Yukawa Potential and the Propagator Term

Transcription

1 PHY304 Partcle Physcs 4 Dr C N Booth Yukawa Potental an the Propagator Term Conser the electrostatc potental about a charge pont partcle Ths s gven by φ = 0, e whch has the soluton φ = Ths escrbes the potental for a force meate by massless 4πε 0 r partcles, the photons For a partcle wth mass, the relatvstc equaton E = p c + m c 4 can be converte nto a wave equaton by the substtutons E ; p x x etc Hence, φ = ( m c 4 c ) φ Or, n the statc, tme nepenent case, ths leas to m c φ = 0, (whch gves φ = 0 for the massless case, as requre) For a pont source wth sphercal symmetry, the fferental operator can be wrtten as 1 φ 1 r ( r ) φ φ, so r r r r r r ( rφ)= m c rφ wth soluton φ = g e r R r where g s a constant (the couplng strength) an R = mc s the range of the force Ths s known as the Yukawa form of the potental, an was orgnally ntrouce to escrbe the nuclear nteracton between protons an neutrons ue to pon exchange Usng ths form of potental an the Born approxmaton leas, after some manpulaton (see the homework!), to a matrx element gven by M f = 4πg q + m c 1 Returnng to our normal conventon of settng c = 1, the terms n the enomnator gve q + m an ths s calle the propagator term It arses from the exchange of a vrtual boson whose rest mass (as a physcal partcle) s m The cross-secton s proportonal to M 1 ( q + m ) f 13

2 PHY304 Partcle Physcs 5 Dr C N Booth Invarance Prncples an Conservaton Laws Wthout nvarance prncples, there woul be no laws of physcs! We rely on the results of experments remanng the same from ay to ay an place to place An nvarance prncple reflects a basc symmetry, an s always ntmately relate to a conservaton law (an to a quantty that cannot be etermne absolutely) The proofs presente n the lectures rely on some key results from quantum mechancs It s not necessary to be able to reprouce the followng proofs, but you must be able to use the results Frst, operators corresponng to physcal observables are Hermtan A ψ = ψ * A for any wave-functon ψ The proof s as follows ( ) * That s, they obey The expectaton value of an observable a 3 = ψ* Aψ r must be real ( A ) ( A ) 3 * 3 ψ* ψ r = ψ ψ r (1) We can expan ψ n terms of another set of wave-functons, φ ψ = 3 * 3 c* φ* A cjφ j r = cjφj A cφ r j j c* c φ* A φ 3 r = cc* φ A φ * 3 r j cφ Substtute ths nto (1) ( ) j j j j j 3 * 3 * r r, But c an c j are arbtrary, so ( φ Aφ j = φj Aφ ) an as ths s for any φ j, ( ) * φ * A = Aφ, as requre Next, we use ths entty to show the mportant result that operators whch commute wth the Hamltonan (the operator for total energy) Ĥ correspon to quanttes whch are conserve, that s they are constants of the moton Conser the tme-epenence of explctly on tme t (e a  s not Â( t ) ) 3 = ψ* Aψ r where we assume that  oes not epen a ψ* 3 Then * ψ = A A t ψ + ψ t t r ψ We make use of the tme-epenent Schronger s equaton: Ĥψ = ψ Hence H ψ * = ψ wth ts complex conjugate = ( H ψ * ) But Ĥ s a Hermtan operator, so from the above result ψ * = ψ * H a Hence = ψ* HAψ ψ* AHψ ( ) 3 = ψ* H, A ψ 3 t r r = 0 f Ĥ an  commute 14

3 Some classcal nvarance prncples are relate to the nature of space-tme Invarance of the Hamltonan (the operator or expresson for total energy) uner a translaton for an solate, multpartcle system leas rectly to the conservaton of the total momentum of the system Ths can be emonstrate classcally, but we wll take a quantum mechancal approach, efnng an operator D whch prouces a translaton of the wavefuncton through δx: D ψ(x) = ψ (x) = ψ(x + δx) Then t can be shown that D exp ( P δx / ) where P s the momentum operator P s sa to act as a generator of translatons Now snce the energy of an solate system cannot be affecte by a translaton of the whole system, D must commute wth the Hamltonan operator H, e [ D, H ] = 0; t must therefore also be true that [ P, H ] = 0, an so P has egenvalues whch are constants of the moton We therefore have three equvalent statements: ) The Hamltonan s nvarant uner spatal translatons (Equvalently, t s mpossble to etermne absolute postons) ) The momentum operator commutes wth the Hamltonan ) Momentum s conserve n an solate system Another conserve quantty s electrc charge, corresponng to an nvarance of physcal systems uner a translaton n the electrostatc potental In classcal electrostatcs, absolute potental s arbtrary the physcs only epens on potental fferences Assumng ths fact remans true, we can conser what woul be the consequences of the possblty of creatng an estroyng electrc charge By hypothess, the energy requre to create a charge Q at a potental Φ 1 woul be W, nepenent of Φ But we coul move the charge to another pont at Φ, lberatng an energy (Φ 1 Φ )Q, before estroyng t wth an energy release W: a net energy gan of (Φ 1 Φ )Q The ablty to create or estroy charge thus volates conservaton of energy Invertng the argument, conservaton of energy together wth nvarance wth respect to a change n electrc potental automatcally requres charge to be conserve Agan, an nvarance prncple mples a conservaton law Quantum mechancally, we may efne a charge operator Q whch, when t operates on a wavefuncton ψ q escrbng a system of total charge q, returns an egenvalue of q Q ψ q = q ψ q If q s conserve, Q an H must commute, an (as wll be shown n the lecture) ths s assure by nvarance uner a global phase (or gauge) transformaton ψ q = exp ( ε Q ) ψ q where ε s an arbtrary real parameter Ths s very closely analogous to the relatonshp between conservaton of momentum an nvarance uner splacement, as wll be emonstrate n the lectures (Invarance uner a local gauge transformaton, where ε can epen on space an tme, has much greater consequences, but that s beyon the scope of ths lecture course) 15

4 The above contnuous transformatons le to atve conservaton laws the sum of all charges or momenta s conserve There are also screte or scontnuous transformatons, whch lea to multplcatve conservaton laws An mportant group of these are party P, charge conjugaton C an tme reversal T The party operator nverts spatal coornates It therefore transforms x nto x, p nto p etc In other wors, polar vectors change sgn; axal vectors, such as angular momentum J, o not Now P ψ(x) = ψ( x) P ψ(x) = P ψ( x) = ψ(x) The party operator thus has egenvalues of ±1 If P ψ(x) = +ψ(x) the wavefuncton s sa to have even party, whle f P ψ(x) = ψ(x) t has o party (Note that wavefunctons o not have to be egenfunctons of party) The sphercal harmoncs Y l m (θ, φ) (met n atomc physcs an elsewhere) are examples of egenfunctons of the party operator (If they are not famlar, look them up!) By conserng a reflecton n the orgn, t shoul be clear that n sphercal polar coornates, the party operator causes r r (unchange) θ π θ φ π + φ, an by nspecton of the form of the sphercal harmoncs t can be seen that Y l m changes sgn f l s o an remans the same f t s even, e P Y l m = ( 1) l Y l m Invarance wth respect to P leas to multplcatve conservaton laws Eg Conser a + b c + The ntal state wavefuncton can be wrtten as ψ = ψ a ψ b ψ l, where ψ a an ψ b are nternal wavefunctons for partcles a an b, an ψ l s the wavefuncton escrbng ther relatve moton whch epens on the angular momentum l The P operator affects each factor, so Pψ = Pψ a Pψ b Pψ l If the ntrnsc partes of the partcles are gven by Pψ a = π a ψ a, etc, where π a s just a number ±1, then Pψ = π a π b ( 1) l ψ or π = π a π b ( 1) l e the party of a multpartcle system s gven by the prouct of the ntrnsc partes of the nvual partcles an the party of the wavefuncton escrbng ther relatve moton A smlar expresson can be wrtten for the fnal state Thus, f the nteracton responsble for the above process s nvarant uner party (as the electromagnetc nteracton s) then π a π b ( 1) l = π c π ( 1) l where l s the fnal relatve angular momentum the allowe (change n) angular momentum epens on the ntrnsc partes of ntal an fnal state partcles Another screte transformaton s charge conjugaton, C, whch changes a partcle nto ts antpartcle Ths reverses the charge, magnetc moment, baryon number an lepton number of the partcle Tme reversal, T, reverses the tme coornate However, as wll be shown n the lectures, T oes not satsfy the smple egenvalue equaton T ψ(t) = ψ( t) = a ψ(t) (In fact, efnng T lke ths not only oes not have the esre effect of causng momentum to be reverse whle leavng energy unchange, t results n a wavefuncton whch oes not obey Schronger s equaton) Instea T must be efne by T ψ(t) = ψ*( t) 16

5 The strong an electromagnetc nteractons are nvarant uner C, P an T transformatons Ths s not true of the weak nteracton, as can be seen by conserng neutrnos (whch are only nvolve n weak nteractons) It s observe that neutrnos are always left-hane, e ther spn s antparallel to ther recton of moton The P operator reverses momentum but not spn, so when apple to a neutrno woul prouce a rght-hane neutrno, whch s not observe Smlarly C apple to a neutrno prouces an unobserve left-hane antneutrno Weak nteractons therefore volate C an P The combnaton CP, however, apple to a left-hane neutrno prouces a rght-hane antneutrno, whch s observe Therefore (to a goo approxmaton) weak nteractons are nvarant uner the combne transformaton CP The weak nteracton, an all other nteractons, are exactly nvarant uner the combnaton CPT Summary Invarance Conserve Quantty Gravtaton, weak, electromagnetc an strong nteractons are nepenent of: translaton n space lnear momentum rotatons n space angular momentum translatons n tme energy EM gauge transformaton electrc charge CPT (prouct of partes below) Gravtaton, electromagnetc an strong nteractons are nepenent of: spatal nverson P spatal party charge conjugaton C charge party tme reversal T tme party 17