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1aCambridge :bCambridge University Press,c2002. 4c©2002. a1 online resource (230 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier0 aCover -- Half-title -- Title -- Copyright -- Contents -- Preface -- 1 Historical survey -- 1.1 Development from 1817 to 1926 -- 1.1.1 Carlini's pioneering work -- 1.1.2 The work by Liouville and Green -- 1.1.3 Jacobi's contribution towards making Carlini's work known -- 1.1.4 Scheibner's alternative to Carlini's treatment of planetary motion -- 1.1.5 Publications 1895-1912 -- 1.1.6 First traces of a connection formula -- 1.1.7 Publications 1915-1921 -- 1.1.8 Both connection formulas are derived in explicit form -- 1.1.9 The method is rediscovered in quantum mechanics -- 1.2 Development after 1926 -- 2 Description of the phase-integral method -- 2.1 Form of the wave function and the q-equation -- 2.2 Phase-integral approximation generated from an unspecified base function -- 2.3 F-matrix method -- 2.3.1 Exact solution expressed in terms of the F-matrix -- 2.3.2 General relations satisfied by the F-matrix -- 2.3.3 F-matrix corresponding to the encircling of a simple zero of… -- 2.3.4 Basic estimates -- 2.3.5 Stokes and anti-Stokes lines -- 2.3.6 Symbols facilitating the tracing of a wave function in the complex z-plane -- 2.3.7 Removal of a boundary condition from the real z-axis to an anti-Stokes line -- 2.3.8 Dependence of the F-matrix on the lower limit of integration in the phase integral -- 2.4 F-matrix connecting points on opposite sides of a well-isolated turning point, and expressions for the wave function in… -- 2.4.1 Symmetry relations and estimates of the F-matrix elements -- 2.4.2 Parameterization of the matrix… -- 2.4.2.1 Changes of Alpha, Beta and Gamma when x1 moves in the classically forbidden region -- 2.4.2.2 Changes of Alpha, Beta and Gamma when x2 moves in the classically allowed region -- 2.4.2.3 Limiting values of Alpha, Beta and Gamma -- 2.4.3 Wave function on opposite sides of a well-isolated turning point.8 a2.4.4 Power and limitation of the parameterization method -- 2.5 Phase-integral connection formulas for a real, smooth, single-hump potential barrier -- 2.5.1 Exact expressions for the wave function on both sides of the barrier -- 2.5.2 Phase-integral connection formulas for a real barrier -- 2.5.2.1 Wave function given as an outgoing wave to the left of the barrier -- 2.5.2.2 Wave function given as a standing wave to the left of the barrier -- 3 Problems with solutions -- 3.1 Base function for the radial Schrödinger equation when the physical potential has at the most a Coulomb singularity at… -- 3.2 Base function and wave function close to the origin when the physical potential is repulsive and strongly singular at… -- 3.3 Reflectionless potential -- 3.4 Stokes and anti-Stokes lines -- 3.5 Properties of the phase-integral approximation along an anti-Stokes line -- 3.6 Properties of the phase-integral approximation along a path on which the absolute value of exp[iw(z)] is monotonic in… -- 3.7 Determination of the Stokes constants associated with the three anti-Stokes lines that emerge from a well-isolated… -- 3.8 Connection formula for tracing a phase-integral wave function from a Stokes line emerging from a simple transition zero… -- 3.9 Connection formula for tracing a phase-integral wave function from an anti-Stokes line emerging from a simple transition… -- 3.10 Connection formula for tracing a phase-integral wave function from a classically forbidden to a classically allowed… -- 3.11 One-directional nature of the connection formula for tracing a phase-integral wave function from a classically… -- 3.12 Connection formulas for tracing a phase-integral wave function from a classically allowed to a classically forbidden… -- 3.13 One-directional nature of the connection formulas for tracing a phase-integral wave function from a classically allowed….8 a3.14 Value at the turning point of the wave function associated with the connection formula for tracing a phase-integral… -- 3.15 Value at the turning point of the wave function associated with a connection formula for tracing the phase-integral… -- 3.16 Illustration of the accuracy of the approximate formulas for the value of the wave function at a turning point -- 3.17 Expressions for the a-coefficients associated with the Airy functions -- 3.18 Expressions for the parameters Alpha, Beta, and Gamma when… -- 3.19 Solutions of the Airy differential equation that at a fixed point on one side of the turning point are represented by a… -- 3.20 Connection formulas and their one-directional nature demonstrated for the Airy differential equation -- 3.21 Dependence of the phase of the wave function in a classically allowed region on the value of the logarithmic derivative… -- 3.22 Phase of the wave function in the classically allowed regions adjacent to a real, symmetric potential barrier, when the… -- 3.23 Eigenvalue problem for a quantal particle in a broad, symmetric potential well between two symmetric potential barriers… -- 3.24 Dependence of the phase of the wave function in a classically allowed region on the position of the point x1 in an… -- 3.25 Phase-shift formula -- 3.26 Distance between near-lying energy levels in different types of physical systems, expressed either in terms of the… -- 3.27 Arbitrary-order quantization condition for a particle in a single-well potential, derived on the assumption that the… -- 3.28 Arbitrary-order quantization condition for a particle in a single-well potential, derived without the assumption that… -- 3.29 Displacement of the energy levels due to compression of an atom (simple treatment) -- 3.30 Displacement of the energy levels due to compression of an atom (alternative treatment).8 a3.31 Quantization condition for a particle in a smooth potential well, limited on one side by an impenetrable wall and on… -- 3.32 Energy spectrum of a non-relativistic particle in a potential proportional to… -- 3.33 Determination of a one-dimensional, smooth, single-well potential from the energy spectrum of the bound states -- 3.34 Determination of a radial, smooth, single-well potential from the energy spectrum of the bound states -- 3.35 Determination of the radial, single-well potential, when the energy eigenvalues are… -- 3.36 Exact formula for the normalization integral for the wave function pertaining to a bound state of a particle in a… -- 3.37 Phase-integral formula for the normalized radial wave function pertaining to a bound state of a particle in a radial… -- 3.38 Radial wave function psi(z) for an s-electron in a classically allowed region containing the origin, when the… -- 3.39 Quantization condition, and value of the normalized wave function at the origin expressed in terms of the level density… -- 3.40 Expectation value of an unspecified function f(z) for a non-relativistic particle in a bound state -- 3.41 Some cases in which the phase-integral expectation value formula yields the expectation value exactly in the… -- 3.42 Expectation value of the kinetic energy of a non-relativistic particle in a bound state. Verification of the virial… -- 3.43 Phase-integral calculation of quantal matrix elements -- 3.44 Connection formula for a complex potential barrier -- 3.45 Connection formula for a real, single-hump potential barrier -- 3.46 Energy levels of a particle in a smooth double-well potential, when no symmetry requirement is imposed -- 3.47 Energy levels of a particle in a smooth, symmetric, double-well potential.8 a3.48 Determination of the quasi-stationary energy levels of a particle in a radial potential with a thick single-hump barrier -- 3.49 Transmission coefficient for a particle penetrating a real single-hump potential barrier -- 3.50 Transmission coefficient for a particle penetrating a real, symmetric, superdense double-hump potential barrier -- References -- Author index -- Subject index. aA mathematical approximation method important for many branches of theoretical physics, applied mathematics and engineering. aDescription based on publisher supplied metadata and other sources. aElectronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries. 0aWKB approximation.952517 4aElectronic books.9525181 aFröman, Per Olof.95251908iPrint version:aFröman, NannytPhysical Problems Solved by the Phase-Integral MethoddCambridge : Cambridge University Press,c2002z97805218120922 aProQuest (Firm)40uhttps://ebookcentral.proquest.com/lib/thebc/detail.action?docID=202083zClick to View aEBK cEBK c76384d76384 00104070aAFbAFd2019-11-26l0pEBKAF0005880r2019-11-26w2019-11-26yEBK 00104070aNPbNPd2019-11-26l0pEBKNP0005880r2019-11-26w2019-11-26yEBK 00104070aEGbEGl0yEBK 00104070aSDbSDl0yEBK 00104070aMObMOl0yEBK 00104070aTNbTNl0yEBK 00104070aLYbLYl0yEBK 00104070aDZbDZl0yEBK 00104070aCYbCYl0yEBK