Scattering Theory The Quantum Theory of Nonrelativistic Collisions

by Taylor, John R.

• ISBN13: 9780486450131
• ISBN10: 0486450139
• Format: Paperback
• Copyright: 2006-05-26
• Publisher: Dover Publications

Summary

This graduate-level text is intended for any student of physics who requires grounding in the quantum theory of nonrelativistic scattering. The time-dependent approach is emphasized, including the use of time-dependent formalism to define all basic concepts and time-independent theory as a tool for computation. 1983 edition.

Table of Contents

Introduction	1	(5)
Mathematical Preliminaries	6	(15)
The Hilbert Space of State Vectors	7	(2)
Subspaces	9	(1)
Operators and Inverses	10	(3)
Unitary Operators	13	(1)
Isometric Operators	14	(2)
Convergence of Vectors	16	(3)
Operator Limits	19	(2)
The Scattering Operator for a Single Particle	21	(17)
Classical Scattering	22	(3)
Quantum Scattering	25	(3)
The Asymptotic Condition	28	(3)
Orthogonality and Asymptotic Completeness	31	(3)
The Scattering Operator	34	(2)
Unitarity	36	(2)
Cross Sections in Terms of the S Matrix	38	(18)
Conservation of Energy	39	(3)
The On-Shell T Matrix and Scattering Amplitude	42	(2)
The Classical Cross Section	44	(2)
Definition of the Quantum Cross Section	46	(3)
Calculation of the Quantum Cross Section	49	(4)
The Optical Theorem	53	(3)
Scattering of Two Spinless Particles	56	(13)
Two-Particle Wave Functions	57	(3)
The Two-Particle S Operator	60	(2)
Conservation of Energy-Momentum and the T Matrix	62	(2)
Cross Sections in Various Frames	64	(2)
The Center-of-Mass Cross Section	66	(3)
Scattering of Two Particles with Spin	69	(12)
The Hilbert Space for Particles with Spin	70	(2)
The S Operator for Particles with Spin	72	(1)
The Amplitudes and Amplitude Matrix	73	(3)
Sums and Averages Over Spins	76	(2)
The In and Out Spinors	78	(3)
Invariance Principles and Conservation Laws	81	(28)
Translational Invariance and Conservation of Momentum	82	(1)
Rotational Invariance and Conservation of Angular Momentum	83	(2)
The Partial-Wave Series for Spinless Particles	85	(5)
Parity	90	(1)
Time Reversal	91	(4)
Invariance Principles for Particles with Spin; Momentum-Space Analysis	95	(8)
Invariance Principles for Particles with Spin; Angular-Momentum Analysis	103	(6)
More About Particles with Spin	109	(19)
Polarization and the Density Matrix	109	(4)
The In and Out Density Matrices	113	(1)
Polarization Experiments in (Spin 1/2) --- (Spin 0) Scattering	114	(5)
The Helicity Formalism	119	(5)
Some Useful Formulas	124	(4)
The Green's Operator and the T Operator	128	(15)
The Green's Operator	129	(5)
The T Operator	134	(1)
Relation to the Moller Operators	135	(3)
Relation to the Scattering Operator	138	(5)
The Born Series	143	(21)
The Born Series	144	(3)
The Born Approximation	147	(3)
The Yukawa Potential	150	(3)
Scattering of Electrons off Atoms	153	(4)
Interpretation of the Born Series in Terms of Feynman Diagrams	157	(7)
The Stationary Scattering States	164	(16)
Definition and Properties of the Stationary Scattering States	165	(3)
Equations for the Stationary Scattering Vectors	168	(2)
The Stationary Wave Functions	170	(3)
A Spatial Description of the Scattering Process	173	(7)
The Partial-Wave Stationary States	180	(32)
The Partial-Wave S Matrix	181	(1)
The Free Radial Wave Functions	182	(3)
The Partial-Wave Scattering States	185	(3)
The Partial-Wave Lippmann-Schwinger Equation	188	(3)
Properties of the Partial-Wave Amplitude	191	(6)
The Regular Solution	197	(1)
The Variable Phase Method	197	(4)
Iterative Solution for the Regular Wave Function	201	(3)
The Jost Function	204	(3)
The Partial-Wave Born Series	207	(5)
Analytic Properties of the Partial-Wave Amplitude	212	(26)
Analytic Functions of a Complex Variable	213	(2)
Analytic Properties of the Regular Solution	215	(2)
Analytic Properties of the Jost Function and S Matrix	217	(6)
Bound States and Poles of the S Matrix	223	(3)
Levinson's Theorem	226	(3)
Threshold Behavior and Effective Range Formulas	229	(3)
Zeros of the Jost Function at Threshold	232	(6)
Resonances	238	(21)
Resonances and Poles of the S Matrix	240	(5)
Bound States and Resonances	245	(4)
Time Delay	249	(4)
Decay of a Resonant State	253	(6)
Additional Topics in Single-Channel Scattering	259	(26)
Coulomb Scattering	259	(7)
Coulomb Plus Short-Range Potentials	266	(4)
The Distorted-Wave Born Approximation	270	(3)
Variational Methods	273	(7)
The K Matrix	280	(5)
Dispersion Relations and Complex Angular Momenta	285	(30)
Partial-Wave Dispersion Relations	287	(4)
Forward Dispersion Relations	291	(3)
Nonforward Dispersion Relations	294	(3)
The Mandelstam Representation	297	(5)
Complex Angular Momenta	302	(4)
Regge Poles	306	(2)
The Watson Transform	308	(7)
The Scattering Operator in Multichannel Scattering	315	(23)
Channels	316	(5)
Channel Hamiltonians and Asymptotic States	321	(5)
Orthogonality and Asymptotic Completeness	326	(5)
A Little More Mathematics	331	(3)
The Scattering Operator	334	(4)
Cross Sections and Invariance Principles in Multichannel Scattering	338	(20)
The Momentum-Space Basis Vectors	339	(3)
Conservation of Energy and the On-Shell T Matrix	342	(4)
Cross Sections	346	(5)
Rotational Invariance	351	(2)
Time-Reversal Invariance	353	(5)
Fundamentals of Time-Independent Multichannel Scattering	358	(14)
The Stationary Scattering States	359	(2)
The Lippmann-Schwinger Equations	361	(2)
The T Operators	363	(1)
The Born Approximation; Elastic Scattering	364	(3)
The Born Approximation; Excitation	367	(5)
Properties of the Multichannel Stationary Wave Functions	372	(19)
Asymptotic Form of the Stationary Wave Functions; Collisions Without Rearrangement	373	(5)
Asymptotic Form of the Stationary Wave Functions; Rearrangement Collisions	378	(2)
Expansion in Terms of Target States	380	(3)
The Optical Potential	383	(8)
Analytic Properties and Multichannel Resonances	391	(27)
Analytic Properties	392	(4)
Proof of Analytic Properties	396	(9)
Bound States	405	(2)
Resonances	407	(6)
Decay of a Multichannel Resonance	413	(5)
Two More Topics in Multichannel Scattering	418	(16)
The Distorted-Wave Born Approximation	418	(6)
Final-State Interactions	424	(10)
Identical Particles	434	(29)
The Formalism of Identical Particles	435	(6)
Scattering of Two Identical Particles	441	(7)
Multichannel Scattering with Identical Particles	448	(2)
Transition Probabilities and Cross Sections	450	(4)
Electron-Hydrogen Scattering	454	(9)
References	463	(4)
Index	467