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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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