Interactive Quantum Mechanics

Transcription

1 Interactive Quantum Mechanics

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3 S. Brandt H.D. Dahmen T. Stroh Interactive Quantum Mechanics Quantum Experiments on the Computer Second Edition With CD-ROM, 128 Figures, and 344 Exercises

4 Siegmund Brandt Physics Department Siegen University Siegen Germany Hans Dieter Dahmen Physics Department Siegen University Siegen Germany Tilo Stroh Physics Department Siegen University Siegen Germany Additional material to this book can be downloaded from ISBN e-isbn DOI / Springer New York Dordrecht Heidelberg London Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface to the Second Edition For the present edition the concept of the book and of INTERQUANTA, the accompanying Interactive Program of Quantum Mechanics, (IQ, for short), was left unchanged. However, the physics scope of the text and the capabilities of the program were widened appreciably. The most conspicuous addition to IQ is the capability to produce and display movies of quantum-mechanical phenomena. So far, IQ presented time dependence as a series of graphs in one frame. While such plots (which can, of course, still be shown) lead to a good understanding of the phenomenon under study and can be examined quantitatively at leisure, the new movies give a more direct impression of what happens as time passes. For such movies, as for the conventional simulations, the parameters defining the physical phenomenon and the graphical appearance can be changed interactively. Movies can be produced and also stored for many quantum-mechanical problems such as bound states and scattering states in various one-dimensional potentials, wave packets in three dimensions (free or in a harmonic-oscillator potential), and two-particle systems (distinguishable particles, identical fermions, identical bosons). Concerning the physics scope, these are the main additions: One-dimensional bound and scattering states are now discussed and computed also for piecewise linear potentials. These, as opposed to the usual step potentials (which are piecewise constant), allow for much better approximations of arbitrary smooth potentials. Another interesting addition to one-dimensional quantum mechanics is the juxtaposition of quantum-mechanical wave packets with classical phase-space distributions. The treatment of quantum mechanics in three dimensions is extended by the hybridization of bound states and by the simulation of magnetic resonance. In the present edition the number of data sets (we call them descriptors), defining a complete simulation either presented as conventional plot or as movie is more than tripled. In this way users have a much richer choice of ready-made examples from which to start their exploits. Moreover, solution descriptors are now provided for the exercises. Siegen, Germany May 2010 Siegmund Brandt Hans Dieter Dahmen Tilo Stroh v

6 Preface to the First Edition This book can be regarded as a concise introduction to basic quantum mechanics: free particle, bound states, and scattering in one and in three dimensions, two-particle systems, special functions of mathematical physics. But the book can also be seen as an extensive user s guide for INTERQUANTA, the Interactive Program of Quantum Mechanics, which we will abbreviate henceforth as IQ. The book also contains a large number of exercises. The program can be used in two ways. By working through (at least a part of) these exercises, the user of IQ explores a computer laboratory in quantum mechanics by performing computer experiments. A simpler way to use IQ is to study one or several of the ready-made demonstrations. In each demonstration the user is taken through one chapter of quantum mechanics. Graphics illustrating quantum-mechanical problems that are solved by the program are shown, while short explanatory texts are either also displayed or can be listened to. INTERQUANTA has a user interface based on tools provided by the Java programming language. With this interface using the program is essentially self-explanatory. In addition, extensive help functions are provided not only on technical questions but also on quantum-mechanical concepts. All in all using INTERQUANTA is not more difficult than surfing the Internet. The modern user interface is the main improvement over older versions of IQ. 1 Moreover, new physics topics are added and there are also new graphical features. The present version of INTERQUANTA is easily installed and run on personal computers (running under Windows or Linux) or Macintosh (running under Mac OS X). We do hope that by using INTERQUANTA on their own computer many students will gain experience with different quantum phenomena without having to do tedious calculations. From this experience an intuition for this important but abstract field of modern science can be developed. Siegen, Germany February 2003 Siegmund Brandt Hans Dieter Dahmen Tilo Stroh 1 S. Brandt and H. D. Dahmen, Quantum Mechanics on the Personal Computer, Springer, Berlin 1989, 1992, and 1994; Quantum Mechanics on the Macintosh, Springer, New York 1991 and 1995; Pasocon de manebu ryoushi nikigacu, Springer, Tokyo 1992; Quantenmechanik auf dem Personalcomputer, Springer, Berlin 1993 vi

7 Contents Preface to the Second Edition v Preface to the First Edition vi 1 Introduction Interquanta The Structure of This Book The Demonstrations The Computer Laboratory Literature Free Particle Motion in One Dimension Physical Concepts Planck s Constant. Schrödinger s Equation for a Free Particle The Wave Packet. Group Velocity. Normalization Probability-Current Density. Continuity Equation Quantile Position. Quantile Trajectory Relation to Bohm s Equation of Motion Analogies in Optics Analogies in Classical Mechanics: The Phase-Space Probability Density A First Session with the Computer Starting IQ An Automatic Demonstration A First Dialog The Free Quantum-Mechanical Gaussian Wave Packet The Subpanel Physics Comp. Coord The Subpanel Physics Wave Packet The Subpanel Physics Quantile The Subpanel Movie The Free Optical Gaussian Wave Packet Quantile Trajectories vii

8 viii Contents 2.6 The Spectral Function of a Gaussian Wave Packet The Wave Packet as a Sum of Harmonic Waves The Phase-Space Distribution of Classical Mechanics Classical Phase-Space Distribution: Covariance Ellipse Exercises Bound States in One Dimension Physical Concepts Schrödinger s Equation with a Potential. Eigenfunctions. Eigenvalues Normalization. Discrete Spectra. Orthonormality The Infinitely Deep Square-Well Potential The Harmonic Oscillator The Step Potential The Piecewise Linear Potential Time-Dependent Solutions Harmonic Particle Motion. Coherent States. Squeezed States Quantile Motion in the Harmonic-Oscillator Potential Harmonic Motion of a Classical Phase-Space Distribution Particle Motion in a Deep Square Well Eigenstates in the Infinitely Deep Square-Well Potential and in the Harmonic-Oscillator Potential Eigenstates in the Step Potential Eigenstates in the Step Potential Quasiperiodic Eigenstates in the Piecewise Linear Potential Eigenstates in the Piecewise Linear Potential Quasiperiodic Harmonic Particle Motion Harmonic Oscillator: Quantile Trajectories Classical Phase-Space Distribution: Harmonic Motion Harmonic Motion of Classical Phase-Space Distribution: Covariance Ellipse Particle Motion in the Infinitely Deep Square-Well Potential Exercises Scattering in One Dimension Physical Concepts Stationary Scattering States. Continuum Eigenstates and Eigenvalues. Continuous Spectra Time-Dependent Solutions of the Schrödinger Equation

9 Contents ix Right-Moving and Left-Moving Stationary Waves of a Free Particle Orthogonality and Continuum Normalization of Stationary Waves of a Free Particle. Completeness Boundary Conditions for Stationary Scattering Solutions in Step Potentials Stationary Scattering Solutions in Step Potentials Constituent Waves Normalization of Continuum Eigenstates Harmonic Waves in a Step Potential Time-Dependent Scattering Solutions in a Step Potential Generalization to Piecewise Linear Potentials Transmission and Reflection. Unitarity. The Argand Diagram The Tunnel Effect Resonances Phase Shifts upon Reflection at a Steep Rise or Deep Fall of the Potential Transmission Resonances upon Reflection at More- and Less-Dense Media The Quantum-Well Device and the Quantum-Effect Device Stationary States in a Linear Potential Wave Packet in a Linear Potential Quantile Motion in a Linear Potential Classical Phase-Space Density in a Linear Potential Classical Phase-Space Density Reflected by a High Potential Wall Stationary Scattering States in the Step Potential and in the Piecewise Linear Potential Time-Dependent Scattering by the Step Potential and by the Piecewise Linear Potential Transmission and Reflection. The Argand Diagram Stationary Wave in a Linear Potential Gaussian Wave Packet in a Linear Potential Quantile Trajectories in a Linear Potential Classical Phase-Space Density in a Linear Potential Classical Phase-Space Distribution: Covariance Ellipse Classical Phase-Space Density Reflected by a High Potential Wall Exercises

10 x Contents 4.12 Analogies in Optics Reflection and Refraction of Stationary Electromagnetic Waves Time-Dependent Scattering of Light Transmission, Reflection, and Argand Diagram for a Light Wave Exercises A Two-Particle System: Coupled Harmonic Oscillators Physical Concepts The Two-Particle System Initial Condition for Distinguishable Particles Time-Dependent Wave Functions and Probability Distributions for Distinguishable Particles Marginal Distributions for Distinguishable Particles Wave Functions for Indistinguishable Particles. Symmetrization for Bosons. Antisymmetrization for Fermions Marginal Distributions of the Probability Densities of Bosons and Fermions Normal Oscillations Stationary States Time Dependence of Global Variables Joint Probability Densities Marginal Distributions Exercises Free Particle Motion in Three Dimensions Physical Concepts The Schrödinger Equation of a Free Particle in Three Dimensions. The Momentum Operator The Wave Packet. Group Velocity. Normalization. The Probability Ellipsoid Angular Momentum. Spherical Harmonics The Stationary Schrödinger Equation in Polar Coordinates. Separation of Variables. Spherical Bessel Functions. Continuum Normalization. Completeness Partial-Wave Decomposition of the Plane Wave Partial-Wave Decomposition of the Gaussian Wave Packet The 3D Harmonic Plane Wave The Plane Wave Decomposed into Spherical Waves The 3D Gaussian Wave Packet

11 Contents xi 6.5 The Probability Ellipsoid Angular-Momentum Decomposition of a Wave Packet Exercises Bound States in Three Dimensions Physical Concepts The Schrödinger Equation for a Particle under the Action of a Force. The Centrifugal Barrier. The Effective Potential Bound States. Scattering States. Discrete and Continuous Spectra The Infinitely Deep Square-Well Potential The Spherical Step Potential The Harmonic Oscillator The Coulomb Potential. The Hydrogen Atom Harmonic Particle Motion Radial Wave Functions in Simple Potentials Radial Wave Functions in the Step Potential Probability Densities Contour Lines of the Probability Density Contour Surface of the Probability Density Harmonic Particle Motion Exercises Scattering in Three Dimensions Physical Concepts Radial Scattering Wave Functions Boundary and Continuity Conditions. Solution of the System of Inhomogeneous Linear Equations for the Coefficients Scattering of a Plane Harmonic Wave Scattering Amplitude and Phase. Unitarity. The Argand Diagram Coulomb Scattering Radial Wave Functions Stationary Wave Functions and Scattered Waves Differential Cross Sections Scattering Amplitude. Phase Shift. Partial and Total Cross Sections Coulomb Scattering: Radial Wave Function Coulomb Scattering: 3D Wave Function Exercises

12 xii Contents 9 Spin and Magnetic Resonance Physical Concepts Spin Operators. Eigenvectors and Eigenvalues Magnetic Moment and Its Motion in a Magnetic Field. Pauli Equation Magnetic Resonance Rabi Formula The Spin-Expectation Vector near and at Resonance Resonance Form of the Rabi Amplitude Exercises Hybridization Physical Concepts Hybrid States in the Coulomb Potential Some Qualitative Details of Hybridization Hybridization Parameters and Orientations of Highly Symmetric Hybrid States Hybrid Wave Functions and Probability Densities Contour Lines of Hybrid Wave Functions and Probability Densities Contour Surfaces of Hybrid Probability Densities Exercises Special Functions of Mathematical Physics Basic Formulae Hermite Polynomials Harmonic-Oscillator Eigenfunctions Legendre Polynomials and Legendre Functions Spherical Harmonics Bessel Functions Spherical Bessel Functions Airy Functions Laguerre Polynomials Radial Eigenfunctions of the Harmonic Oscillator Radial Eigenfunctions of the Hydrogen Atom Gaussian Distribution and Error Function Binomial Distribution and Poisson Distribution Hermite Polynomials and Related Functions Legendre Polynomials and Related Functions Spherical Harmonics: Surface over Cartesian Grid Spherical Harmonics: 2D Polar Diagram Spherical Harmonics: Polar Diagram in 3D Bessel Functions and Related Functions

13 Contents xiii 11.8 Bessel Function and Modified Bessel Function with Real Index Airy Functions Laguerre Polynomials Laguerre Polynomials as Function of x and the Upper Index α Gaussian Distribution Error Function and Complementary Error Function Bivariate Gaussian Distribution Bivariate Gaussian: Covariance Ellipse Binomial Distribution Poisson Distribution Simple Functions of a Complex Variable Exercises Additional Material and Hints for the Solution of Exercises Units and Orders of Magnitude Definitions SI Units Scaled Units Atomic and Subatomic Units Data-Table Units Special Scales Argand Diagrams and Unitarity for One-Dimensional Problems Probability Conservation and the Unitarity of the Scattering Matrix Time Reversal and the Scattering Matrix Diagonalization of the Scattering Matrix Argand Diagrams Resonances Hints and Answers to the Exercises A A Systematic Guide to IQ A.1 Overview A.1.1 Starting IQ A.1.2 Introductory Demonstration A.1.3 Selecting a Descriptor File A.1.4 Selecting a Descriptor and Producing a Plot A.1.5 Creating and Running a Movie A.1.6 Printing a Plot A.1.7 Changing Colors and Line Widths A.1.8 Changing Parameters A.1.9 Saving a Changed Descriptor

14 xiv Contents A.1.10 Creating a Mother Descriptor A.1.11 Editing Descriptor Files A.1.12 Printing a Set of Plots A.1.13 Running a Demonstration A.1.14 Customizing A.1.15 Help and Context-Sensitive Help A.2 Coordinate Systems and Transformations A.2.1 The Different Coordinate Systems A.2.2 Defining the Transformations A.3 The Different Types of Plot A.3.1 Surface over Cartesian Grid in 3D A.3.2 Surface over Polar Grid in 3D A.3.3 2D Function Graph A.3.4 Contour-Line Plot in 2D A.3.5 Contour-Surface Plot in 3D A.3.6 Polar Diagram in 3D A.3.7 Probability-Ellipsoid Plot A.3.8 3D Column Plot A.4 Parameters The Subpanel Movie A.5 Parameters The Subpanel Physics A.5.1 The Subpanel Physics Comp. Coord A.5.2 The Subpanel Multiple Plot A.6 Parameters The Subpanel Graphics A.6.1 The Subpanel Graphics Geometry A.6.2 The Subpanel Graphics Accuracy A.6.3 The Subpanel Graphics Hidden Lines A.7 Parameters The Subpanel Background A.7.1 The Subpanel Background Box A.7.2 The Subpanel Background Scales A.7.3 The Subpanel Background Arrows A.7.4 The Subpanel Background Texts A.8 Parameters The Subpanel Format A.9 Coding Mathematical Symbols and Formulae A.10 A Combined Plot and Its Mother Descriptor A.10.1 The Subpanel Type and Format A.10.2 The Subpanel Table of Descriptors A.10.3 Special Cases A.11 Details of Printing A.11.1 Preview. Colors and Line Widths A.11.2 Using a System Printer A.11.3 Creating PostScript Files: IQ Export A.12 Preparing a Demonstration

15 Contents xv B How to Install IQ B.1 Contents of the CD-ROM B.2 Computer Systems on which INTERQUANTA Can Be Used 365 B.3 Installation with Options. The File ReadMe.txt B.4 Quick Installation for the Impatient User Index

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