Introduction to Quantum Mechanics

380 Pages·1960·7.064 MB·English

by  Dicke, Robert H. & Wittke, James P.

Introdutcot ion
QUANTUM
MECHANICS
Introducttoi on
QUANTUM
MECHANICS
by
ROBERT H. DICKE
PrincUentiovner sity
and
JAMES P.W ITTKE
Laboratories
R.C.A.
PrinceNtewo nJ,er sey
ADDISON-WESLEY PUBLISHING COMPANY, INC.
RAEDIMNAGS,USE SATCTPHSLA OAT LOL ODNON
• •
THISB OOKI SA N ADDISON-WESLEY
WORLDS TUDENSTE RIES EDITION
SeconPdr inting
1966
A compleatned unabridgreedp rinotf theo riginAamle rican
work,t hisW orlSdt udeSnetr ieedist imoany b es olodn liy nt hose
countrtioew sh iciht i sc onsignbeydA ddisoWne-sleoyr i tasu thor­
izedt raded istributIotrm sa.y notb e re-exporftreodm t he
countrtyo whichi th asb eenc onsignaendd,i tm ay notb es old
int heU niteSdt ateosf A mericoar i tsp ossessions.
Copyright
© 1960
ADDISON-WESLPEUYB LISHICNOGM PANYI,N C.
ALL RIGHTS RESERVED. THIS BOOK, OR PARTS
THEREOF, MAY NOT BE REPRODUCED IN ANY FORM
WITHOUT WRITTEN PERMISSION OF THE PUBLISHER.
LibroafCr oyn grCeastsaC laorgNd o .
60-11852
PREFACE
Atp resent, mqeucahnatnpuirmco sv iudswe ist thh bee smto dewle
havoeft hpeh ysical wionpr alrdt iaconuftdl h,ase ru ,b microwsocrolpdi c
oft haet omT.h ibso oiksa ni ntrodutoc titpohhney siccoanlc eapntds
mathemaftoirucmlaalt ioofnn osn relatqiuvainstmtueimcc h aniTcos .
getth meo sptr offirto tmh ibso oakf ,a miliwairtbihat syi c undergraduate­
levpehly siicnsc,l uadtionmgi c physics, aenldec cltarsosmiacganle tism,
mechaniisrc esq,u iAr kendo.w leodfdg ieff ereanntdii natle cgarlaclu lus,
ands omfea miliwairtdihit ffye reenqtuiaatlii soa nlssn,oe eded.
Thed ecistiolo inm tihtte e xtton onrelatqiuvainstmtueimcc h anics
enabulset doe xpltohrbeea sicco nceopfqt usa ntmuemc hanciacrse fully
while abvootithdh iceno gm plicaantdui nosnast isafsapcetocoffitr esyl d
theyo.rI ti so ur fibremlt iheaftt h ecsoem plicsahtoiubolenmd se to nly
aftaenru nderstoafna,dn idpn hgy sifceaellf ionnrgo, n relatqiuavni­stic
tumm echahnaibsce se anc quired.
Thitse xitsd esigned fporarin mi anrtirloyd uccotuorirsnqye u antum
mechanaittc hsfie r st-gyreaadru laetvebe ultt) h fier spto rtiisoa nl so
suitafboalr se e niorc-oluervsTeehl.i t se xcta nb ed iviidnetdto h ree
sectiIonnt sh.fie r sste cttihofienr ,s tth rcehea ptweerp so,i onutt the
inadeqoufca lcays sciocnaclet poet xsp lmaainnay t omic-scale phenomena
ands uggheoswtt h bea sciocn ceopfct lsa ssicalm umsebtce ah latneircesd
toe xpleaxipne rimoebsnetravla tiTohners e.a sonabolfae wnaevses­
mechantihceaoolrfp y a rtbiechlaev iisso hro wn,i tia ssne ndeh owc lassi­
calalnyo malous sbuechahas qv uiaonrt,i zoafet nieornag nyda ngular
momentaunmd "tthuen netlhrion"ug g paho tenbtairarbliy e praa rticle,
carne suflrtoa mf ormalwihiscmih n heregnitvbleoyst hw aav aen da
partaiscpleetc ott h een titthyac tl assiwcoaullbldecy o nsidaesar ed
particle.
Thes ecosnedc toifto hne bCohoakp,t 4et rhsr ou10g,sh t awrittsth w o
chaptwehriscl ha tyh eg roundfwoorr km tohrefe o rmpaols,t ulational
apparcothoq uantmuemc hawnhiicfcsoh l lWoewt sh.ep nl aScceh rodinger
wavmee chanoina c mso rdee finaenddr igorboaussai nsd,w eu se toi t
disscs uusch baassipceo cftm se chanaismc esa sureamnegnutlm,oa ­r
mentuamn,dt hbee havoifao p ra rtiinca ls ep heriscyamlmleyt ric force
field.
Thesfier sttw os ectiionwn hsi,ct hh eem phaissoi nsa g raosfpt he
fundamepnrtianlca inpdtl heesmi art hemaftoircmaull awtiitaohm n i,n i­
mumo ft hien volmvaetdh emamtainciaplu ltahtaistooo n fst oebns cure
thpeh ysoifct shm ea tttehre,r effoorramen i ntrodutcott hiseou nb ject
ona l evtehlaa tw ell-pruenpdaerregdr acdauhnaa tned lTeh.ip sa rt of
v
vi PREFACE
thbeo otkh uisss uitfaoburls eie n a nu ndergracdouuarItsteep .r ovides
sufficiberneta adntddh e pbtoht hf atmoi litahrreei azdewe irtt hh bea sic
ideaansdm athematicalo feq xuparnetmsuesmci hoanna incdts of orm
thbea sfiosdr e epuenrd erstlaantdeiWrne.gb elitehvaietti sd esirable
top reseqnuntat umm echanaitac nsa dvanucnedde rgraduate level,
whenepvoesrs ifbotlrhe r,re eea soFnisr.qs uta,n tmuemc hanisis cusca h
bastiocoo lfm odeprhny stihcaistt u ss e shboeau clqdu iarsee adr alsiy s
feasiSbelceo.nt dh,eq uantum-mecphiacntoiufcrt aehl ew orlidsi n
manyw ays fosroe tioge nv erycdoanyc epctosid fiaeisdnc lassmiec­al
chantihcasct o nsidetriamibeslr ee quirdeed vetltooh paf ta miliarity
whicphe rmait trsu tlhyo rogurgahos fpt ·h feu lsli gnifiocfta hnceco en ­
ceptFsi.n aelvleyyr,ey a firn dmso rient hwea yo fi deaansdt echniques
crowdtihnegwi aryi nttohg er aducautrer icuClleuamtr.hl iyns,e wm a­
teriabfe a cccaon mmodoantlbeyydd i scaorldmdia ntge roiria nlt roducing
newm ateriali nte hacero luiroesfsre t udy.
Thet hisredc toifot nh eb ookc,o mpritshieln agse ti gchhta pters,
represae cnotnss idebrraobaldeeo nfti hnvegi ewpaonidno tft hsec ope
oft hper obltehmast bceh aann dlWeedi .n troadlutceer nraetpirvees enta­
tioannsd g eometriinctaelr preotfat thiefo onrsm alainsddm i scuss
methoodfts r ansfofrrmooimnn rege presetnota antoitohCnea rn.o nical
transformatthieorinersl aattnoitdo r na nsfonrsom fat thrieeo presenta­
tioanr ter eatAepdp.r oximmaettihooandr sde e velwohpiecdph e rmai t
tremenedxopuasn isnit ohnne u mboefrp roblwehmiscc ha nb eh andled
witsho maes sucre;at nheasrete h eanp pltioie mdp orctlaanstos fpe rso b­
lemssu,c ahss cattaenrdip nrgo blceomnsc ertnhiienn gt eraocfat ni on
atowmi tsht ro(ncgl assical) efileelcdTtshr.ro omuaggthnhoeteute tix ct ,
emphaissgi isv teona lgebtreacihcn iaqnudte hse,pi orw earn de legance
arcel easrhloyw n.
Ofs pecial imipsto hrelt aascnthc aep tdeera,l wiintgqh u antum­
statimsetcihcaanlHi ecrswe.e d evetlhotepe chnitqhuaaetrs pe l aying
ane ver-incrroeliaensm iondge rpnh ysiUcnst.ni olw t,h ehsaev ien,
genebreaelnn,e gleicnt etdoe nxq tusa nttuh�omr y.
Toa vopiods smiibsllee aidmipnrgen ssas,li colu rvients h ibso ohka ve
beecna refcuolmlpyu atnedpd l otted.
Fotrh ree adwehro ltiobk reosa hdieusnn derstoafnt hdseiu nbgj beyc t
seeihnogwi td evelonpoeitdn t, h hei ndsoifgp hrte senktn-odwalye dge
buats i ta ppeatrote hdo osnet hfeo reforfto hnidtse velobpriannogcf h
physitchfiser, s t chaprteefre rgteionmvc aeenssyo f t hkee yp apeirns
thed evelopomfet nhteb asviice wpoainndtc so nceopfqt usa ntum
mechanics.
Exercisgeisva etant r heeen dosfm oscth aptTehresfy.u rtihlelru strate
variaosupse acntdcs o nsequoeftn hcqee usa ntum-mechanoifc al picture
natuarnedg ivae q uantit"afteifevolcer"e rtasapienc otfsn atuarned,
PREFACE vii
devealw oopr king ciantp hasebo illuiottfipy or no bleWmeb se.l ieve this
cacno me tohnrloyu gh conpsrcaiceitnnitt chiueeos uoesf t haes sociated
mathemattoioclasl.
Wew isthoa cknowltehdaegd ev iocfPe r ofeSs.Ts roerimaastn o t he
mosdte sircaobnlteae nndto rganizoaftt hiiboson o kT.h eh elpM ro.f
N.V andVe re inne dittihnoegr igniontaoelns w hicthh itse xitsb ased
isa lsaoc knowleTdhgiebsdo .oh ka sb enefiftreodtm h ede taicloemd­
mentosfM r.Y ounKgi mw,h or eatdh mea nuscrWiep wti.sa hl stoo
acknowltehdecg oen sidehrealabpnl dee ncouraggeimveebnnyto ur
wivtehsr ougthhopeur te paroaftt hiiboson o k.
Finaalw loyr,md a yb es aiadb onuott atGieonne.r ally,re psryem­bols
senting anruwemr bietrtsei nt ali(inac )as,n dv ectqoura ntiatriein e s
boldfraocmea( an;) o peraatroiern sl ightrfoamcacena pit(aA,l)a sn d
m-atriacrieensb oldfsaacnsese rifA tt imietis sn o otb vijouushsto w
(a).
ag ivseynm bol bsehc oounlsdi ;dt ehriiessed spectirauilenlC yh apt1e3r,
ont ransforomfra etpiroensse nItnas tuicaohmn bsi.g ucoausswe esh ,a ve
triteodc hootshete y pfeo ntth awto ulmda ket hse ignifiocfat nhcee
expresasthi aonncd l eatroet shte reIatdi eshr o.p etdh atth e choices
helrpa,t htehrah ni ndtehrre,e adienh1 i sst uodfyt hibso ok.
March R..HD.
1960
J.P.W.
CONTENTS
CHAPTERI NTRODUCTION 1
1.
1-1Qu antmuemc hanais cyss,to efdm y namics 1
1-2E videonftc heie n adeqoufca lcays mseicchaaln ics 3
1-3S omnee cescshaarrya ctoefqr uiasnttituchmse ory 14
1-4S ummary 18
.
CHAPT2E.RW AVEM ECHANICS 21
2-1T hew ave-padrutailciltey 21
2-2T hew avfeu nction 23
.
2-3T heu ncertraeilnattyi on 27
2-4W avpea ckets 31
2-5S ummary 34
.
CHAPT3E.RS cHRODINEGQEURA'TsI ON 36
3-1T hee quatoifmo ont ioofan w avfeu nction 36
3-2O ne-dimemnostiioonnaa lp p oatsetnh tiilall 40
3-3O ne-dimemnostiioronenafl:le cbtyai noi nn finiwtiedbleay r rie4r6
3-4O ne-dimemnostiioionnana p lo tenwteilall 50
3-5P artfliucxl e 60
3-6S umma.r y 62
CHAPTERF OURITEERC HNIQAUNEDES X PECTAVTAILOUNE S
4. . 64
4-1T heF ouriinetre gral 64
.
4-2T heK ronedcekleatrna td h Dei rdaecl ta function 66
4-3E igenveaqluuaet ions 69
4-4E xpectvaatliuoens 71
4-5S ummray. 75
CHAPTERR EVIEOWFC LASSIMCEACLH ANICS 77
5.
5-lI ntroduction 77
.
5-2 Generacloiozreddia nnaLdta egsr anegqeu'ast ions 77
5-3H amilton's equations 82
5-4P oisbsroanc kets 85
5-5C anontircaanls formations 86
5-6S ummray. 88
CHAPTERO PERATFOoRR MALISM
6. 90
Postuolfaq tueasn tmuemc ha.n ics 90
6-1
6-2A lgebmreatihco ds 103
6-3M any-pasrytsitcelmes 109
6-4S umma.r y 121
ix
CNOETNTS
X
CHAPTE7R. M EASUREMENT 115
7-T1h mee anoifmn ega surement 151
7-P2h otpoonl ari.z ation 116
7-S3u mma.r y 121
CHAPT8ER. T HEC oRRESPONDPERNICNEC IPLE 122
8-T1h ree laotfqi uoann tmuemc hatnoci lcass mseicchaaln ics1 22
8-T2h ter ansfirtoqimuo ann tmuemc hatnoci lcass mseicchaaln i1c22s
8-T3h ceo rresppornidneacnnictdpeh lu een certraeilnattyi on1.2 9
8-4 mTihnei muunmc ertwaaivfneut nyc tion 131
8-T5h uen certpariinntacynit dph lseei mhpalrem oonsicci lla1to3r2
.
8-S6u mmary 143
.
CHAPTE9R.A NGULAMRo MENTUM 173
9-O1r bital anguolpaerr-amtoomresn tum 173
9-O2r biatnaglu lar-mwoamvfeeun ntcutmi ons 124
9-A3n gumloamre nitnug me neral 148
9-A4d diotfia onng umloamre nta 149
9-C5l asosp erators 125
T
9-S6u mmya.r 145
CHAPTE1R0 .CE NTRFAOLR CES 165
1-01Q ualibteahtaiwvviieto ahrna ttrapcottievnet ial 165
1-02T hhey droagteonmi c 185
1-03T he three-dimensional oscilla1t66o r
1-04T hfer peaer ticle 170
1-05P arity 172
10-6S ummary. 147
CHAPTE1R1 M.A TRIRXE PRESENTATIONS 176
1-11M atrriexp reseonfwt aavtfeiu onncsta inoodpn esr ators 176
1-12M atrailxg ebra 177
.
1-13T ypoefms a trriexp resentation 108
1-14I nfinmiatter ices 148
.
1-15S umm.a ry 168
CHAPTE1R2 S.PI NA NGULAMRo MENTUM 198
12-M1a trriexp reseonfat nagtuiloanr -moopmeernattuomr s 189
1-22S ystweimtsshp ionn e-half 194
12-E3l ectrporne-csepsisni on 195
1-24P aramargenestoinca nce 199
1-25S ummary 205
.
CHAPTE1R3 T.R ANSFORMAOTFIR OENPSR ESENTATIONS 208
13-I1n troduction 208
.
1-32A g eometarniaclaolg ues-pHaiclebe rt 210
1-33E igenevqaulautei ons 212
13-G4r opurpo peorfutn iiettsar rayn sformations 214
CONTENTS xi
13-5C ontinmuaoturs.i ces 214
13-6C anontircaanls formations 217
13-7S umma.r y 222
CHAPT1E4R.A PPROXIMMAETTIHOOND S. 226
14-1T hen eefdoa rp proximmeatthioodns 226
14-2T ime-indeppeerntduerntbtha etoiroyn 226
14-3T ime-dependent perturbation the2o3r7 y
14-4V ariattieocnhanli ques 242
14-5T he WKB me.t hod 245
14-6S umma.r y 253
CHAPT1E5.R I NTERACTIOAN S TRWOIENTLGHE CTROMAGNETI25C9 FIELD.
15--T1h eH amiltoofnap i aanr tiinac nle el ectromfiaeglnde tic2 59
15--2M otioofan f reele ecitnra uo nni form mfiaeglnde tic 260
15--3T hew eak-Zfieeelmdea ffne c.t 265
15-4T heg -fa.c tor 269
15--5T hes trongZ-efieemleadffn e ct 270
15--6I nteraocfat nia otno meilce ctwriotanh p laenlee ctromagnetic
wave. 272
15--7S elecrtuiloens 278
15--8S umma.r y 282
CHAPT1E6R.S CATTERING 285
16-1P hysiccoanlc epts 285
16-2T heB oranp proximation 291
16-3P artwiaavle s 297
16-4S umma.r y 308
CHAPT1E7R.I DENTIPCAARLT ICLES 311
17-1T hep article-exchange operator 311
17-2T heP auplrii nciple 312
.
17-3T hes pin-indHeapmeinldteonnti an 315
17-4E ffecofts pisny mmeotnrt yhe en erogfay s tate 318
17-5V alebnicned iintn hghe y dromgoelne cule 324
17-6P araan-odr tho-hydrogen 327
.
17-7S ystoefmm so rteh atnw poa rticles 329
17-8S umma.r y 329
CHAPT1E8R.Q UANTUTMA-TSISTMIECCAHLA NICS 331
18-1I ntrodu.c tion 331
18-2T hed ensmiattyr ix 332
.
18-3T hee quatoifo nm ootfti hodene nsity matrix 337
18-4O rdearnedddi sorednesreemdb les 338
18-5S tatioennasreym bles 342
18-6S ystoefmn so nintepraarcttiicnlge s 345
18-7I degaal.s 349
18-8S ummary 354
.
TABLoEF A TOMCIoCN sTANTS 360
.
INDEX 361
.