The Wentzel-Kramers-Brillouin Method

Substituting this expansion in Equation 27-29 and equating the coefficients of the successive powers of h/2iri to zero, we obtain the equations 

The first two terms when substituted in Equation 27-28 lead to the expression 

The probability distribution function to this degree of approximation is therefore 

The approximation given in Equation 27-34 is obviously not valid near the classical turning points of the motion, at which W — V. This is related to the fact that the expansion in Equation 27-30 is not a convergent series but is only an asymptotic representation of y, accurate at a distance from the points at which W = V. 

For systems of the type under discussion, the second term introduces half-quantum numbers i.e., with y = y0 + 

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DIV.2a. I. Prigogine et G. Garikian, Sur le calcul des niveaux energetiques par la methode de Wentzel-Kramers-Brillouin et son application a I hydrogene liquide (On the calculation of the energy levels by the Wentzel-Kramers-Brillouin method and its application to liquid hydrogen), J. Phys. 330-332 (1947). 

In the approximation of the Wentzel-Kramers-Brillouin method, we take fi = k(l)unf a) from the Landau theory on the hydrodynamic instability of the plane flame. The relative change in the disturbance because of the stretch-effect is equal to that in the tangential velocity component... 

The effects of X7 may be treated using ordinary non-degenerate perturbation theory or, as in Dunham s original work, by means of the Wentzel-Kramers-Brillouin method... 

The calculation of the electronic-transmission factor currently involves three different methods, viz. the Landau-Zener formula, Fermi s golden mle [35], and electron tunneling formalism such as the Wentzel-Kramer-Brillouin method [36]. We used the Landau-Zener formula [37,38] to calculate it ... 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

Field emission is a tunneling phenomenon in solids and is quantitatively explained by quantum mechanics. Also, field emission is often used as an auxiliary technique in STM experiments (see Part II). Furthermore, field-emission spectroscopy, as a vacuum-tunneling spectroscopy method (Plummer et al., 1975a), provides information about the electronic states of the tunneling tip. Details will be discussed in Chapter 4. For an understanding of the field-emission phenomenon, the article of Good and Muller (1956) in Handhuch der Physik is still useful. The following is a simplified analysis of the field-emission phenomenon based on a semiclassical method, or the Wentzel-Kramers-Brillouin (WKB) approximation (see Landau and Lifshitz, 1977). 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

In another series of papers [26] Shin has used the WKB (Wentzel-Kramers-Brillouin) method for evaluating the vibrational transition matrix element, employing various forms of interaction potential. Comparisons are made with quantum-mechanical solutions. 

This quasiperiodicity can be explained as follows. By applying the fitting procedure, based on the WKB (Wentzel-Kramers-Brillouin) method, one can find that the smallest positive root x = x(jS+1) of the Hermite polynomial Hes+i(x) is approximately equal to... 

Refractive-Index Profile. The refractive-index profile was approximated by the conventional power law. The output pulse width from the GI POF was calculated by the Wentzel-Kramers-Brillouin (WKB) method (10) in which both modal and material dispersions were taken into account as shown in Equations (3), (4), and (5). Here, aintemodai cTintramodai, and CTtotai signify the root mean square pulse width due to the modal dispersion, intramodal (material) dispersion, and both dispersions, respectively. 

For high radial number n or orbital momentum /, the Wentzel-Kramers-Brillouin (WKB) method is known to provide an excellent approximation. In fact, the WKB approximation works already surprisingly well for small values of n or /. We refer here to the review of Quigg and Rosner [17], who present a detailed study of the WKB approximation for confining potentials. Let us simply recall here some basic results. In the WKB approximation, the eigenenergies of S-waves are given by... 

The crucial factor defining the refractive index profile in GI POFs is the coefficient g, and the optimum value for maximizing the bandwidth can be determined from the modal and material dispersions [79-81]. From analyses using the Wentzel-Kramers-Brillouin (WKB) method, the modal dispersion niate-rial dispersion total dispersion fftotai can be expressed as follows ... 

Still another method, namely use of the WKB (Wentzel-Kramers-Brillouin) theory, has been suggested by Englman and Ranfagni (1980) for use with multidimensional or complex energy surfaces. 

Numerical methods for finding bound-state solutions of Equation 1.15 are described in Section 1.3.1.3. However, in conceptual terms a considerable amount may be understood in terms of semiclassical arguments [7]. In semiclassical methods, the Schrodinger equation is expanded semiclassically in powers of h. The resulting first-order JWKB (Jeffreys-Wentzel-Kramers-Brillouin) quantization condition gives remarkably accurate results for the vibration-rotation energies E l of diatomic molecules ... 

For an arbitrary continuous potential barrier, an approximate general expression for transition probability (barrier permeability) can be derived using the approach of ZWAAN-KEMBIE /60/, which is a generalization of the familiar BWK (BRILLOUIN-WENTZEL-KRAMERS) method/6l/. 

This approach is named after its founders—Wentzel (1926), Kramers (1926), Brillouin (1926a,b), and Jeffreys (1925). The WKBJ method is one of the powerful approximate approaches of quantum mechanics. Although in the present discussion we are concerned only with its application to obtaining approximate eigenvalues for bound states of the onedimensional Schrodinger equation, such application does not cover all of its range and its force. 

Named after G. Wentzel [Zeits. f. Phys, 55, 518 (1926)], H. A. Kramers [Zeits. f Phys, 39, 828 (1926)] and L. Brillouin [Comptes Rendus 183, 24 (1926)] who independently applied this method to problems involving the Schrodinger equation in the early days of quantum mechanics. 

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