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The Time-Independent Schrödinger Equation

Gary E. Bowman

in Essential Quantum Mechanics

Published in print:
2007
Published Online:
January 2008
ISBN:
9780199228928
eISBN:
9780191711206
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199228928.003.0009
Subject:
Physics, Condensed Matter Physics / Materials

After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions ... More


Basics on Bose‐Einstein condensation

D. GuÉry‐Odelin and T. Lahaye

in Ultracold Gases and Quantum Information: Lecture Notes of the Les Houches Summer School in Singapore: Volume 91, July 2009

Published in print:
2011
Published Online:
September 2011
ISBN:
9780199603657
eISBN:
9780191729515
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199603657.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces basic notions about Bose-Einstein condensates (BECs). One first studies ideal Bose gases, putting an emphasis on the role of the trapping geometry and of the dimensionality. ... More


Multi-Instantons In Quantum Mechanics

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0043
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The concepts of multi-instanton configurations and contributions are non-trivial because classical equations are non-linear and thus in general a linear combination of solutions is not a solution. ... More


Degenerate Classical Minima And Instantons

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0041
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter considers a somewhat different situation in which instantons play a role: classical actions with degenerate isolated minima. The simplest examples are provided by simple quantum ... More


The Structure of Quantum States

Gary E. Bowman

in Essential Quantum Mechanics

Published in print:
2007
Published Online:
January 2008
ISBN:
9780199228928
eISBN:
9780191711206
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199228928.003.0004
Subject:
Physics, Condensed Matter Physics / Materials

The quantum state is the fundamental entity in quantum mechanics. This chapter develops the basic mathematical structure of quantum states by developing the correspondence between states and ordinary ... More


Bound states in one dimension

Efstratios Manousakis

in Practical Quantum Mechanics: Modern Tools and Applications

Published in print:
2015
Published Online:
December 2015
ISBN:
9780198749349
eISBN:
9780191813474
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198749349.003.0008
Subject:
Physics, Atomic, Laser, and Optical Physics

In this and in the following chapter, it reviews an important part of undergraduate material, namely solving for bound states and for scattering in one-dimension for the case where the potential is ... More


One-Dimensional Jost Solutions: The S-Matrix Revisited

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691148373
eISBN:
9781400885404
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691148373.003.0026
Subject:
Mathematics, Applied Mathematics

This chapter examines the properties of one-dimensional Jost solutions for S-matrix problems. It first considers how the left–right transmission and reflections coefficients can be expressed in terms ... More


Quantum Tunneling

Frank S. Levin

in Surfing the Quantum World

Published in print:
2017
Published Online:
November 2017
ISBN:
9780198808275
eISBN:
9780191846014
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198808275.003.0014
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

Quantum tunneling, wherein a quanject has a non-zero probability of tunneling into and then exiting a barrier of finite width and height, is the subject of Chapter 13. The description for the ... More


Scattering in one dimension

Efstratios Manousakis

in Practical Quantum Mechanics: Modern Tools and Applications

Published in print:
2015
Published Online:
December 2015
ISBN:
9780198749349
eISBN:
9780191813474
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198749349.003.0009
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter discusses scattering in one dimension. It considers simple examples, it discusses tunneling and it analyzes the analytic structure of the scattering amplitude. It also discusses ... More


The Classical-to-Semiclassical Connection

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691148373
eISBN:
9781400885404
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691148373.003.0021
Subject:
Mathematics, Applied Mathematics

This chapter discusses the connection between the classical and semiclassical domains of scattering. Scattering phenomena may be described via three regimes: the scattering of waves by objects with ... More


The electron

L. Solymar, D. Walsh, and R. R. A. Syms

in Electrical Properties of Materials

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198829942
eISBN:
9780191868504
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198829942.003.0003
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

Discusses with some rigour the properties of electrons, based on the Schrodinger equation. Introduces the concepts of wave function, quantum-mechanical operators, and wave packets. Examples cover the ... More


Ocean Acoustics

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691148373
eISBN:
9781400885404
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691148373.003.0012
Subject:
Mathematics, Applied Mathematics

This chapter deals with the mathematics of ocean acoustics. A number of environmental factors affect the transmission of sound in the ocean, including the depth and configuration of the bottom, the ... More


Superposition

William J. Mullin

in Quantum Weirdness

Published in print:
2017
Published Online:
March 2017
ISBN:
9780198795131
eISBN:
9780191836480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198795131.003.0004
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

A quantum wave function can be a superposition of multiple waves. A double-well potential has a wave function with elements in each well. The lowest energy state has a symmetric wave function, and ... More


Introduction to the WKB(J) Approximation: All Things Airy

John A. Adam

in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

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


Application to Few-Mode Systems

Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

in Phase Space Methods for Degenerate Quantum Gases

Published in print:
2014
Published Online:
April 2015
ISBN:
9780199562749
eISBN:
9780191747311
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562749.003.0010
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter presents three examples of applying of phase space methods in systems with only a few important modes. The boson application is to interferometry in two-mode Bose–Einstein condensates, ... More


Time to Absorption in Discounted Reinforcement Models

Robin Pemantle

in Social Dynamics

Published in print:
2014
Published Online:
June 2014
ISBN:
9780199652822
eISBN:
9780191779367
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199652822.003.0013
Subject:
Philosophy, Philosophy of Science

Reinforcement schemes are a class of non-Markovian stochastic processes. Their non-Markovian nature allows them to model some kind of memory of the past. One subclass of such models are those in ... More


Network Formation by Reinforcement Learning: The Long and Medium Run

Robin Pemantle

in Social Dynamics

Published in print:
2014
Published Online:
June 2014
ISBN:
9780199652822
eISBN:
9780191779367
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199652822.003.0012
Subject:
Philosophy, Philosophy of Science

This chapter investigates a simple stochastic model of social network formation by the process of reinforcement learning with discounting of the past. In the limit, for any value of the discounting ... More


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