estimate analytically the role of diﬀerent parameters and their inﬂuence on boundedness or unboundedness of a critically stable quantum system with much higher precision than provided by the WKB approximation, which often fails miserably for systems on the border of stability. In addition, it was shown that ﬁve QLM iterations are usually enough to obtain both the wave function and energy with extreme accuracy of ten signiﬁcant ﬁgures.
Acknowledgement. The research was supported by the Bilateral Cooperation Program at the Ministry of education, science and sport of Slovenia (RK), by the Israeli Science Foundation grant 131/00 (VBM) and by the U.S. National Science Foundation (PHY-0139959) (FT).
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