The Picture Book of Quantum Mechanics

464 Pages·2001·19.318 MB·English

by  Siegmund Brandt, Hans Dieter Dahmen (auth.)

Frequently UsedSymbols
a Bohr radius H Hamiltonian
a(xo, PO, XS, PS) analyzingamplitude j probabilitycurrentdensity
AI, An, ... amplitudefactorsinregious jfJp) sphericalBesselfunction
BI, Bn, ... I, II, ...ofspace
k wavenumber
B magnetic-inductionfield e
angular-momentum
C speedoflight quantumnumber
C correlationcoefficient V:ni Laguerrepolynomials
d~m' Wignerfunction L angularmomentum
D~m' Wignerfunction i. angular-momentumoperator
dcrjdQ differentialscattering m quantumnumber
crosssection ofZcomponent
ofangularmomentum
e elementarycharge
Ec complexelectric fieldstrength m,M mass
E energy M(x, t) amplitudefunction
M magnetization
En energyeigenvalue
n refractiveindex
Ekin kineticenergy
E electricfieldstrength n principalquantumnumber
f(k) spectralfunctionwithrespectto ne(P) sphericalNeumannfunction
wavenumber n unit vector
f(p) spectralfunction withrespectto P momentum
momentum
P momentumoperator
f(7'}) scatteringamplitude
(p) momentumexpectationvalue
[e partialscatteringamplitude
p momentumvector
lem(e,l/)) directionaldistribution p
vectoroperatorofmomentum
g gyromagneticfactor
Pe Legendre polynomial
h pm
Planck'scoustant e associated Legendrefunction
h = hj(2rr)
r relativecoordinate
h~+)(p)
sphericalHankelfunction
r radialdistance
ofthefirst kind
h~-)(p) r positionvector
sphericalHankelfunction
ofthesecondkind R center-of-masscoordinat
R(r), Re(k,r) radialwavefunction 11e(r) scatteredpartialwave
e
Rne radialeigenfunction iJ, polarangle
S spinquantumnumber iJ scatteringangle
=
S (SI, S2, S3) spin-vectoroperator A wavelength
Se scattering-matrixelement ILo vacuum permeability
time IL reducedmass
T oscillationperiod JL magnetic moment
T transmissioncoefficient p probabilitydensity
T kineticenergy pel classicalphase-spaceprobab
density
TT, TR transition-matrixelements
(TO widthofground state
U voltage
ofharmonicoscillator
VO group velocity
(T\, (T2, (T3 Pauli matrices
v phasevelocity
p
(Tk widthinwavenumber
V potential (energy)
(Te partialcrosssection
Veeff effectivepotential
(Tp widthinmomentum
w averageenergy density
(Tx width inposition
W Wignerdistribution
(Ttot total crosssection
W energy
cp(x) stationarywavefunction
We, Wem coefficientsintheangular
cpp(r) stationaryharmonic
decomposition ofawavepacket
wavefunction
x position
cp statevectorofstationarysta
(x) positionexpectationvalue
cP,f1> azimuthalangle
fem sphericalharmonic
X magnetic susceptibility
Z atomicnumber
X generalspinstate
ex fine-structureconstant
1/J(x,t) time-dependentwavefuncth
de scatteringphaseshift
1/Jp(r,t) harmonicwavefunction
L1k wave-numberuncertainty
t statevector
L1p momentumuncertainty
to angularfrequency
L1x positionuncertainty
Q solidangle
cO vacuum permittivity
1 V nabla(ordel)operator
11\, 11-\ spin basestates
2
V Laplaceoperator
11k(r) scatteredwave
THE PIG UREBOOKOF
QUANTUM
MECHANICS
Third Edition
Springer Science+ Business Media, LLC
I'elmund arantU I Hans I'M. la"men
THE PICTURE BOOK OF
OUANTUM
MECHANICS
Third Editian
With 21 2 IIlustrations
Springer
Siegmund Brandt Hans Oieter Oahmen
Physics Oepartment Physics Oepartment
University of Siegen University of Siegen
0-57068 Siegen 0-57068 Siegen
Germany Germany
[email protected] [email protected]
Caver iIIustratian: Large figure-The hydrogen atom in the eigenstate n =3 , 1= 3, m =O . (Shown is
=
the surface p(x, y, y) 0.0002 of constant probability density.) Small figures-Polar diagrams of the
absolute squares of various spherical harmonics.
library of Congress Cataloging-in-Publication Data
Brandt, Siegmund.
The picture book of quantum mechanics/Siegmund Brandt, Hans Dieter Dahmen.-
3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4612-6532-0 ISBN 978-1-4613-0167-7 (eBook)
DOI 10.1007/978-1-4613-0167-7
1. Quantum theory. 2. Wave mechanics. 1. Title: Quantum mechanics. II. Dahmen,
Hans Dieter, 1936- III. Title.
QC174.12 .B73 2000
530. I 2--dc21 00-061867
Printed on acid-free paper.
First edition © 1985, Wiley, New Vork.
© 2001, 1995 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 2001
Softcover reprint of the hardcover 3rd edition 2001
AII 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, 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 of general descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that such
names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be
used freely by anyone.
Production managed by Francine McNeill; manufacturing supervised by Jacqui Ashri.
Photocomposed copy prepared from the authors' I8.TEJX files.
9 8 7 6 5 4 3 2 1
Additional material to this book can be downloaded Crom http://extras.springer.com.
15BN 978-1-4612-6532-0 SPIN 10780678
To Renate Brandt
and Ute Dahmen
Preface to theThird Edition
Forthethirdedition,thePictureBookwasextendedintwoways.ACD-ROM
wasincluded,and achapter onquantilemotionwasadded.
On the CD-ROM, all the figures in the book are available in full color.
Readers can look at (or print) individual figures and also posters in which
relatedfiguresandtheircaptionsaregrouped together.The material, ifprinted
ontransparenciesorintheformoflargeposters, canalsobeusedasateaching
aid.The useoftheCD-ROM isdescribed onpage xvii.
With the quantile concept taken from mathematical statistics, it is possi
ble to define trajectories and velocities not only for point particles but also
for probability distributions.In this way, for instance,the velocity of a wave
packet undergoingatunnel effectcan bediscussed.
The realizationoftheCD-ROM wouldnothavebeenpossible without the
ideas,theexpertise,and thediligenceofDr.Erion GjonajandMr.TiloStroh.
The computer typesetting and the layout of the text were performed by
Mr. Stroh. Our warmest thanks go to both of them. We would also like to
thank Dr. Thomas von Foerster of Springer-Verlag New York, Inc., for his
continuinginterestand formaking thisedition possible.
Siegen, April 2000 Siegmund Brandt
HansDieter Dahmen
vii
Preface totheSecond Edition
Inthe second edition, aim andstyle ofthePictureBookwere leftunchanged.
It is our aim to explain and exemplify the concepts and results of quantum
mechanics by visualization through computer graphics and, in parallel, by
the discussion of the relevant physical laws and mathematical formulae.The
scope ofthe book, however,was widened appreciably.
The most importantextension isthechapterabout spin andmagneticres
onance. In three-dimensional quantum mechanics,the presentation of wave
packet motion on elliptical and hyperbolic Kepler orbits should help to es
tablish the correspondence and the differences between the classical and the
quantum-mechanicaldescription ofplanetary motion.
Maybe thebest-knownquantum-mechanicalconcept isthatofuncertainty
introduced byHeisenberg, whoformulated thecelebrated uncertainty relation
L.\x/sp ~ II/2 for the product of the uncertainties of a particle in position
and in momentum. Also, in classical mechanics position and momentum of
a particle may be known only up to some uncertainty so that a probability
densityinphase space (spannedbyposition x and momentum p) isneeded to
describe the particle. For many physical situations, we present and compare
the time development of this classical phase-space probability density with
the probability density ofquantum mechanics.
An additional tool we use is the analyzing amplitude introduced in Ap
pendix C. It allows the definition of a directional distribution that is very
helpful in visualizing angular momentum and spin wavefunctions.
Togenerate thecomputergraphics ofthe firstedition ofthePicture Book,
we developed an interactive program on quantum mechanics. A modernized
version, which we call INTERQUANTA (abbreviated IQ), together with an
accompanying text has been published by Springer-Verlag in various edi
tions.v" It is apleasure toacknowledge the generous help provided by IBM
Germany in the development of IQ. In particular, we want to thank Dr. U.
Groh forhis competenthelp intheearly phase ofthe work. At various stages
oftheproject, wewerehelpedconsiderablybyfriendsandstudents inSiegen.
Wewould particularly like to thank THoStroh for his many valuable contri
butions.
ix
x PrefacetotheSecondEdition
Allcomputer-drawnfiguresinthepresenteditionwereproducedusing the
publishedversion ofIQorrecentextensions realized with the help of Sergei
Boris, Anli Shundi, and Tilo Stroh.The computertypesetting and the layout
ofthetextwere donebyUteBenderandAnli Shundi. Wewould liketothank
them for their excellentwork.
Last but not least, we thank Drs. H.-U. Daniel, T. von Foerster, and
H.1. Kolsch ofSpringer-Verlag for their constantinterestand support.
Siegen, May 1994 SiegmundBrandt
Hans DieterDahmen
IS. Brandt and H.D. Dahmen, Quantum Mechanics on the Personal Computer, 3rd ed.,
Springer-Verlag,Berlin, Heidelberg, NewYork,1994.
2S.BrandtandH.D.Dahmen, QuantumMechanicsontheMacintosh,Springer-Verlag,New
York,Berlin,Heidelberg, 1991.
3S. Brandt and H.D. Dahmen, Pasocon de Manebu Ryoushi Nikigacu, Springer-Verlag,
Tokyo,1992.
4S. Brandt and H.D. Dahmen, Quantenmechanik aufdem Personalcomputer, Springer
Verlag,Berlin,Heidelberg, NewYork,1993.