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Physical Problems Solved by the Phase-Integral Method.

By: Contributor(s): Publisher: Cambridge : Cambridge University Press, 2002Copyright date: ©2002Description: 1 online resource (230 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780511157714
Subject(s): Genre/Form: Additional physical formats: Print version:: Physical Problems Solved by the Phase-Integral MethodDDC classification:
  • 530.124
LOC classification:
  • QC20.7.W53 F77 2002
Online resources:
Contents:
Cover -- 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.
2.4.4 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….
3.14 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).
3.31 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.
3.48 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.
Summary: A mathematical approximation method important for many branches of theoretical physics, applied mathematics and engineering.
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Cover -- 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.

2.4.4 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….

3.14 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).

3.31 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.

3.48 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.

A mathematical approximation method important for many branches of theoretical physics, applied mathematics and engineering.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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