Interactive Quantum Mechanics: Quantum Experiments on the Computer

394 Pages·2011·10.45 MB·English

by  S. Brandt

Interactive Quantum Mechanics

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
Siegmund Brandt Hans Dieter Dahmen
Physics Department Physics Department
Siegen University Siegen University
57068 Siegen 57068 Siegen
Germany Germany
[e-mail] [e-mail]

Tilo Stroh
Physics Department
Siegen University
57068 Siegen
Germany
[e-mail]

Additional material to this book can be downloaded from http://extras.springer.com
ISBN 978-1-4419-7423-5 e-ISBN 978-1-4 419-7424-2
DOI 10.1007/978-1-4419-7424-2
Springer New York Dordrecht Heidelberg Lo ndon

© Springer Science+Business Media, LLC 2011
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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 capabili-
ties of the program were widened appreciably.
ThemostconspicuousadditiontoIQisthecapability toproduceanddis-
play movies of quantum-mechanical phenomena. So far, IQ presented time
dependence as a series of graphs in one frame. While such plots (which can,
ofcourse,stillbeshown)leadtoagoodunderstandingofthephenomenonun-
der 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,
asfortheconventionalsimulations,theparameters definingthephysicalphe-
nomenon and the graphical appearance can be changed interactively. Movies
canbeproducedandalsostoredformanyquantum-mechanicalproblemssuch
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, iden-
tical bosons).
Concerning the physics scope, these are the main additions: One-dimen-
sional 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 ar-
bitrary smooth potentials. Another interesting addition to one-dimensional
quantummechanicsisthejuxtapositionofquantum-mechanicalwavepackets
with classical phase-space distributions. The treatment of quantum mechan-
ics in three dimensions is extended by the hybridization of bound states and
bythe 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 nowprovided for the exercises.
Siegen, Germany SiegmundBrandt
May 2010 HansDieter Dahmen
Tilo Stroh
v
Preface to the First Edition
Thisbookcanberegardedasaconciseintroductiontobasicquantummechan-
ics: free particle, boundstates, and scattering in one and in three dimensions,
two-particlesystems,specialfunctionsofmathematicalphysics. Butthebook
can also be seen as an extensive user’s guide for INTERQUANTA, the Inter-
active Programof QuantumMechanics, whichwe 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 exer-
cises, 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 illustrat-
ingquantum-mechanical problemsthat are solvedbythe programare shown,
while shortexplanatory texts are either also displayed orcan be listened to.
INTERQUANTAhasauserinterfacebasedontoolsprovidedbytheJava
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
usingINTERQUANTAisnot more difficult than surfingthe Internet.
Themodernuserinterfaceisthemainimprovementoverolderversionsof
IQ.1 Moreover,newphysicstopicsareaddedandtherearealsonewgraphical
features.
The present version of INTERQUANTA is easily installed and run on
personalcomputers(runningunderWindowsorLinux)orMacintosh(running
under Mac OSX).
We do hope that by using INTERQUANTA on their own computer many
studentswillgainexperiencewithdifferentquantumphenomenawithouthav-
ing to do tedious calculations. From this experience an intuition for this im-
portant but abstract field of modern science can be developed.
Siegen, Germany SiegmundBrandt
February 2003 HansDieter Dahmen
Tilo Stroh
1 S. Brandt and H.D. Dahmen, Quantum Mechanics on the Personal Computer, Springer,
Berlin1989,1992,and1994;QuantumMechanicsontheMacintosh,Springer,NewYork
1991 and 1995; Pasocon de manebu ryoushi nikigacu, Springer, Tokyo 1992; Quanten-
mechanikaufdemPersonalcomputer,Springer,Berlin1993
vi
Contents
Preface to the Second Edition . . . . . . . . . . . . . . . . . . v
Preface to the First Edition . . . . . . . . . . . . . . . . . . . vi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Interquanta . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Structure of ThisBook . . . . . . . . . . . . . . . . . 2
1.3 The Demonstrations . . . . . . . . . . . . . . . . . . . . . 3
1.4 The Computer Laboratory . . . . . . . . . . . . . . . . . . 3
1.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Free Particle Motion in One Dimension . . . . . . . . . . . . . 5
2.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Planck’s Constant. Schrödinger’s Equation
for a Free Particle . . . . . . . . . . . . . . . . . . 5
2.1.2 TheWave Packet. GroupVelocity. Normalization . 6
2.1.3 Probability-Current Density. ContinuityEquation . 7
2.1.4 Quantile Position. Quantile Trajectory . . . . . . . 8
2.1.5 Relation to Bohm’sEquationof Motion . . . . . . . 9
2.1.6 Analogiesin Optics . . . . . . . . . . . . . . . . . 10
2.1.7 Analogiesin Classical Mechanics:
ThePhase-Space Probability Density . . . . . . . . 11
2.2 A First Sessionwith the Computer . . . . . . . . . . . . . 15
2.2.1 Starting IQ . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 AnAutomatic Demonstration . . . . . . . . . . . . 16
2.2.3 AFirst Dialog . . . . . . . . . . . . . . . . . . . . 16
2.3 The Free Quantum-Mechanical GaussianWave Packet . . . 17
2.3.1 TheSubpanel Physics—Comp.Coord. . . . . . . . . 18
2.3.2 TheSubpanelPhysics—WavePacket . . . . . . . . . 19
2.3.3 TheSubpanelPhysics—Quantile . . . . . . . . . . . 19
2.3.4 TheSubpanel Movie . . . . . . . . . . . . . . . . . 20
2.4 The Free Optical GaussianWave Packet . . . . . . . . . . . 20
2.5 Quantile Trajectories . . . . . . . . . . . . . . . . . . . . 21
vii
viii Contents
2.6 The Spectral Functionof a GaussianWave Packet . . . . . . 22
2.7 The Wave Packet as a Sum of Harmonic Waves . . . . . . . 23
2.8 The Phase-Space Distribution ofClassical Mechanics . . . . 25
2.9 Classical Phase-Space Distribution: Covariance Ellipse . . . 26
2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 BoundStates in One Dimension . . . . . . . . . . . . . . . . . 32
3.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1 Schrödinger’sEquation with a Potential.
Eigenfunctions. Eigenvalues . . . . . . . . . . . . 32
3.1.2 Normalization. Discrete Spectra. Orthonormality . . 33
3.1.3 TheInfinitely Deep Square-Well Potential . . . . . 33
3.1.4 TheHarmonic Oscillator . . . . . . . . . . . . . . 34
3.1.5 TheStep Potential . . . . . . . . . . . . . . . . . . 34
3.1.6 ThePiecewise Linear Potential . . . . . . . . . . . 36
3.1.7 Time-Dependent Solutions . . . . . . . . . . . . . 37
3.1.8 Harmonic Particle Motion. Coherent States.
Squeezed States . . . . . . . . . . . . . . . . . . . 37
3.1.9 Quantile Motionin the Harmonic-Oscillator Potential 38
3.1.10 Harmonic Motion of a Classical
Phase-Space Distribution . . . . . . . . . . . . . . 38
3.1.11 Particle Motion in a Deep Square Well . . . . . . . 41
3.2 Eigenstates inthe Infinitely Deep Square-Well Potential
and in the Harmonic-Oscillator Potential . . . . . . . . . . 43
3.3 Eigenstates inthe Step Potential . . . . . . . . . . . . . . . 45
3.4 Eigenstates inthe Step Potential – Quasiperiodic . . . . . . 46
3.5 Eigenstates inthe Piecewise Linear Potential . . . . . . . . 47
3.6 Eigenstates inthe Piecewise Linear Potential –Quasiperiodic 48
3.7 Harmonic Particle Motion . . . . . . . . . . . . . . . . . . 49
3.8 Harmonic Oscillator: Quantile Trajectories . . . . . . . . . 51
3.9 Classical Phase-Space Distribution: Harmonic Motion . . . 52
3.10 Harmonic Motionof Classical Phase-Space Distribution:
Covariance Ellipse . . . . . . . . . . . . . . . . . . . . . 53
3.11 Particle Motionin the Infinitely Deep Square-Well Potential 54
3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Scattering in One Dimension . . . . . . . . . . . . . . . . . . 63
4.1 Physical Concepts . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Stationary Scattering States.
ContinuumEigenstates andEigenvalues.
ContinuousSpectra . . . . . . . . . . . . . . . . . 63
4.1.2 Time-Dependent Solutions
of the Schrödinger Equation . . . . . . . . . . . . . 64
Contents ix
4.1.3 Right-MovingandLeft-Moving
Stationary Waves of a Free Particle . . . . . . . . . 64
4.1.4 Orthogonality andContinuumNormalization
of Stationary Waves ofa Free Particle. Completeness 65
4.1.5 BoundaryConditions
for Stationary Scattering Solutionsin Step Potentials 66
4.1.6 Stationary Scattering Solutionsin Step Potentials . . 67
4.1.7 ConstituentWaves . . . . . . . . . . . . . . . . . 68
4.1.8 Normalization ofContinuumEigenstates . . . . . . 68
4.1.9 Harmonic Waves in a Step Potential . . . . . . . . 68
4.1.10 Time-Dependent Scattering Solutions
in a Step Potential . . . . . . . . . . . . . . . . . . 69
4.1.11 Generalization toPiecewise Linear Potentials . . . . 69
4.1.12 Transmissionand Reflection. Unitarity.
TheArgand Diagram . . . . . . . . . . . . . . . . 70
4.1.13 TheTunnel Effect . . . . . . . . . . . . . . . . . . 71
4.1.14 Resonances . . . . . . . . . . . . . . . . . . . . . 72
4.1.15 PhaseShifts uponReflection at a Steep Rise
or Deep Fall ofthe Potential . . . . . . . . . . . . . 72
4.1.16 TransmissionResonances uponReflection
at ‘More- and Less-DenseMedia’ . . . . . . . . . . 74
4.1.17 TheQuantum-Well Device
andthe Quantum-Effect Device . . . . . . . . . . . 75
4.1.18 Stationary States in a Linear Potential . . . . . . . . 77
4.1.19 Wave Packet in a Linear Potential . . . . . . . . . . 77
4.1.20 Quantile Motionin a Linear Potential . . . . . . . . 78
4.1.21 Classical Phase-Space Density ina Linear Potential 78
4.1.22 Classical Phase-Space Density
Reflected bya High Potential Wall . . . . . . . . . 79
4.2 Stationary Scattering States in the Step Potential
and in the Piecewise Linear Potential . . . . . . . . . . . . 81
4.3 Time-Dependent Scattering bythe Step Potential
and bythe Piecewise Linear Potential . . . . . . . . . . . . 83
4.4 TransmissionandReflection. The ArgandDiagram . . . . . 87
4.5 Stationary Wave in a Linear Potential . . . . . . . . . . . . 90
4.6 GaussianWave Packet in a Linear Potential . . . . . . . . . 91
4.7 Quantile Trajectories in a Linear Potential . . . . . . . . . . 92
4.8 Classical Phase-Space Density in a Linear Potential . . . . . 92
4.9 Classical Phase-Space Distribution: Covariance Ellipse . . . 94
4.10 Classical Phase-Space Density
Reflected bya HighPotential Wall . . . . . . . . . . . . . 95
4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 97