11

Relativity and quantum mechanics

The early twentietb ceutury saw two major revolutiollS in the way physicists unders tand the world One was qllantum mechanics itself and lhe otlll was the theory of relativity Jmponanl results also clnergcd whe n these two ideas were brought together some of these have been refelTed to in earlier chaplers- in particular tbe fact thaI fundame ntal pmlicJes sneh as electrons h(lve intnnsic angular momentum (spin) was stated to be a rela ti Vistic effec L This chapter explores the relationship between relativity Hlel quantum mechanics morc deeply

A full unde rstanding of relativistic quantum mechamcs is well outSide the scope of thi s book but many of the important results c an be understood at Ihis leve l and these will be discussed in tllis chapter AIlee a sho rt summary of the main results of special relativily we show how combiuing lhis with [he lime-dependent Schrodingcr equalion leads to a new wave equation known as the Dirac equalion We show how the Dirac equation requi res particles sllch as electrons La ha ve inllinsic angular momerllum (spin) and we explore some of its other consequences The ehapter conclUdes with au ou tl ine of some more advanced Ideas known as quantum field theory

OUf treatme nt is confined to Lhe quantum elTec ts associaled wilh special rda uvllY The reconciliation of quanlum mechanics with general relati vity is stili a lOpic o f aClive resea rch and few If any generall y accepted resulls have emerged from it so far

111 Basic Results in Special Relativity

Spec ial relativity modUles class ical (ie nonquantulll ) kinematics a nd dynamics to enco mpass phenomena that become increasingly s ignifi cant when particles mo ve al spe-eds comparable (0 the speed or lighl V-k will a~ ~ume lllat the reader is familiar with lllt mai n resull s of special relaivily and thi s section will be reqri -ctcd to a summary of those needed i ur ou r later di scussion

I The lme matics of relaivjy mmiddote governed by the Lorenl z transfo rmltl lion which relategt the position and time coordinates (x y 2 I) of an event observed in ( Ille ineni(ll

f frame 01 efe fence [0 those (x Z I) observed from anu ther moving al co nstant

247 246 Quantum Mechanics

vcloci ry v in the x dircc rion reialivl (0 the firs t We have

(1raquor X - lit X ~ ~=== y = V 1= (1 1) =-shyV - 1 ve

where c is (he speed of lighL The momentuill p and energy E of a particle also transform by a Lorentz transformation where x y z alld CI in (11 1) arc replaced by Pr Py pz and fi e respectively the energy inc luding the rest-mass ene rgy mel In th e case of a frce particle the energy and IDomenlum are re laled by

pound2 = c7 t- m2c4 ( 112)

Equation 012) is an example of a Lorel1lz iIvariant- ie it has th e same form in all inerliltl] fra mes of reference as can he verified by applying the Lorentz trausformation Lo the componenls of p and E and substituting into (1 L 2) Also ii we defiue pound by E = Ine2 +E the nonrelativl stic limit is when pound laquo11(2 in whicll case ( J 12) reduces to f = p22m the NcwOnL an express ion fo r kinetic energy

We now conside r the case where the pltHtic le is not free but subjecl La an elec tromagne tic field This fi eld can in tum be represented by a sealar potenlial ~~ (r) and a vecLor potential A () wbere the e lectric field is 0 = - V cent +a1 at and the magne tic fl eld is B = V x A The components of A along with IP Lhen form a fourmiddot vec tor ~ i lll ilar to tha t form ed by p and E It can tlleu be shown thaI (he behav iour of a parLicle with charge q is deLermined by

(E _ qltraquo 2= (p-qA) c +11c ( 3)

112 The Dirac Equation

To develop rel)liviSlic quanl1lm theory we lqok for a wave equa tion that re lates to (113) in a man1)er analogous 10 Lhe relahOll belween the Newtonian expression for kjne tic eneLgy and the SehrodiJlger equation-d chapte rs 2 and 3 This was flrst done successfully for a particle such as an electron by P A M Dirac in 1928 and om treatment basica lly foll ows his approach (n the same way as Lb e Schrodinger equal ion cannot be derived from classical mechanics bccall~C 1 i ~ essentiall y new physics any reiati visllC cgualion can ouly be guessed aL by a process of lnduc tion and its truth or otherwise must be establ ished by tes ting ItS consequences against experllllcnt Following DiI we stalt this process by considering thc tilne-dependent Schrodinge r equat ion

() (IlA)in at I = HVI

and begin as we d id in the llo llldativisLic C(lse by cOllsuJcrili g Llle case of J free

partiC le

Relativiry and quanlum mecuJIlics

A free particle

Ve consider a particle such as all elec tron wilh mass 111 (dropping Lhe subscrivt to keep the notation as simple as possible) in ltl field -free region of space - i e where A - if = O FollowlIlg the principles of postulate 3 in chapter 4 we assume that the energy opera(Q r I-I CUll be expressed in Lerms of the momentulO operator P in lhe same way as E IS re lated to p in the cJassicallimiL Hence using (J J 2) and (11A)

Ii avr = llPc + fI2el I ( 1l5) at Before we can proceed we have (0 know how to deal with an expression such as

Lhe ri ght-hand side of (115) which has Lhe form of an o perator that is a square rOO l o f ano ther operator There is no dcnlli le prescription for handling this bUL we do know Ulat in ordcr Lo preserve Lore nL invariance positional coorchnares and Lime musL appear in a sinular way in any relati vis tic theory So because the left-hand side of (11 S) is linear in Ji if we are to make the standard replacement PI = - ilt j etc the right-hand side should be linear in these quantities Apply ing thi s pnnciple and following Dirac we write

a Imiddot I ihat lJI = call~ +C(X2 Py +CG3Pl + 3mc lJI ( 116)

where the (XI and 3 are dimension less CJlJantities that are independent of posillon and lime

for (11 5) and (116) 10 be cons istent thc squares of the operators a ll the right shyhand sides of theseuro equ ations should be equivalent Thal is

[caIPx +ca2P cet)- + 3mc212 = c2 p2 + 1I12A (I 17)

Multiplying olll the lefl-hand side of ( 11 7) and equaLing corresponding Lerms leads [0

a =af - af = I a [ ~ + ~~ = ~~ + ~~ = ~a + ~~ = O (l18)

aJ + Ja[ = aJ +Jal ~ a fJ +la = 0

This is obviously not possible if a l and 3 a rc scaJ ar numbers and Dirac showed thaI the simplest expressions for a l and 3 consistent with (118) are a set of 4 x 4 maLrices

0 0 0 -i] 000 ] o 0 00 01 0 ia=UI = 0100 0 - 0 0 [ [0 00 i 0 0 0

fo a I 0] I 0 0 0] IJOO-l 0 0 0a 3 fJ = (119)1 0 0 0 00 -1 0[l0-1 ) deg 00 0 - 1

248 249 Quantum Mechanics

T he reader shoul d check Ihat (bese expressions have the prope rties sel out in (118) Equation (1 17) c ornplcltH~lIled hy the defi llliions ( LJ 9) constitutes the Dirac equal ion for a free plrlicJe We analyze it furt her to draw Oll t jts phYi ical sjgni fic ance and to e~plore jPi tolulio ns

A remarkable fea ture o f the cxpre)sio l1s (1 19) is that tJ1C three 4 )( 4 1l1l lnccs representing a Ie all made II p fro m 2 x 2 Pauli spin mai llecs il ltrod uced 1n chapter 6 ilnd defined in (615) as

fO - i1 (1 11 0) = [~~l ~ li O J = l~ ~I j while J can be ex pressed in erlll s of the 2 x 2 unit maLrjx (represenled by I) Thus

) 12 3 llJl I) C = [0i 0i) = and iJ = [~ ~l

wllcre a = ax etc The fo llow ing properti es of the Pauli spin malricci which can be eltlsily proved by direct substitutio n wilJ be used shortl y

2 =0bull2 = 0y1 = a (l1 12) axOy = - Oy Ox = iOl

[ 01 [- 0] iOxOy = 0 - Oy Ox = 0 i

Similar results ho ld fo r c yc li ~ pennutations or the Carlesi au coord inates The fac l Ihal the Dirac equa ti on is a matrix equation imp lies Ihat V is a veclOr

formed out of fou r functions or pos ition and lime

( 111 3)VI = [~~l = [V)VI) 1shyIJf4

where 0+ and IJI- are two-component vec tors defined by ( I L 13) We now use the aboye to rewrite the Dirac equation (116) as

JlJI (o middot p)c V_+ mc If = i fi~ ( 111 4)

iJI_ ()amiddot p c yt - mc Y- =i LJt

When we int roduced the nonre lalivislie Sehrodinger equation in lhapre rs 2 ltlnd we separated oul the l ime dependence anJ den ved the time-independent equa tion by pu lling vr = I exp( -Eth) Followi ng Lhe same procedure he re we get

2(0- middot p)ClL + mc u+ = EI+ ( I I 15)

(0 middot J cu+ - mc2 1L =- Eu_

Relativity alld quallltim mechanics

We note aga in ltJal aJi the lenns in ( 11 15) ltIre 2 x 2 rnatrice so Ihal each of these f-qualions actuaJl y represents two equations

We uow look tor SOJu tlO llS o f (1 l 15) that will correspond to the (in t independent pan or tht wave func tion of a rela tivisti c free part icle We first usc the second o r (11 15) to express _ in lenns or u-shy

14 _ r _ - ~(o p )u+ (I IJ G)

We subs titute thlS 11110 the fi rst o f (1 1 J5) to gel

(7 p)cu = (E - mc)(E + mc) II (1lJ 7)

Expclllding the first tert(l on tlie leftmiddot hand side in Cartesian coord inates M d rea rrangshying we get

2 ~2 ~ - 2 2 2 Ox Px + (Ox Oy + Uy O( ) PxPy-t oJe Ut = (pound - In C Ut (1118)

where the symbol +0 illlpUes repea ting (he previous expressio ns twice with cyclic permutation o f the Cartesian coordinates Using ( 11 12) we can rewri te (1 J18) as

2jJ2C2U+ = ( F 2 _m c4 )u+ (1119)

or _fi2c2V 2U_ = (pound2 _ m2c) u + (1 120)

where we have used the differeJHiai operator represen tation of momentu ill TI)ls equation has pJane wave solu tions for

II = v+exp (ikmiddotr) (1121)

where

v+ = [~~l and V and )2 are CO nSlJnrs SO that Vt is an eigenvector of spin in a direc tIO n de te rmined by IJle rela ti ve values of VJ and V2 We a lso have

2 4pound 2= fi2c2k2 i m c ( 1122)

This is identical to ( 112) provided p = filL which is JUSt the de Brog lie re lation relating momcn tUlTl a nd wave vector introduced in chapte r 1 The components of u_ can now be obta ined by substitu ting (1 12 1) into ( 11 16) Fo( convemence we choose the direction of the z axis to bt parallel ro k so OWl

(0 middot))11+ 0[ ( - iii z) I cxp k~ ~ Kv exp(ik (1123 )

whcrl~ usi ng OI IO) V I I

v_ U = (1 24) [ [ 1 ~ )j - 12

248 249 Quantum Mechanics

T he reader shoul d check Ihat (bese expressions have the prope rties sel out in (118) Equation (1 17) c ornplcltH~lIled hy the defi llliions ( LJ 9) constitutes the Dirac equal ion for a free plrlicJe We analyze it furt her to draw Oll t jts phYi ical sjgni fic ance and to e~plore jPi tolulio ns

A remarkable fea ture o f the cxpre)sio l1s (1 19) is that tJ1C three 4 )( 4 1l1l lnccs representing a Ie all made II p fro m 2 x 2 Pauli spin mai llecs il ltrod uced 1n chapter 6 ilnd defined in (615) as

fO - i1 (1 11 0) = [~~l ~ li O J = l~ ~I j while J can be ex pressed in erlll s of the 2 x 2 unit maLrjx (represenled by I) Thus

) 12 3 llJl I) C = [0i 0i) = and iJ = [~ ~l

wllcre a = ax etc The fo llow ing properti es of the Pauli spin malricci which can be eltlsily proved by direct substitutio n wilJ be used shortl y

2 =0bull2 = 0y1 = a (l1 12) axOy = - Oy Ox = iOl

[ 01 [- 0] iOxOy = 0 - Oy Ox = 0 i

Similar results ho ld fo r c yc li ~ pennutations or the Carlesi au coord inates The fac l Ihal the Dirac equa ti on is a matrix equation imp lies Ihat V is a veclOr

formed out of fou r functions or pos ition and lime

( 111 3)VI = [~~l = [V)VI) 1shyIJf4

where 0+ and IJI- are two-component vec tors defined by ( I L 13) We now use the aboye to rewrite the Dirac equation (116) as

JlJI (o middot p)c V_+ mc If = i fi~ ( 111 4)

iJI_ ()amiddot p c yt - mc Y- =i LJt

When we int roduced the nonre lalivislie Sehrodinger equation in lhapre rs 2 ltlnd we separated oul the l ime dependence anJ den ved the time-independent equa tion by pu lling vr = I exp( -Eth) Followi ng Lhe same procedure he re we get

2(0- middot p)ClL + mc u+ = EI+ ( I I 15)

(0 middot J cu+ - mc2 1L =- Eu_

Relativity alld quallltim mechanics

We note aga in ltJal aJi the lenns in ( 11 15) ltIre 2 x 2 rnatrice so Ihal each of these f-qualions actuaJl y represents two equations

We uow look tor SOJu tlO llS o f (1 l 15) that will correspond to the (in t independent pan or tht wave func tion of a rela tivisti c free part icle We first usc the second o r (11 15) to express _ in lenns or u-shy

14 _ r _ - ~(o p )u+ (I IJ G)

We subs titute thlS 11110 the fi rst o f (1 1 J5) to gel

(7 p)cu = (E - mc)(E + mc) II (1lJ 7)

Expclllding the first tert(l on tlie leftmiddot hand side in Cartesian coord inates M d rea rrangshying we get

2 ~2 ~ - 2 2 2 Ox Px + (Ox Oy + Uy O( ) PxPy-t oJe Ut = (pound - In C Ut (1118)

where the symbol +0 illlpUes repea ting (he previous expressio ns twice with cyclic permutation o f the Cartesian coordinates Using ( 11 12) we can rewri te (1 J18) as

2jJ2C2U+ = ( F 2 _m c4 )u+ (1119)

or _fi2c2V 2U_ = (pound2 _ m2c) u + (1 120)

where we have used the differeJHiai operator represen tation of momentu ill TI)ls equation has pJane wave solu tions for

II = v+exp (ikmiddotr) (1121)

where

v+ = [~~l and V and )2 are CO nSlJnrs SO that Vt is an eigenvector of spin in a direc tIO n de te rmined by IJle rela ti ve values of VJ and V2 We a lso have

2 4pound 2= fi2c2k2 i m c ( 1122)

This is identical to ( 112) provided p = filL which is JUSt the de Brog lie re lation relating momcn tUlTl a nd wave vector introduced in chapte r 1 The components of u_ can now be obta ined by substitu ting (1 12 1) into ( 11 16) Fo( convemence we choose the direction of the z axis to bt parallel ro k so OWl

(0 middot))11+ 0[ ( - iii z) I cxp k~ ~ Kv exp(ik (1123 )

whcrl~ usi ng OI IO) V I I

v_ U = (1 24) [ [ 1 ~ )j - 12

250 Quantum Mechanics

dlHl l herefore

bullbull_ _ c [ ] (1125)mrl+ E -ll cxp(ikz

Collec ting resulLS fWIll 0113) ( I 121) and ( I 125) we have ~

~ = r 1exp[i(k Lil ( i)1 1126)

l --t l VI _ r

IIJ-t E 2

We firs t note lilal (he wave function has the form of a pl ane wave which is an eige nfunctio n of the momentum operato( - Uiv as well as (he energy operator (hi s result is [he same as found in the nonrelaLiyisLi c ltase discussed in ch(l plcr 4 We ~ow consider how (1126) relates to the idea of spin developed jn chapler 6 The wave fun ellon lJI is nOI in genera l an eigenvector of spin beca use v+ amI v _ 1l1ay be eigenvectors of diffe rent spin components However but the re are two particular cases where VI does represent a spm eigenstate The first of Ihese is lhe norue lativistic limit where IVlte lt lt me2 we can then p ut v_ equal to ze ro so tha t v+ multiplies the whole wave [unction T he part icle is lhen in an eigenstate of spin whose direction is dettnn ined by the relati ve values of Vt and V2 We note tha t this description is

identical to lhal developed in chapter 6 where we bad to add Ole property of spin by hand and we Cltln now see thaI spin is emerges in ltl natural way from Dirac s relati vistic theory when taken to il s nonrelarivistic limit The second case where 1JI is a spin e igenvector is when the ax is of quantization is in the direction of the particle momentum We have taken thi s to be the zdirection so e ither V2 = 0 when the spin is par (lllci to t or VI = 0 whcn the spin is anI ipdral1eJ 10 z in bot h cases v+ 110] or [0 1J mult iplies the whole wave fu nc tion Slates like thi s where the spill is parallel or antipa ra llel to the parti cle momentu m are known as sla les of defin ite hcliciry and he li city plays an importanl role in the analys ts of the beha viour of high ene rgy e lectro ns and otJler sp in half parlJcles

Worked RXlJnpJe 111 Confiml expl ici ll y that lhe e( pression give n in ( I I 26) 1lt a soluti on to the Dirac equation as se t out in ( 116) with pound ~~ ivcn by (1 1 22)

Solut ion Writ (11 26) as II = vex pji(kz - poundt j li)l ~llId subs tilutmg into (116) oc ~l t

E ll cxdLk t i l~ mc2v

Relativity alld quol1lWI me lI1ics rlt

j c

pound [

V z

lUIre VI

- hh 17J(J - Ii

[ 0 0 I O J - r ] [ 0 0 II] [ V10 00 shy 1 III 0 1 0012

== lite 111(shy + lIIe ru shyI 0 0 0 tgt1 ~yent ll U 0 j 0 ~v 11 0- 1 00 1shy J 1100 - 1 -~ -

D bull +t II F bull

r ~ 0 J~ mrl4F - m( VI bull filkl~

l 1 ( I mr ~ l

(nkc - $~Emc2 ) I (- like - m~middotpound Jl( 2 ) 1 ~

[J ~r~~4~--- 1 IL-- I EJ

m ~f

J E I

-~12 JIIIC-pound

which ho lds if and on ly if E = (hlllc1 ~ m2r4) 12

A particle in an electromagnetic field

We now consider how to ex tend our treatment to the case where the particle is lIot

frcC bUI subject 10 an electromagneti c fi eld represented by a scalar and a vec tor potenLial as in ( I L 3) Assumi ng toe panicle to have the electronic charge - e we generali ze (1 16) us ing (114) 0 ge

iii ~ = lea (Px+ eAx) +ea(Py- eAr) +ca3(P + eA ) + 3c + VJ~ ( 1127)

This is [he general fo rm of the li me-dependen t Dirac equalion

Following tJ)e same procedure as in the free-particle case the equivalent of ( 11 15) is

(j CP + eA )cu _ + mc 2u+ + Vl~ - EIl+ ( 1128)

amiddot (P + eA )w+ - nuu_V IL = ELL

where V = - ecfJ The equiva ienl o r (11 16) is now

C )u- =- c l1(u (P+eA)II+ (1129)

A particl e in a m agn eli c fie ld

We fi rs t consider (h e case where the sca lar pote nLial V IS zero Th e equivalent of ( 1117) is ben obtained by subSiituting ( 1129) into (1128) to gel

([0- (I + eA)I ) c u+ = (pound - c ju 1_ ( 1130)

Expressing be lefl-llltlnd s ide of ( 11 30) in Cartesian coordinates we find Ihat it cO1lains telms of two Iypes Fjrl t

6 ()(J eAJ211 t l = (Pr + eAJ~cu l-

2S2 Qual1flll11 Merhanics

using (I I 12) Second

~ ~ ~ 2

[OOy(P +eA)(Py+eAy)+ aya(P I eAy) (P + eA )1c u+

=ieojAJy - PyAx +PxAy- AyP1c2u bull

2[ au a J JIl ] =eliOc JU -o - - -a (A u+) +(A)I1+ ) --4-shydy y ax u

JA y (JA ) 2=enJJz ( ax - Ty r II t

= ena~J3z(2 + (1 1 3 1)

where we llave used 0112) and the d ifferc lllbl operaLOr represenlalion PI = -ilia j ax elc Using these results along with their equivalents for Ihe o ther Cartes ian coord inates ( 1130) becomes

- )2 2 24( (P +eA ) - eliamiddotn c u+= (E - 11 c )11+ (1 131)

a nd lL can be ob1ained using ( 1129) We shall conce ntra le on the no nrclat iv is ti c Ii mit wherc we ca n put u_ = 0 aud

I - 2 e )-(P +eA) --SmiddotB += poundU+ ( l 33)( 2m m

whe re S = ~ hO represents the spin angu la r mome ntum and I = E -- me2 is the nonrclati vis tic energy

The firs t te rm ill (1132) o r (1 133) is what is ex pec ted for a cha rged pa rti cle in a magnet ic fie ld bllt the second term is no t However this has the same fo rm as

an operator re presenLi ng the ene rgy of interac Li on between a magnet ic fie ld and a

particle Vho~(- ang ular momentu m is re pre sEnted by the o perato r Sa nd which has a m agne tic moment o r magn itude enj Once again this is exactl y what we proposed for a spin-hale parlicle in chapLe r 6 and we nole Lh fl l the spin g -factor which was

g ivCtllhe value Lwo o n the basis of experime nt nOw eme rges as a nalmal conseque nce of the Dirac equaLi o n

A particle sn bjec t to a scala - potentia l

We now aim 10 the case or a parlJd~ movlllg in a scalar pote nti al o nly so thaL Lhe

vector pote ntIal is zero FollOWing the same procedu re as before (1128) and (1129) lead to

r~ (2 ~ middotl 2

l(u P) 1 (umiddotP)+V u+= (E -mc )u+ (l134) 11( + C - v

[un hc l SII IOIL Il (1 I l( I II ~ h lo~ III he Wide tp Cit middot l-- f yenIor 10 f N I ~i( il ~reclll(ll witb (llpoundlCri llllm 11c1e CJ I f) ~ I I~W II h Jlt~ lI l lI JO t lite ll U~ rJ liltll i on ( Ih( eI~tloma~ l lcl k hdd V h lL h i s JI Sl lIs~ r1 tlwI I Ilcr I th l dtnpttT

Relaflviry and quantum mechanics 253

We de fine F c2 (mc2 +E - V) and consider the fi rs t te rm wi thi n the square t)rackets in ( L 134) Expressing Ihis in Cartesian coord inates leads LO Lwo types of terms Firs t

apFPr = JPJj3 (1 135) and second

a~ayAFIJ ayalFyFPJ = iG( FP - PyFJ)

ltif _ dF )= ntgt ( -Jx Py - Jy Px

= nO (VF x P) (1136)

where we have used ( 11 12) and the differenti al o pe rato r represenla tions o f Pr a nd Py Using these results along with the ir equivale nts fo r the othe r Cartes i ~n compo nents (1129) becomes

[PF p + V + (zamiddot (V F x p)J II = (pound -me)I1+

Remembe ring that P is a diffe re ntial operator the firs t term in the above expression can be expanded to give

[Fft2 + (PF) middotf + V + namiddot (VI x P)] u+ = (pound - mc2)u+ ( 1137)

Once lttgain we cO llcenrrate on the no nre la tivislic limit and de fine e as E - lIIe2 so thal

cI = -0------ shy2mcL + pound - V

1 ( E V) -I 2m 1+ 2111 c 2

~ - ( V -- )I 1+shy ( 1138) - 2m 2tnc ~

so that P I = - iliV F = (- ilij4m2c2)VV = (I 4m 2(2)pV and (1137) becomes

V - E 1 I - 1rP~ +v + _P +- (PV) P+ --lmiddota(Vv xp) + = eu+ (1139)t Lm 411l ( 4mc 4m-c

We now focus o ur a Ltention to the case of a spherica lly symme tlic pote nti a l V( r) so lhat

1 aV _ l Jv VV x P =--rx P = - - L ( 1140) r d r r ar

whe re L is the orbital a ngu la r momentum OpCfltor T he final Lerm ill squa re LHac kets ill (I 119) then has the f pfll

Ji 1 I c1v ~ A

--- - LmiddotS2 1112C1 r ()

254 255

Quanwm Mechallics

which is idenlicallo the spin -orb it term introduced in chapler 6 (641) M oreover whe reas our earl ier trcalmellt requi red a faclor of two [0 be inlTod uced inlo lhe defi ni tion Ilf the spin mag netic IllOtllC1l1 and again in the spill -orb it expression (Thomas precess ion) bOlh of these are incl uded JULOmatLCalLy 10 (I 40) The f~lc t

l l3t this is a necessary consequence of requiring cons istency between rela tiv ilY and quantum mechanjc~ is another of the triumphs of the Dirac equation f he remaJll ing two (Crill in ( 11 40) represent further rela li vislic correctio ns that are o f the same order as the spi ll-- orbil term but w llieh g~ncra ll y do not add significantly (Q the spectral structure

We conclude lhat the idea of spin developed in chapter 6 emerges as all inevilable consequence of Lbe DifflC equ ation iu th e nonrelativistic limit while tili s equation itself is required to e ll sure consistency be tween quantum mechanics and re la tivity a t all energIes We emphasize again lbat the electron is not spinning in any physical sense The whol e coneep t o f angular momentum and its conservation in a spherically symmetric potential is a co nsequence of our experience with large-scale classical systems It ac tlla ll y breaks down for fundamental particles where quantum effects a re important bu t can be re instated if we assign tlus in trinsic angu lar momentum to them Of course o ne impJication is lha t a ll fundameuld pal1icle~ shou ld have the same va lue of spin and thi s is true for lhe electron the proton and neulrOn and aU the quarks Other particles such as the photo n and the V31i ollS bosons associated with the strong all d weak Interactions have integer spin but a re subject to diffe rent equations It is also ill te res ting to note that the emergence of sp in from the Dirac equation in the presellce o f a scalal potential and the consequen t energy contribution IS independent of the fltlct th at V (r) is an electrostalic potent ial ltl nd therefore of the spin-orbit couplin g model

The fully relativistic form of the Dirac equation (11 34) can be solved exactly in the case o f the hydrogen atom in zero n field where V(r) = - Ze2 j (47rEor) (cL chapte r 3) We sha ll not give the details of this here but s impJy quote tile result

E =me (I + a====-- ]_ 2 (lIAI )

[n- j- ~+ J(j+ )2-ail where Hand j are the principal quantum number (d chapter 3) and the LOlal angular momentu m quantum number respec ti veiy (j 1 - cf chapter 6) and

0 == p2 j(4iteonc ) 1S Ihe fine structure constant

Vorked Example lL2 Sbow lhat in Ihe case of a parLlcle subject LO a spbencally sYITlI)l etnc pvLclllial tll e elw r~y r -ompatible WI th the totul Ilngulu- n1() menlum Soluli ol1l A sllflicrerll condili on for lwo dY ll1ni(ll variubles LO be compatible is LlWl lhelr operators COrrIl11l11C The Cngul l[ nomenLu rn operILor can hI wrillen as

J~ i I S

I =R x PI ~ - ttl

2

Relativiry and qualfum mechanics

where

L = [~ ~l while the Hamilt oni an is

Ii = u middot F+3l1lc2 + V ()

IllS eas il y hown lila l j commutes with 3mc2 and V (r) as 1111 1ILCer is sphencaJiy symmetric Now conSIder

lexPL ~ lexo (yP - PH -- iha ) py

using th e standard cornmutation relations for posllion and momentU[f] (chapter 4) Also

luPy t]= ([0 ay] [ax 0]- [a 011 0ay1) P Oy OJ ~ o ax 0 ogt ( (]y oj

0 aya-aa] [0 aJ = [ o ~middot = - 2 = - 21aJPyltJyax - OxltJy (]z 0 Pr

lJ sing [he commutatiOn relalions for [he PauJj spm malrices After multiplicatIOn by iii tile last term cilncels the One evalualed earlier Similar pairs of eyual and opXls ite lerms resu lt when the other COmponenlS of amiddot P and j are evaluated whieh proves thai the wtal angular momentum is compatjble with the HamiILouian We no te however [hm neither t nor S are individUil lly co rnpt(ible wilh ft

113 Antiparticles

Tile classical relativjstic relation (112) expresses the square of the tolal energy E of a f ree partic le in lerms of the square of its momentum p It follows that here is no necessary res triction on the sign of E and the eq uation has a full set of solutions [or all nega tive values of E less than - mc2 as well as for positive values of E

2 grea ter than mc Classically these are rcjec red ltIS being unphysical bur in the quantum -mechanical case negative energy stutes could in principle be reached by a quamum trans ilion and spontaneous trans itions to sta tes of ever-lower energy might be expected bur are of course never observed

It is cJear from ( 1122) (ha l Ule D irac equation for a free particle does indeed possess sOlutjons wilh

F = - V h2c1k Z +m2c

To overcome the problem 01middot trlIlsjlions Lo ne~a(i ve-encrgy states Djrac made the radical sugges tion th[l these were all al ready fi lled by electrons so thaI trans itions into them would be prevented by the Pauli exclusion prjuciple He then cOlls idacd the conseq uences of all electron being excited rrom one of these lil led slates with energy - (1(2 I- E) into a state of energy (mc2 + pound) under the inAuence o a photon

254 255

Quanwm Mechallics

which is idenlicallo the spin -orb it term introduced in chapler 6 (641) M oreover whe reas our earl ier trcalmellt requi red a faclor of two [0 be inlTod uced inlo lhe defi ni tion Ilf the spin mag netic IllOtllC1l1 and again in the spill -orb it expression (Thomas precess ion) bOlh of these are incl uded JULOmatLCalLy 10 (I 40) The f~lc t

l l3t this is a necessary consequence of requiring cons istency between rela tiv ilY and quantum mechanjc~ is another of the triumphs of the Dirac equation f he remaJll ing two (Crill in ( 11 40) represent further rela li vislic correctio ns that are o f the same order as the spi ll-- orbil term but w llieh g~ncra ll y do not add significantly (Q the spectral structure

We conclude lhat the idea of spin developed in chapter 6 emerges as all inevilable consequence of Lbe DifflC equ ation iu th e nonrelativistic limit while tili s equation itself is required to e ll sure consistency be tween quantum mechanics and re la tivity a t all energIes We emphasize again lbat the electron is not spinning in any physical sense The whol e coneep t o f angular momentum and its conservation in a spherically symmetric potential is a co nsequence of our experience with large-scale classical systems It ac tlla ll y breaks down for fundamental particles where quantum effects a re important bu t can be re instated if we assign tlus in trinsic angu lar momentum to them Of course o ne impJication is lha t a ll fundameuld pal1icle~ shou ld have the same va lue of spin and thi s is true for lhe electron the proton and neulrOn and aU the quarks Other particles such as the photo n and the V31i ollS bosons associated with the strong all d weak Interactions have integer spin but a re subject to diffe rent equations It is also ill te res ting to note that the emergence of sp in from the Dirac equation in the presellce o f a scalal potential and the consequen t energy contribution IS independent of the fltlct th at V (r) is an electrostalic potent ial ltl nd therefore of the spin-orbit couplin g model

The fully relativistic form of the Dirac equation (11 34) can be solved exactly in the case o f the hydrogen atom in zero n field where V(r) = - Ze2 j (47rEor) (cL chapte r 3) We sha ll not give the details of this here but s impJy quote tile result

E =me (I + a====-- ]_ 2 (lIAI )

[n- j- ~+ J(j+ )2-ail where Hand j are the principal quantum number (d chapter 3) and the LOlal angular momentu m quantum number respec ti veiy (j 1 - cf chapter 6) and

0 == p2 j(4iteonc ) 1S Ihe fine structure constant

Vorked Example lL2 Sbow lhat in Ihe case of a parLlcle subject LO a spbencally sYITlI)l etnc pvLclllial tll e elw r~y r -ompatible WI th the totul Ilngulu- n1() menlum Soluli ol1l A sllflicrerll condili on for lwo dY ll1ni(ll variubles LO be compatible is LlWl lhelr operators COrrIl11l11C The Cngul l[ nomenLu rn operILor can hI wrillen as

J~ i I S

I =R x PI ~ - ttl

2

Relativiry and qualfum mechanics

where

L = [~ ~l while the Hamilt oni an is

Ii = u middot F+3l1lc2 + V ()

IllS eas il y hown lila l j commutes with 3mc2 and V (r) as 1111 1ILCer is sphencaJiy symmetric Now conSIder

lexPL ~ lexo (yP - PH -- iha ) py

using th e standard cornmutation relations for posllion and momentU[f] (chapter 4) Also

luPy t]= ([0 ay] [ax 0]- [a 011 0ay1) P Oy OJ ~ o ax 0 ogt ( (]y oj

0 aya-aa] [0 aJ = [ o ~middot = - 2 = - 21aJPyltJyax - OxltJy (]z 0 Pr

lJ sing [he commutatiOn relalions for [he PauJj spm malrices After multiplicatIOn by iii tile last term cilncels the One evalualed earlier Similar pairs of eyual and opXls ite lerms resu lt when the other COmponenlS of amiddot P and j are evaluated whieh proves thai the wtal angular momentum is compatjble with the HamiILouian We no te however [hm neither t nor S are individUil lly co rnpt(ible wilh ft

113 Antiparticles

Tile classical relativjstic relation (112) expresses the square of the tolal energy E of a f ree partic le in lerms of the square of its momentum p It follows that here is no necessary res triction on the sign of E and the eq uation has a full set of solutions [or all nega tive values of E less than - mc2 as well as for positive values of E

2 grea ter than mc Classically these are rcjec red ltIS being unphysical bur in the quantum -mechanical case negative energy stutes could in principle be reached by a quamum trans ilion and spontaneous trans itions to sta tes of ever-lower energy might be expected bur are of course never observed

It is cJear from ( 1122) (ha l Ule D irac equation for a free particle does indeed possess sOlutjons wilh

F = - V h2c1k Z +m2c

To overcome the problem 01middot trlIlsjlions Lo ne~a(i ve-encrgy states Djrac made the radical sugges tion th[l these were all al ready fi lled by electrons so thaI trans itions into them would be prevented by the Pauli exclusion prjuciple He then cOlls idacd the conseq uences of all electron being excited rrom one of these lil led slates with energy - (1(2 I- E) into a state of energy (mc2 + pound) under the inAuence o a photon

256 257 Quantum M ecfr(lIl ics

E p-p

~~- l ~ ~s~

fI(

1

me

FIGURE 111 An electron ean be exdted from one of llle nlled negHive energy states Lo creale a f ree eleclrOn o f posit ive energy and a vacancy The properties o f the negative-energy

sea conlltlinjng a vacancy are lhe same as those of a positron

of energy 2(mc2 +pound) The excited electron has positi ve energy and therefore behaves quite normall y bUllhe filled sea of negatjve energy sla tes now has a vacancy This means IhilL the IOla l energy of the negative-energy slJ tes has been increased by IIe2 + t and tl e iI net rnomenllrn is - p wbere p is the momentum of the excited electron Referring to figure 1 1 we consider how this momentum changes under the action of an applied electric field All the nega tive ly charged e lectrons will acce lerate in a directio n opposi te to that of the fie ld taki ng the vacancy w ith them As a rcsul t the net moment um -p increases ill the same direction as the field It fo llows IhaL the neglllve-e tle rgy sea plus a vacancy will behave just like a positively charged electro n This pa rti cle which had not been observed when Dirac developed hi s theory is called a positro n and the phOlOU has lherefore c reated an electronshypositron patL The ex perin1ental observation of the positIOn a few years after Ih is prediction (by Anderson in 1933) was auother great Sll ccess for Diracs theory A fu rther conseque nce is tJlat aLi spin-half panicles (protons quarks e lc) should have

analogous antiparticles ancllhis has also been confirmed~ Despite Its predicuve success however there a re problems with Dirac s earl y

model Itl particular the sea of occupied negative ~tate~ has no observable properti es unti l ( vacancy is c relhd This mUSl mea n that the infinite se t of part icles has no grav itational mass and no charge despite tJle facllhat our analysis of the expected hchaviour of tile vnlJncy IS1gtumed thai al l the e lectrons responded lO an applied ekctrit field Another feaLure of Dlracs mode l is that there is symmetry between the pOSllron and tht e- hcLro n ~ a theory [hat pos ited elec trons as being associated wilh

bulls UII I JI~~IIIlJotnt s [1 hi ahl C a rc Im~sJ en Ihe pl1yoll- 1)[ ~t-IIIII h l 11gt1 s wilemiddot puslti ve-charge (arncrs

(ClIutt from Ihe ccit11 I eit-cfrt lllS r m oheJWI~e h1I ball(k

Relativity and quanfW1t mechanics

v3eancir-s iu an otherwise fi ll ed sea of positrons would make identical predictions Jat e r middotq ulIltum-field theories dis pense with the idea of 31 fi Ded ~rl and simply postulate that particle and ltlnlipartjclc pairs are ncicd states of the Dirac field whose ground slate is the vacu um We give a brief mtroducLion to the idtns of fi eld theory 10 Section 115

114 Other Wave Equations

If instead o f following Dirac we opera te on ( ll 5) by

-Jic1p 2 + miii ~ 2c ]

we gel a2

Ii aX +]cP+mc]v = O (1142)

This js known as the Klein-Gordoll equation which was actually di~covered before the Dirac eqnalion It was initially thought not La be relevant as the parti cle probabi lity density associated with il is not necessari ly posit ive Howevcf it was later realized that this parti cle density could be imerpreted as ltl charge density its sign depending on whetber particles or antiparticles were dom inant Solutions to the Dirac equation are also solll1i on~ to the Klein-Gordon equa tion bIll the latter possesses anoUler se t of solut ions with no intlinsic angular momentum h can Ihe reforc used to describe lhe propel1ies of particles with zero spin

The re lativistic equations that describe the photon are of course Maxwells equatioll s These have to be further quantized to reveal the properties of the photon and Ihis is brieHy di scussed in (he nex t section

115 Quantum Field Theory and the Spin-Statistics Theorem

The spin-stati stics theorem Slates that the wave functi ons of parlicles with integer spin are syrrunetric wuh repect 10 eltc han~l of the labels on the par(j cles- V I 12) = Vf(2 1) - and obey Bose-Einstein stlltisti cs (see chapter 10) In contrast pnrt icles with hal f integer spin have anti symmeuic wave funct ions-IJI( 12) = - 1(2 1)shyobey Fermi-DIrac SIll isLics and are cousequent ly subjelt to the exclusion principle The spln-s(atistics theorem was shown by Pauli ill 1940 to fonow from some qui te deep symmetry properties of relalivisllC quantum fi eld theory Many Lheocelical physicists over (he years have believed (hal such a simple con nection between two apparently qUIre separa le propcfLies of rhe rllndamenral particles should have

259 QUOWIln Mechanics25R

a simple proof A number of altenll)ls to find sllch a proof have bctn m~dl but so far none has emergecilha t hlS veen gene rally accepted Pauli s proor is well beyond the comp3ss of tillS book hut we wi1l give a short tnlroduClion to some of the basic ideas of fH~ Jd theories anti expllin how these give some suppurt to the va lidity of the

spin-slarbpoundics tl lcorcm Q uanlUm mechanics as developed up [0 now has (ake Jl the exis lence of particles

SUcil as Ille elec tron as given Til contrast the qnanLum r-ield approach builds them iUlo the fo(mCllism oj the theory ilself T he ~ t arlin g point is the vtlcuum-space and time with nOmalleTami no radial ion which we represen t by lhe quantum-mechanical sl(lle vec(or 10) We then defi ne a creal ion operator 01 which operates on 10) to c reate the state 11 ) wllicll represen ts the vacuum plus one particle in a Slale with a pan icular momentum a lld ~pin T he opera tor ii know n as an annihilation operatorremoves

pJnicles rrom the staLc To develop a fi eld theory ror the electron from the Dirac equa tion we conshy

sider the case of a free clec trun with momentum V = hk so that E = plusmn Eo == plusmn J(m 2c- l- p2C2) A gene ral solution of the Dirac equation is a linea l combination

of the wave functions corresponding to -Eo and lherefore IHlS tbe form

IJ ~ 10 I exp( - iEo Ii) + b 2 exp(iEolIi) ex p(ik r ) ( 1143)

whe re Ur and U2 are tbe time-independent parts of four- component vectors (cf (11 13)) and (i and b ~ are co maants We note from our earJi er discussion IJ)al Itt and U2 are orthogonal and we elfl also assume Ont they are norma li zed Thilt is

T tU u r = U1 U2 = 1

( 1144 )UUl = U~Ut = 0

where the supersc ript t ind ica tes Hermitian conjugate (ef the di sClission of maHix mechan ics III chapLer 6) The probabi lities of findiug the syslem in the stales with energy E = Eo and E = - Eo are lall and Ib12

respectively From our ea rlie r discuss ioLl we expect laf and Ibl2 to be the probability of the system containing an elec tron of momentum p and a posi tron o(momenHlIu - p respec ti vely If we confine ourse lves La low-ene rgy stales and use the timc-dependent Sehrodinger

equal10n the expectatIon value of pound is given by

(1145)()= (IJmd )= Eo(a a - W )

whIle that of the total charge is

( 11 46)(Q ) = -e(II ) = - e(o + bb )

USillg ( 11 lt1 lt1)

Tjll- JU sqa l l l ~ clIHit)n and anniollllllUn operHOR (or ever IIl1oJcd vl llC or 1111 momen(ull1 nd Spill lt0 ii 101 may be l ~ hflIc J 1CCOf(il ur ly These lahels ale ~mHncd In our d ~ lt U lSlOll 10 a~SlS I (larilY

Reloliviry al1d qllQIllU111 mechalic~

OUf li reory wou hJ agree with the experimental observJtion of e lectrons and pos itrons ir the 61gns of Ule second termS lll the rijht~ hand sides of (l 145) and (1 146) were reve rsed As we shall see thi s is just what quantum fie ld theory can achieve To move to a quantum field theory wcmiddot replace the constant- - and v by opera tors 0 and bi and tJle ir complex conjugates by J r and b which are the He rcJlti an conjugates o r aand ampt As laquolways wben we extend our theory in to a new area we do so illducli vely testing the predictions of the new theory against expenment As oa is a measure of tile probability of the syslem being in a s talt of positive E the opcrHof ata is taken to re present the number of electrons while the number of positrons is represented by the operator LtL Il follows thaI 0 1 and ht are rea tio ll opera tors re lating to e lec trons and posilrons respectively ltlud with the propertI es d Iscussed above while ii and bare [he corresponding annihilation operntors

To proceed fmther we have to postu late JUore specific properties of the c reation and anruJJiialioI1 operators We encountered sin-Lilar operators ( then ca ll ed raising and lowerrng operalOrs) when we discussed the harmonic osci ll ator in chapter 4

and angular momentum in chapters 5 and n In that case the comrnutaLion relaLio n ii t oJ = 1 plus the cOlld ilion thaI a ll ene rgy levels had to be positi ve led to the energy spectrum EI = (11 + ~) hw However jf we were to assun]e that the sa me condi tion holds in the present case the firslte rm in (l145) would lead to ltl ladde r of positive energy levels but the second term wou ld produce a ladder or evermiddotdecreasing negat ive-energy levels whjeh is just what we are tr ying to avoid

Dirac and Jordan showed that in order to produce a field theory for fermions that is cOi1sistent with the Dirac equation the opera to rs a b af and poundgtt must obey anticommutation relations wh ich are similar to commutation relations but with a positive rather 1113n a negative sign Tha i is

aal +ata =bbt + amppound = 1

They al so postulaled that

-- I-T -fb - I 0aa =a a = ou + a = o r(l2 T a2(i l = e tc (1147)

whe re the subsc ri pts 1 and 2 indicate differen l momentum or sp in SLates The quanti Lies He = atJ and HI = j)1 bare to representlhe total number of electro ns r nd pos it rons respecti vely

We now show tllat these rei allOns imply the Pauli cxclusion principle_ We conside r the e ffect of ope rating on the slafe 111) which COl1 lains 11 eicLtrons and we note thal identital argumenls can be applied to positron slates From ( 1117) wehave

0 = 1iI) = oln - I ) = In - 2)

( 11 48)

using D irac notatiun (d chapter 6) Hence either I - 1) 0 1 In- 2) must represent Ihe Vlcuu m state implying that 1 = 0 or 11 = l Thns the stale lS e ilhe r empty or con tains a single parll(lmiddot which is just what is requlled by the Pauli exclusion

(0 QqnrrlfU Me(b(lIIicS

rnnClplc hi chaptcr 10 we lt howed thlH the excl us ion princ iplc wns CoIlSC4u cm t (If anlJ -ymmcll Y 01 Uu wave fU IlC tiPIl lnd we Cri ll de monstra te- II1IS in the plc~nt

contex t algt Le i I I ~) Ic pf-e nl a stal ~ w here the d iffe ren t iatcs l abelled I ami 2 each conlain onl tkclron We can ge nerfll c II1Is by the opcrltio n o r the crea tion

0pt fltHOIlt on the ucuum gtI lle 101

II 2) ~ 010)

He nce

12 1) =aiaIIO) = 11 2) (1149)

where the hlst s tep u se~ 0147) We call now s lIlllmari ze lhc e lIccls or upe rating on the parLilt ie s tates w ith the

c reation and anllihila(io n operators We have

JIU) - 0 a ll ) = 10)

al ll )= O 0 10)= 11 ) (II 50)

whic h d ll cnly lc~l d [0

10) = hiOI 0

NII) = a a ll ) = 11 ) (115)

so as expected the eigenvaJue o f Nt are li t = 0 a nd lI t = 1 Similar arguments using [ and 1 I produc t corresponding res ults for the positron Slates

RelUrning now Lo lhe propenics o f the Slate ( 11 43) the total e ne rgy (11 45)

becomes

(E) - Eul - bh) = ~o(iit Ii + [ [ - I) - Eo(n + n) ( 1l52)

where we have asu illed thiJl the system IS ill a ll eigens tale o f Nr and lf with e igcnvltdues II and ti p respec ti ve ly We can si Illilmly express the (olal electric c harge

(cr (I ) 46)) (Q) = - c( Ii +b1 ) - - c(n - np) (11 53)

Where mthe final s teps we have omitted a lcrm - Eo in ( 1152) and -e in (1 1 53) In order to ensure Ih ( (he e nergy nd c harge of the vac uum st1te arc both zero

ltiuch Itrl-pu lO t tClins nrt infi nite whe n totalled over all th[ energy s lales the ubtlac1ton or suc h Infini t l~s III o rde r to obtain a physic ) result is a COI11I IOI1 fLoatllre of more advanced J-tPCC IS 111 qU illllUm middotfi e ld theory Given the above we ltee th11 (pound ) l just the ~IIIII of thc Cnerr il of the c1cctfOn ~ lu1 positrons which are both positive

Mo reo H (Q) is j ust the expel-It d n~t cha rr Stl lllntari l I1l~ ~ htl shown thai a fie ld theory thu lgtSOCI ~Hcs posillvt ~n( rg )

w it h Ihe eXiste nce or hUlh t lcl lrons and pos itrons equ l n~ thr crealion and all n ihishy

lallon opelttor to obey antiCOlnlOlltation relations Thi s Tn turn ensures lhat the oi upulton nU l1lbel$ 01 the tUlles are ei the r zero o r one so th at the Paul exciusioLl

pnnCip le is obeyed

Relat illfY (lnd quonlUm lIIecwlliCJ 2(11

Althoug h we have - hnw n that a fldtJ theory can he developed thai i con~istcnt with the Dime Ctjuation and Ihe Pluli CcillsH)n rrinclr le 11 113 ha been ll rgely an ind uctive nrgumenl ralhe r lh ] a proof A I ~o WI ha ve so 1r said uothi n$ abOUI II J(~ properties of pltlftickt WiLh Integer cpin v here Bo c-ElOstcln l li slics ar( expec t cd 10 ap ply If fit ld quanli7ntjon is nppliell to the K k inmiddotGordon CCluatHln in Ihe sp in-llto cacc il lS fu und lhal pos iti ve e nergies for hoth part icles amJ lttmiparLic1es Ire obli lOcd

1Umllig thai Ule creatIOn and anllllllialion opew lors ob) com mutation rat her thilll a nlicomm ulaliO Il relali o ns The~e permit mulliple o(C upalion o f the s ta tes and he nce BosemiddotEillst~in s tati s tiCs Howeve r to complete the proof wc also have to show lilnl

fermion-type solutio lls La Ihe KJ e ill-Gordo n equatio n that are nOI also so lutions to the Dirac equatio n are not a lJ owed and Ihi s is considerabl y harder A further probkm

IS thal the Dirc equation relates on ly 10 spin h~d f md fhe Klein-Gordon q Ultl II On only to sp in-zero panicles while 111 e spi n-s tClli s ti cs theorem refers gene ra ll y to even nd odtJ nll rHoc) or hal f-ill legers HOrcver parl ic les with ~p in gltHc t 111111 one-half lIl -ty be properl y described as tig lHl y bound composites of spin-llalr particles whic h wnuJd then obey the spin-s tati stics Ihenrem

fll lldnnentaJ fea ture of lttil thelte Ipproachcs lO the spin -s tati s tics thcore m is thnt they re ly On the re lntivistic regime to predic t ltI res ult hat appli es to tY itcUs WIhTe rCla lJ viSlic crfeCli arc o therWise negli g ible This llil prompted sume quite

differe nt approaches to the proble m notably iJ suggesti on by Ucrry and Robbins ill 1997 lhat the anti sYlllmetry of the Iwo-fernlloll wave fUllcti oll ma y be assoclltlI ed wi th a geo metrical phase faclOr resulti ng fro m the topology of thc system

As was poiJlled out in chap ter 1 quant um mechanics began w ith the ide a o r (he qUa ntum of e lecLrornagnetl c rndi Hion (or photon) To develop thi s runher we would have to quami7e Ma)we lJ s equa ti on to producc a quallltim ve rsion or Ihe

clec trolll aJ netic 11cld This can be done and IL IIlvo lves deflning o pera tors Illal CftH te annihilate e hclrornagnc lc fie ld qumu~1 ie photon This Jlifers rrO lll thc quantum

fie ld theory develo ped above in that till an ti co lll Jl1l1lnt io n relations arc replaced by commut ation re lations so thut the prOplTjcs or (he operaturs arc sim ilar to those

developed ill the theory of the h1rmoIllC i)sc illa tor As a res ult photols Ire boon anJ not suhject 10 the the exclus ion principle Photons ltIlso have the e nergy alld

momentum gwcn by the Planck and de Broie rcbt ions and are [ounJ 10 1Jave

total-spin quantum number j 1 The s tal e wi lh Iff = 1= 1 conespond 10 ri g ht and le ft circu larly polanzed plane waves the -ta te wi th Ill) 0 would con t-pond to a long itud inally polarized ckctromag ne tic wavc a nd js forbidden The imeraclio n

be lwccn Ihe fields replcscilting nWtlel nnJ r1t lb tion ttn nlo he huil IIltO quan tu m field th tory and new re~ul f S hlVC bcr-Il predictcd that have Ix n co pe(imentally confifmcd

i s middotc imlicatcd ciJrlltl Ihe L-( llll 11 l tJ vJnlge of q uantum Ilel J henry is Ihal il includes the existence or rarlklts jn the fo rrnallsm j~ q uanta or lhe field whercl5 preVIOusly wc assumed Ihl l)IlC ll e 1) say 11 c lecun ll w hosl quantllm properties

we le described by the Schrod ll [lC equau ul T he (ollcepl of the qllanti ztu fi e ld also enables all dl tcrIlailve 31H1 pedlar s cktper ulldcrs t~lI1di l1 j of the C(IlCtpt fr illiJlqillgui )~t11bility Rathel til ~aT inr that [J Imiddotticleo t rttlti 2 are idc l1tll ~I1 Wt

call Simply say thaI (be Jle ld COIl I llm two CXCil ltlIi o ns which we do not ath mr I It

262 Q II QnlllfiI Mechan ics

label An nn~ l ogy IS sometimes drawn between hav in1 lwO identical pNHld coins and lWOpounds in a b(l nk account Ll lhe Vater case the two unit50 have no mdi vidua l Idellli l) 1 nd fie ld theory provides a simJlar conct-ptual basis for the description of a fi eld conl ajnj ug two quanta

Problems 111 Show by ub~tllulion dlUl lhe matrices givellill (119) ha( the prope rti es sel OUI ill ( 11 8)

112 Show lll)t th e zer(l anguOlI -momentum energy eigenfunctions [or an lIlfi11lle sphen eal well oblailled ill chpler 3 are also elgenfuIlcllOm o r (he Duac equation and determine lhe cOLTesponuir g eIgenvalues

11 3 Expand lhe re la ljy isllC expreSS io n fo r the hydr0fen -alom tl)f~y levels ( 1I A l ) III PQwers or (J 2

-sSI lJIirg a to be small SIi()w thal the te m l of order a lS [he same as the nOIl -relativ lWc e opres~aO Il for

the hydll~en ato m elllqy levels (E) obtained in thapter 3 and that the ne lterm equals

_I IIC]~~ (3 ____)2 n- 4 J t 12

Show that this resulls 10 s p in -Qrb lt s plitting of the form de rived in Chapter 6 aud agrees with the cpreltsions den ved in pro blems 7 7 aud 7 8 mthe case where = 2 and = J

114 Ohtaill Ml expresslo ll for the wave function o f a free pHt ie le with negat l c energy b~ usi ng the i rSI of (II J 5) to eprc lts II III terms of _ and hence obtain a versIOn of (1 1 21) appropn ale 10 tillS prohlem Hellce show IhBt lhe we fun ct IO ns wilh oppositely signed [ are orlhrogona l

U5 E llai n wlw lite held operator repregtenting the scatterin g of particles fr om Siaies labelled J ald 2 into stal(s luhllicd3 ~l1d lt1 iM

deklc

(0 QqnrrlfU Me(b(lIIicS

rnnClplc hi chaptcr 10 we lt howed thlH the excl us ion princ iplc wns CoIlSC4u cm t (If anlJ -ymmcll Y 01 Uu wave fU IlC tiPIl lnd we Cri ll de monstra te- II1IS in the plc~nt

contex t algt Le i I I ~) Ic pf-e nl a stal ~ w here the d iffe ren t iatcs l abelled I ami 2 each conlain onl tkclron We can ge nerfll c II1Is by the opcrltio n o r the crea tion

0pt fltHOIlt on the ucuum gtI lle 101

II 2) ~ 010)

He nce

12 1) =aiaIIO) = 11 2) (1149)

where the hlst s tep u se~ 0147) We call now s lIlllmari ze lhc e lIccls or upe rating on the parLilt ie s tates w ith the

c reation and anllihila(io n operators We have

JIU) - 0 a ll ) = 10)

al ll )= O 0 10)= 11 ) (II 50)

whic h d ll cnly lc~l d [0

10) = hiOI 0

NII) = a a ll ) = 11 ) (115)

so as expected the eigenvaJue o f Nt are li t = 0 a nd lI t = 1 Similar arguments using [ and 1 I produc t corresponding res ults for the positron Slates

RelUrning now Lo lhe propenics o f the Slate ( 11 43) the total e ne rgy (11 45)

becomes

(E) - Eul - bh) = ~o(iit Ii + [ [ - I) - Eo(n + n) ( 1l52)

where we have asu illed thiJl the system IS ill a ll eigens tale o f Nr and lf with e igcnvltdues II and ti p respec ti ve ly We can si Illilmly express the (olal electric c harge

(cr (I ) 46)) (Q) = - c( Ii +b1 ) - - c(n - np) (11 53)

Where mthe final s teps we have omitted a lcrm - Eo in ( 1152) and -e in (1 1 53) In order to ensure Ih ( (he e nergy nd c harge of the vac uum st1te arc both zero

ltiuch Itrl-pu lO t tClins nrt infi nite whe n totalled over all th[ energy s lales the ubtlac1ton or suc h Infini t l~s III o rde r to obtain a physic ) result is a COI11I IOI1 fLoatllre of more advanced J-tPCC IS 111 qU illllUm middotfi e ld theory Given the above we ltee th11 (pound ) l just the ~IIIII of thc Cnerr il of the c1cctfOn ~ lu1 positrons which are both positive

Mo reo H (Q) is j ust the expel-It d n~t cha rr Stl lllntari l I1l~ ~ htl shown thai a fie ld theory thu lgtSOCI ~Hcs posillvt ~n( rg )

w it h Ihe eXiste nce or hUlh t lcl lrons and pos itrons equ l n~ thr crealion and all n ihishy

lallon opelttor to obey antiCOlnlOlltation relations Thi s Tn turn ensures lhat the oi upulton nU l1lbel$ 01 the tUlles are ei the r zero o r one so th at the Paul exciusioLl

pnnCip le is obeyed

Relat illfY (lnd quonlUm lIIecwlliCJ 2(11

Althoug h we have - hnw n that a fldtJ theory can he developed thai i con~istcnt with the Dime Ctjuation and Ihe Pluli CcillsH)n rrinclr le 11 113 ha been ll rgely an ind uctive nrgumenl ralhe r lh ] a proof A I ~o WI ha ve so 1r said uothi n$ abOUI II J(~ properties of pltlftickt WiLh Integer cpin v here Bo c-ElOstcln l li slics ar( expec t cd 10 ap ply If fit ld quanli7ntjon is nppliell to the K k inmiddotGordon CCluatHln in Ihe sp in-llto cacc il lS fu und lhal pos iti ve e nergies for hoth part icles amJ lttmiparLic1es Ire obli lOcd

1Umllig thai Ule creatIOn and anllllllialion opew lors ob) com mutation rat her thilll a nlicomm ulaliO Il relali o ns The~e permit mulliple o(C upalion o f the s ta tes and he nce BosemiddotEillst~in s tati s tiCs Howeve r to complete the proof wc also have to show lilnl

fermion-type solutio lls La Ihe KJ e ill-Gordo n equatio n that are nOI also so lutions to the Dirac equatio n are not a lJ owed and Ihi s is considerabl y harder A further probkm

IS thal the Dirc equation relates on ly 10 spin h~d f md fhe Klein-Gordon q Ultl II On only to sp in-zero panicles while 111 e spi n-s tClli s ti cs theorem refers gene ra ll y to even nd odtJ nll rHoc) or hal f-ill legers HOrcver parl ic les with ~p in gltHc t 111111 one-half lIl -ty be properl y described as tig lHl y bound composites of spin-llalr particles whic h wnuJd then obey the spin-s tati stics Ihenrem

fll lldnnentaJ fea ture of lttil thelte Ipproachcs lO the spin -s tati s tics thcore m is thnt they re ly On the re lntivistic regime to predic t ltI res ult hat appli es to tY itcUs WIhTe rCla lJ viSlic crfeCli arc o therWise negli g ible This llil prompted sume quite

differe nt approaches to the proble m notably iJ suggesti on by Ucrry and Robbins ill 1997 lhat the anti sYlllmetry of the Iwo-fernlloll wave fUllcti oll ma y be assoclltlI ed wi th a geo metrical phase faclOr resulti ng fro m the topology of thc system

As was poiJlled out in chap ter 1 quant um mechanics began w ith the ide a o r (he qUa ntum of e lecLrornagnetl c rndi Hion (or photon) To develop thi s runher we would have to quami7e Ma)we lJ s equa ti on to producc a quallltim ve rsion or Ihe

clec trolll aJ netic 11cld This can be done and IL IIlvo lves deflning o pera tors Illal CftH te annihilate e hclrornagnc lc fie ld qumu~1 ie photon This Jlifers rrO lll thc quantum

fie ld theory develo ped above in that till an ti co lll Jl1l1lnt io n relations arc replaced by commut ation re lations so thut the prOplTjcs or (he operaturs arc sim ilar to those

developed ill the theory of the h1rmoIllC i)sc illa tor As a res ult photols Ire boon anJ not suhject 10 the the exclus ion principle Photons ltIlso have the e nergy alld

momentum gwcn by the Planck and de Broie rcbt ions and are [ounJ 10 1Jave

total-spin quantum number j 1 The s tal e wi lh Iff = 1= 1 conespond 10 ri g ht and le ft circu larly polanzed plane waves the -ta te wi th Ill) 0 would con t-pond to a long itud inally polarized ckctromag ne tic wavc a nd js forbidden The imeraclio n

be lwccn Ihe fields replcscilting nWtlel nnJ r1t lb tion ttn nlo he huil IIltO quan tu m field th tory and new re~ul f S hlVC bcr-Il predictcd that have Ix n co pe(imentally confifmcd

i s middotc imlicatcd ciJrlltl Ihe L-( llll 11 l tJ vJnlge of q uantum Ilel J henry is Ihal il includes the existence or rarlklts jn the fo rrnallsm j~ q uanta or lhe field whercl5 preVIOusly wc assumed Ihl l)IlC ll e 1) say 11 c lecun ll w hosl quantllm properties

we le described by the Schrod ll [lC equau ul T he (ollcepl of the qllanti ztu fi e ld also enables all dl tcrIlailve 31H1 pedlar s cktper ulldcrs t~lI1di l1 j of the C(IlCtpt fr illiJlqillgui )~t11bility Rathel til ~aT inr that [J Imiddotticleo t rttlti 2 are idc l1tll ~I1 Wt

call Simply say thaI (be Jle ld COIl I llm two CXCil ltlIi o ns which we do not ath mr I It

262 Q II QnlllfiI Mechan ics

label An nn~ l ogy IS sometimes drawn between hav in1 lwO identical pNHld coins and lWOpounds in a b(l nk account Ll lhe Vater case the two unit50 have no mdi vidua l Idellli l) 1 nd fie ld theory provides a simJlar conct-ptual basis for the description of a fi eld conl ajnj ug two quanta

Problems 111 Show by ub~tllulion dlUl lhe matrices givellill (119) ha( the prope rti es sel OUI ill ( 11 8)

112 Show lll)t th e zer(l anguOlI -momentum energy eigenfunctions [or an lIlfi11lle sphen eal well oblailled ill chpler 3 are also elgenfuIlcllOm o r (he Duac equation and determine lhe cOLTesponuir g eIgenvalues

11 3 Expand lhe re la ljy isllC expreSS io n fo r the hydr0fen -alom tl)f~y levels ( 1I A l ) III PQwers or (J 2

-sSI lJIirg a to be small SIi()w thal the te m l of order a lS [he same as the nOIl -relativ lWc e opres~aO Il for

the hydll~en ato m elllqy levels (E) obtained in thapter 3 and that the ne lterm equals

_I IIC]~~ (3 ____)2 n- 4 J t 12

Show that this resulls 10 s p in -Qrb lt s plitting of the form de rived in Chapter 6 aud agrees with the cpreltsions den ved in pro blems 7 7 aud 7 8 mthe case where = 2 and = J

114 Ohtaill Ml expresslo ll for the wave function o f a free pHt ie le with negat l c energy b~ usi ng the i rSI of (II J 5) to eprc lts II III terms of _ and hence obtain a versIOn of (1 1 21) appropn ale 10 tillS prohlem Hellce show IhBt lhe we fun ct IO ns wilh oppositely signed [ are orlhrogona l

U5 E llai n wlw lite held operator repregtenting the scatterin g of particles fr om Siaies labelled J ald 2 into stal(s luhllicd3 ~l1d lt1 iM

deklc

262 Q II QnlllfiI Mechan ics

label An nn~ l ogy IS sometimes drawn between hav in1 lwO identical pNHld coins and lWOpounds in a b(l nk account Ll lhe Vater case the two unit50 have no mdi vidua l Idellli l) 1 nd fie ld theory provides a simJlar conct-ptual basis for the description of a fi eld conl ajnj ug two quanta

Problems 111 Show by ub~tllulion dlUl lhe matrices givellill (119) ha( the prope rti es sel OUI ill ( 11 8)

112 Show lll)t th e zer(l anguOlI -momentum energy eigenfunctions [or an lIlfi11lle sphen eal well oblailled ill chpler 3 are also elgenfuIlcllOm o r (he Duac equation and determine lhe cOLTesponuir g eIgenvalues

11 3 Expand lhe re la ljy isllC expreSS io n fo r the hydr0fen -alom tl)f~y levels ( 1I A l ) III PQwers or (J 2

-sSI lJIirg a to be small SIi()w thal the te m l of order a lS [he same as the nOIl -relativ lWc e opres~aO Il for

the hydll~en ato m elllqy levels (E) obtained in thapter 3 and that the ne lterm equals

_I IIC]~~ (3 ____)2 n- 4 J t 12

Show that this resulls 10 s p in -Qrb lt s plitting of the form de rived in Chapter 6 aud agrees with the cpreltsions den ved in pro blems 7 7 aud 7 8 mthe case where = 2 and = J

114 Ohtaill Ml expresslo ll for the wave function o f a free pHt ie le with negat l c energy b~ usi ng the i rSI of (II J 5) to eprc lts II III terms of _ and hence obtain a versIOn of (1 1 21) appropn ale 10 tillS prohlem Hellce show IhBt lhe we fun ct IO ns wilh oppositely signed [ are orlhrogona l

U5 E llai n wlw lite held operator repregtenting the scatterin g of particles fr om Siaies labelled J ald 2 into stal(s luhllicd3 ~l1d lt1 iM

deklc