2 edition of **Modified Airy function and WKB solutions to the wave equation** found in the catalog.

Modified Airy function and WKB solutions to the wave equation

A. K. Ghatak

- 281 Want to read
- 9 Currently reading

Published
**1991** by National Institute of Standards and Technology in [Gaithersburg, Md.] .

Written in English

- Wave equation -- Numerical solutions.,
- Airy functions.,
- WKB approximation.

**Edition Notes**

Statement | A,K. Ghatak, R.L. Gallawa, I.C. Goyal. |

Series | NIST monograph ;, 176 |

Contributions | Gallawa, R. L., Goyal, I. C., National Institute of Standards and Technology (U.S.) |

Classifications | |
---|---|

LC Classifications | QA927 .G43 1991 |

The Physical Object | |

Pagination | ix, 164 p. : |

Number of Pages | 164 |

ID Numbers | |

Open Library | OL1322778M |

LC Control Number | 92199381 |

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The Wentzel-Kramer-Brillouin (WKB) method has served in problems of quantum mechanics and nonuniform optical waveguides, the discipline treated in the monograph. The monograph discusses the WKB and the Modified Airy Function (MAF) methods in considerable detail, to the end that the reader gains an appreciation of the strengths and weaknesses of :// G/abstract.

Get this from a library. Modified Airy function and WKB solutions to the wave equation. [A K Ghatak; R L Gallawa; I C Goyal; National Institute of Standards and Technology (U.S.)] The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted :// Modified Airy function and WKB solutions to the wave equation A,K.

Ghatak, R.L. Gallawa, I.C. Goyal （NIST monograph, ） National Institute of Standards and Technology, E Airy Function The Airy function plays a central role in the construction of WKB wave functions. Here, we briefly review the essential properties of this function. Moreover, we derive asymptotic expansions in various domains, discuss the Stokes and anti-Stokes lines and address the Stokes phenomenon.

E.I Definition and Differential Equation A.K. Ghatak, R.L. Gallawa and I.C. Goyal, Modified Airy Function and WKB Solutions to the Wave Equation, National Institute of Standards & Technology Monograph #, US Government Printing Office, Washington ().

Google Scholar stability of solutions to certain PDEs, in particular the wave equation in its various guises. Most of the equations of interest arise from physics, and we will use x,y,z as the usual spatial variables, and t for the the time There are important jobs in the literature equation Airy, such book of Vallee Olivier and Soares Manuel [1].

This book contains the relation of the Airy function with special functions; for example, the Bessel function. To study the Schrödinger equation in relation to the Airy equation is advisable to consult [2] and [3]. NationalInstituteofStandardsandTechnologyMonograph ,pages(Nov) CODEN:NIMOEZ MENTPRINTINGOFFICE WASHINGTON equation (from the point of view of the stationary phase approximation), see the book review by Marsden and Weinstein of the book Geometric asymptotics by Guillemin and Sternberg.

It appeared in the Bulletin of the AMS in May The main goal of these lectures is to provide a motivation for the construction of approxi-mate ://~dbaskin/ Accurate solutions to Schrodinger's equation using modified Airy functions is very simple and allows a very accurate description of bound-state wave functions and the corresponding eigenvalues.

It is also demonstrated that the eigenvalues can be determined from the tabulated zeros of the Airy function with as much ease as the WKB method Equation () is referred to as the Airy integral and can be shown to be the solution to a homogeneous differential equation of the type d2y dx2 ¼ xy: () This equation is generally known as the Airy equation or the Airy differential equation.

However, caution must be exercised in The monograph discusses the WKB and the Modified Airy Function (MAF) methods in considerable detail, to the end that the reader gains an appreciation of the strengths and weaknesses of :// A historical background of Airy functions in physics, their derivatives and zeroes are presented in this paper.

As application, we derive the odd and even solutions for the symmetric V-shape potential. The Airy function in Physics. The Airy function was introduced in by Sir George Biddell Airy For the sake of completeness we’ll close out this section with the 2-D and 3-D version of the wave equation.

We’ll not actually be solving this at any point, but since we gave the higher dimensional version of the heat equation (in which we will solve a special case) we’ll give this as well. The 2-D and 3-D version of the wave equation is, The WKB approximation, named after Gregor Wentzel, Hendrik Anthony Kramers, and Leon Brillouin (it is sometimes called the JWKB approximation, where the “J” stands for Harold Jeffreys), is a technique for obtaining approximate solutions to the time-independent Schrödinger equation in one dimension (and for radially symmetric problems in 3D).

Approximate solutions to the scalar wave equation: The decomposition method Article (PDF Available) in Journal of the Optical Society of America A 15(5) May with Reads Airy stress function. Numerous solutions to plane strain and plane stress problems can be determined through the use of a particular stress function technique.

The method employs the Airy stress function and will reduce the general formulation to a single governing equation in terms of a single unknown. The resulting governing equation is WKB_refract.m. Plot WKB solution for refraction of a wave incident at 45 degrees on a erf ramp of refractive index from 1 to 2.

WKB_reflect.m. Plot matched WKB/Airy solution for reflection of a wave incident at 45 degrees on a erf ramp of refractive index from 2 to 1. ://~breth/classes/AM Abstract. Appropriate second order Linear Differential Equations (LDEs) which can be solved exactly by the Modified Airy Function (MAF) method, which normally gives approximate solutions in terms of Airy functions for Schrodinger-like LDEs, have been studied in WKB-wave function, which is by itself an asymptotic series.

In exact-WKB, the position is rst elevated to a complex variable, and the classical potential is used to turn the x2C plane into Stokes graph and regions.4 Once one considers the analytic continuation of the WKB-solution of (x) from one Stokes region to an adjacent one in complex xplane, In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (–92).

The function Ai(x) and the related function Bi(x), called the Airy function of the second kind and sometimes referred to as the Bairy function, are linearly independent solutions to the differential equation and this can only be true if the fractional change in kinetic energy is negligible, i.e., over one de Broglie the change in the kinetic energy is minus the change in the potential energy, the conclusion is that the WKB approximation should be valid when the wave function varies much more rapidly than the potential.

This rapid variation of the wave function relative to the common method is to take the WKB wave functions away from the zeros of the poten-tial, and to glue them together with Airy functions, which approximate the solutions near the zeros (see for example the textbook [9, Section ] or the book [5]).

Usually, the potential in the Schro¨dinger equation is real-valued. However, there are situations In summary, in the overlap region, we find WKB solutions and the asymptotic values of the solution to the Schrodinger equation with a linear approximation to the potential.

Then, by making (78) and (79) match (72) and (73), respectively, the WKB connection Inside vs outside, there is a sign change inside the square root, so that changes the nature of the "phase" $\phi(r)$.

Normally, when you match wave functions you require that $\psi_\mathrm{left}(x) = \psi_\mathrm{right}(x)$ (continuity) and that the derivative changes according to what you get when you integrate the Schrodinger equation: $\int_\mathrm{left}^\mathrm{right}\!dx\, \left(-\frac Web of Science You must be logged in with an active subscription to view :// @article{osti_, title = {Modified semi-classical methods for nonlinear quantum oscillations problems}, author = {Moncrief, Vincent and Marini, Antonella and Maitra, Rachel}, abstractNote = {We develop a modified semi-classical approach to the approximate solution of Schroedinger's equation for certain nonlinear quantum oscillations :// The Semiclassical Approximation to Leading Order.

Consider a particle moving along in a slowly varying one-dimensional potential. By “slowly varying” we mean here that in any small region the wave function is well approximated by a plane wave, and that the wavelength only changes over distances long compared with a ://:_Quantum_Mechanics.

The previous equation is the Bessel equation. At z!0 it becomes the equipotent equation: 1 z @ @z z @R @z s2 z2 R= 0 (18) which can be solved explicitly: R= C 1z s+ C 2z (19) One can seek a solution of (17) in the form R= z 2 s F(z;s) (20) Fsatis es the equation: F00+ 2s+ 1 z F0+ F= 0 (21) The solution of equation (21) can be found in the form ~zakharov/ The wave function from solution of Schrodinger equation by MAF method isr3]: where Ai is the Airy function, A is constant and: in which V .) is potential, Ei is energy level.

xo is the position where Ei = v (x,). The energy level 4 is obtained from the boundary condition of Yi (O)=O. That is, one needs to solve Ei from the equation: Quasilinearization Method and Summation of the WKB Series R.

Krivec1 and V. Mandelzweig2 2 J. Stefan Institute, P.O. BoxLjubljana, Slovenia Racah Institute of Physics, Hebrew University, JerusalemIsrael 1 arXiv:math-ph › 百度文库 › 行业资料.

The Airy function is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field.

For the same reason, it also serves to provide uniform semiclassical approximations near a turning point in the WKB method, when the potential may be locally of the Airy function and the second independent solution of y'' - xy = 0. From Figure we find that the electric field is related to the energy difference on both sides of the well: qE(i,n)Lx =Ei −Ei−n () where the subscript i refers to the ith zero of the Airy function and the subscript n to the nth quantized energy level in ~bart/book/book/chapter1/pdf/ We apply the transmutation method to give a new explanation of the Stokes phenomenon for the Airy differential equation and of the change of the coeffcients in its asymptotic solutions for large values of argument in different parts of the complex plane.

As a transmutation operator, a Weyl type fractional order integral is used. But this scheme is a special case of the so-called Poisson dependent Schr¨odinger equation, for which see Sec.

This section oﬀers an intuitive presentation of the WKB ansatz, which is a description of how the wave function depends on ¯h, and the beginnings of the classical picture associated with it.

The time-independent Schr¨odinger equation that we will use is − ¯h2 2m ∇ Modified Airy Function Method for the Analysis of We consider an approximate solution to the wave equation appropriate to the optical waveguides encountered in practice. The refractive-index profile may be arbitrary, and the geometry may be two or three dimensional.

A circular or a planar waveguide could thus be treated by this Simão Correia, Raphaël Côte, Luis Vega, Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation, Journal de Mathématiques Pures et Appliquées, /, (). Quantum tunnelling or tunneling (see spelling differences) refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount.

This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun. [1] It has important applications to modern devices such as the tunnel {{#invoke:Hatnote|hatnote}} In mathematical physics, the WKB approximation or WKB method is a method for finding approximate solutions to linear partial differential equations with spatially varying coefficients.

It is typically used for a semiclassical calculation in quantum mechanics in which the wavefunction is recast as an exponential function, semiclassically expanded, and then either the.

The U.S. Department of Energy's Office of Scientific and Technical This paper concerns a certain class of two-dimensional solutions to four generic partial differential equations—the Helmholtz, modified Helmholtz, and