*John A. Adam*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0022
- Subject:
- Mathematics, Applied Mathematics

This chapter reexamines the WKB(J) approximation and applies it to some simple one-dimensional potentials, with a focus on the case of a triangular barrier. It first considers the connection formulas ...
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This chapter reexamines the WKB(J) approximation and applies it to some simple one-dimensional potentials, with a focus on the case of a triangular barrier. It first considers the connection formulas and proposes an an alternative approach before discussing tunneling from a physical standpoint. It then turns to the case of a triangular barrier and goes on to explore the phase shift, offering some comments on convergence and the transition to classical scattering. It also describes the asymptotic behavior of the Coulomb wave function and revisits the spherical coordinate system. Finally, it finds the WKB(J) approximation with respect to Coulomb scattering and the formal WKB(J) solutions for the time-independent radial Schrödinger equation, and justifies the Langer transformation by showing how the asymptotic phase of the radial WKB(J) wave function is recovered.Less

This chapter reexamines the WKB(J) approximation and applies it to some simple one-dimensional potentials, with a focus on the case of a triangular barrier. It first considers the connection formulas and proposes an an alternative approach before discussing tunneling from a physical standpoint. It then turns to the case of a triangular barrier and goes on to explore the phase shift, offering some comments on convergence and the transition to classical scattering. It also describes the asymptotic behavior of the Coulomb wave function and revisits the spherical coordinate system. Finally, it finds the WKB(J) approximation with respect to Coulomb scattering and the formal WKB(J) solutions for the time-independent radial Schrödinger equation, and justifies the Langer transformation by showing how the asymptotic phase of the radial WKB(J) wave function is recovered.

*John A. Adam*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0007
- Subject:
- Mathematics, Applied Mathematics

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order ...
More

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order in principle) with spatially varying coefficients. The WKB(J) approximation is closely related to the semiclassical approach in quantum mechanics in which the wavefunction is characterized by a slowly varying amplitude and/or phase. The chapter first introduces an inhomogeneous differential equation, from which the first derivative term is eliminated, before discussing the Liouville transformation and the one-dimensional Schrödinger equation. It then presents a physical interpretation of the WKB(J) approximation and its application to a potential well. It also considers the “patching region” in which the Airy function solution (the local turning point) is valid, the relation between Airy functions and Bessel functions, Airy integral and related topics, and related integrals.Less

This chapter deals with the WKB(J) approximation, commonly used in applied mathematics and mathematical physics to find approximate solutions of linear ordinary differential equations (of any order in principle) with spatially varying coefficients. The WKB(J) approximation is closely related to the semiclassical approach in quantum mechanics in which the wavefunction is characterized by a slowly varying amplitude and/or phase. The chapter first introduces an inhomogeneous differential equation, from which the first derivative term is eliminated, before discussing the Liouville transformation and the one-dimensional Schrödinger equation. It then presents a physical interpretation of the WKB(J) approximation and its application to a potential well. It also considers the “patching region” in which the Airy function solution (the local turning point) is valid, the relation between Airy functions and Bessel functions, Airy integral and related topics, and related integrals.

*John A. Adam*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0001
- Subject:
- Mathematics, Applied Mathematics

This book deals with rays, waves, and scattering and covers many of the mathematical concepts, structures, and techniques used to study them. The subject of rays is explored in an ...
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This book deals with rays, waves, and scattering and covers many of the mathematical concepts, structures, and techniques used to study them. The subject of rays is explored in an atmosphere–sea–earth sequence, while waves are examined via the reverse sequence earth–sea–atmosphere. The book also considers the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics; Kepler's laws of planetary motion, developed from gravitational scattering; surface gravity waves; diffraction; acoustics; electromagnetic scattering, including the Mie solution; and the WKB(J) approximation and its application to some simple one-dimensional potentials. The book concludes with an analysis of the salient properties of Sturm-Liouville systems with particular reference to the time-independent Schrödinger equation. This chapter provides an overview of the rainbow directory, rays, waves, classical and semiclassical scattering, and caustics and diffraction catastrophes.Less

This book deals with rays, waves, and scattering and covers many of the mathematical concepts, structures, and techniques used to study them. The subject of rays is explored in an atmosphere–sea–earth sequence, while waves are examined via the reverse sequence earth–sea–atmosphere. The book also considers the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics; Kepler's laws of planetary motion, developed from gravitational scattering; surface gravity waves; diffraction; acoustics; electromagnetic scattering, including the Mie solution; and the WKB(J) approximation and its application to some simple one-dimensional potentials. The book concludes with an analysis of the salient properties of Sturm-Liouville systems with particular reference to the time-independent Schrödinger equation. This chapter provides an overview of the rainbow directory, rays, waves, classical and semiclassical scattering, and caustics and diffraction catastrophes.