Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER-0 WAVEFUNCTIONS, OBSERVABLES d OPERATORS 0. Represettios i the sptil d mometum spces 0..A Represettio of the wvefuctio i the sptil coordites bsis { positio x } 0..A The Delt Dirc 0..A Comptibility betwee the physicl cocept of mplitude probbility d the ottio used for the ier product. 0..B Represettio of the wvefuctio i the mometum coordites bsis { mometum p } 0..B Represettio of the mometum p stte i spce-coordites bsis { positio x } 0..B Idetifyig the mplitude probbility mometum p s the Fourier trsform of the fuctio ( x) 0..C Tesor Product of Stte Spces 0. The Schrödiger equtio s postulte 0..A The Hmiltoi equtios expressed i the cotiuum sptil coordites. The Schrodiger Equtio. 0..B Iterprettio of the wvefuctio Eistei s view o the grulrity ture of the electromgetic rditio. Mx Bor s probbilistic iterprettio of the wvefuctio. Determiistic evolutio of the wvefuctio Esemble 0..C Normliztio coditio of the wvefuctio Hilbert spce Coservtio of probbility 0..D The Philosophy of Qutum Theory 0.3 Expecttio vlues 0.3.A Expecttio vlue of prticle s positio 0.3.B Expecttio vlue of the prticle s mometum 0.3.C Expecttio (verge) vlues re clculted i esemble of ideticlly prepred systems
0.4 Opertors ssocited to observbles 0.4.A Observbles, eigevlues d eigesttes 0.4.B Defiitio of the qutum mechics opertor F ~ to be ssocited with the observble physicl qutity f 0.4.C Defiitio of the Positio Opertor X ~ 0.4.D Defiitio of the Lier Mometum Opertor P ~ 0.4.D. Represettio of the lier mometum opertor P ~ i the mometum bsis { mometum p } 0.4.D. Represettio of the lier mometum opertor P ~ i the sptil coordites bsis { positio x } 0.4.D3 Costructio of the opertors P ~ ~, P ~, P 3 0.4.E The Hmiltoi opertor 0.4.E. Evlutio of the me eergy i terms of the Hmiltoi opertor 0.4.E. Represettio of the Hmiltoi opertor i the sptil coordite bsis 0.5 Properties of Opertors 0.5.A Correspodece betwee brs d kets 0.5.B Adoit ) opertors 0.5.C Hermiti or self-doit opertors Properties of Hermiti (or self-doit) opertors: - Opertors ssocited to me vlues re Hermiti (or self-doit) - Eigevlues re rel - Eigevectors with differet eigevlues re orthogol 0.5.D Observble Opertors 0.5.E Opertors o ssocited to me vlues 0.6 The commuttor 0.6.A Expressio for the geerlized ucertity priciple 0.6.B Cougte observbles Stdrd devitio of two cougte observbles 0.6.C Properties of opertors tht do commute 0.7 How to prepre the iitil qutum sttes 0.7.A Kowig wht c we predict bout evetul outcomes from mesuremet? 0.7.B After mesuremet, wht c we sy bout the stte? 0.7.C Simulteous mesuremet of observbles 0.7.C Defiitio of comptible (or simulteously mesurble) opertors 0.7.C Coditio for observbles A ~ d B ~ to be comptible 0.7.C3 Complete set of commutig opertors
Refereces: Feym Lectures Vol. III; Chpter 6, 0 Clude Cohe-Toudi, B. Diu, F. Lloe, Qutum Mechics, Wiley. "Itroductio to Qutum Mechics" by Dvid Griffiths; Chpter 3. B. H. Brsde & C. J. Jochi, Qutum Mechics, Pretice Hll, d Ed. 000 3
CHAPTER-0 WAVEFUNCTIONS, OBSERVABLES d OPERATORS Qutum theory is bsed o two mthemticl items: wvefuctios d opertors. The stte of system is represeted by wvefuctio. A exct kowledge of the wvefuctio is the mximum iformtio oe c hve of the system: ll possible iformtio bout the system c be clculted from this wvefuctio. Qutities such s positio, mometum, or eergy, which oe mesures experimetlly, re clled observbles. I clssicl physics, observbles re represeted by ordiry vribles. I qutum mechics observbles re represeted by opertors; i.e. by qutities tht upo opertio o wvefuctio givig ew wvefuctio. This chpter presets three mi sectios: The first icludes descriptio of the sptil-coordites bsis d the mometum-coordites bsis, which re typiclly used to represet qutum stte. The ext describes how to build the qutum mechics opertor correspodig to give observble. The fil sectio ddresses how to build (mthemticl) qutum stte from give set of experimetl results. The key mthemticl cocept used here is the complete set of commutig opertors 0. Represettio of the wvefuctios i the sptil d mometum spces A rbitrry stte c be expded i terms of bse sttes tht coveietly fit the prticulr problem uder study. For geerl descriptios, two bses re frequetly used: the sptil coordite bsis d the lier mometum bsis. These two bsis re ddressed i this sectio. 0..A Represettio of the wvefuctio i the sptil coordites bsis { x, x } Chpter 9 helped to provide some clues o the proper iterprettio of the wvefuctio (the solutios of the Schrodiger equtio.) This cme through the lysis of the prticulr cse of electro movig cross discrete lttice: the wvefuctio is pictured s wve of mplitude probbilities (complex umbers whose mgitude is iterpreted s probbilities). 4
Notice, however, tht whe tkig the limitig cse of the lttice spcig tedig to zero, oe eds up with situtio i which the electro is propgtig through cotiuum lie spce. Thus, this limitig cse tkes us to the study of prticle movig i cotiuum spce. I logy to the discrete lttice, where the loctio of the toms guided the selectio of the stte bsis { }, i the cotiuum spce we cosider the followig cotiuum set, { x, x } Cotiuum sptil-coordites bse () I logy to the discrete cse, x stds for stte i which prticle is locted roud the coordite x. x For every vlue x log the lie oe coceives correspodig stte. If oe icludes ll the poits o the lie, complete bsis set results s idicted i (), which will be used to describe geerl qutum stte d, hece, to describe the oe-dimesio motio of prticle. A give stte specifies the prticulr wy i which the mplitude probbility of prticle is distributed log lie. Oe wy of specifyig this stte is by specifyig ll the mplitudeprobbilities tht the prticle will be foud t ech bse stte x; we write ech of these mplitudes s x. We must give ifiite set of mplitudes, oe for ech vlue of x. Thus, = About the x ottio x x Represettio of the wvefuctio () i the sptil coordites bsis We could use ltertive ottios, like, for exmple, = x A Ψ (x ) ; s to mke this expsio to resemble the expsio of vector i terms of bsevectors x with the correspodig coefficiets A Ψ (x ) plyig the role of weightig-fctor coefficiets. Isted of A Ψ (x ) sometimes (x) is preferred; thus, = x (x) ; Multiplyig the expressio bove by prticulr br x, we should obti, x = ( x ) 5
(This result will be ustified bit more rigorously i the ext sectio below; see the delt Dirc sectio). Tht is, ( x) x Thus, we will use idistictly the followig ottio = x x.= Amplitude probbility tht the prticle iitilly i the stte be foud (immeditely fter mesuremet) t the stte x. (3) Represettio of the x x. wve-fuctio i the (4) sptil coordites bsis umber Number Cutio: x does ot me [x]* (x). Workig with the sptil coordites bse { x } my costitute the oly occsio i which the ottio x becomes cofusig with the defiitio of the sclr product. Note: I Chpter 8 we used the ottio (t) = A (t), where the mplitudes A (t) = were determied by the Hmiltoi equtios d i = H dt. I this chpter, we re usig isted cotiuum bse { x, x } ; ccordigly the mplitudes will be expressed s x, which is lso writte s x. Soo we hve to ddress the how the Hmiltoi equtios look like whe usig cotiuum bse. The Delt Dirc A subproduct of the mthemticl mipultio expressig stte i give bsis is the closure reltioship tht the compoets of the bsis set must comply: the Delt Dirc reltioship. This is illustrted for the cse of the spce-coordites bsis. = x [ x ] 6
x x = x = x [ x ] x x [ x ] x = x x [ x ] ( x ) ( x ) Or, equivletly x = f x x x [ x ] O the other hd, it is ccepted tht for rbitrry fuctio f, the delt fuctio is defied s ( x - x ) f x The lst two expressios re cosistet if, x x = ( x x ). Thus, we hve the followig result: Cse of discrete sttes Delt Kroeecker Cse of cotiuum sttes Delt Dirc (x -x ) (5) m = m x x = ( x - x ) Comptibility betwee the physicl cocept of mplitude probbility d the ottio used for the ier product * x) x We kow tht give stte, the mplitude probbilitiesx tht pper i the expsio = x x re determied by the Hmiltoi equtios. 7
Suppose we hve prticle i the stte d we wt to kow the mplitude probbility (fter give mesuremet process) to fid the prticle t the stte. Tht is, we wt to evlute. There re my pth wy for the stte to trsit to stte. It could do it by pssig first through y of the bse-sttes x. Sice ech stte x geertes pth, d ll these pths hve the sme iitil d fil stte, the, ccordig to the rules estblished i Chpter 7, the totl mplitude probbility will be give by, x = ll x x x First, the sum over smll regio of width would be xx multiplied by. The, sice x vries from to the expressio bove c be writte s, = x x (6) We c here itroduce the ottio used bove, mely x = x Also, sice x = x * oe obtis, = * x) x (7) The mplitude probbility is equl to the ier product betwee the fuctios d 0..B Represettio of the wvefuctio i the mometum coordites bsis { p } I the previous sectio, rbitrry stte ws expressed i terms of the compoets of the spce-coordites bsis { x, x }. Here we preset ltertive bsis wy to express the stte, the cotiuum bse of mometum coordites. We will relize tht we re lredy fmilir with the sttes comprisig this ew bsis. Recll tht i Chpter 5 we itroduced the Fourier trsform F= F(k) of fuctio (x). It ivolved the itroductio of the complex hrmoic fuctios e k, where e k (x) = e ikx for k 8
( x )= F(k) e ikx dk Fourier trsform Bse-fuctio e k evluted t x where the weight coefficiets F(k), referred to s the Fourier trsform of the fuctio, re give by, F(k) = e - i k x' ( x' ) ' However, it is coveiet to express the lst two expressios i terms of the vrible p = k, ( x )= (p) e i ( p / ) x d p (8) where (p) = e Fourier trsform of -i(p/ )x ( x' ) ' (9) Expressios (8) d (9) hve clerer physics iterprettio, p ( x ) = π e i ( p / ) x Accordig to de Broglie, p represets ple wve of defiite lier mometum p. (0) We c re-write expressio (8) i terms of fuctios, ( x )= ( p) Fourier trsform of ( x) e i ( p / ) x d p p = ( p) p dp Fuctio Sum of fuctios () Expressios (8) d () re equivlet. 9
Plcig p i brcket ottio First, guided by expressio (4), = otice tht expressio (0), p (x) = stte p s follows, x x, e i ( p / ) x, suggests to defie mometum π p = x p ( x ) x e i( p / ) x π Represettio of the mometum stte p i the spce-coordite bsis { x } () x p p ( x ) = π e i ( p / )x Amplitude probbility tht prticle, i stte of mometum p, be foud t the coordite x. (3) This is the de Broglie hypothesis i the lguge of mplitude probbilities. Plcig expressio () i brcket ottio I expressio () we c mke the the followig ssocitio, Hece, = (p) p dp p = (p) p dp (4) Fourier trsform ( x ) Expressio (4) gives s lier combitio of the mometum sttes p defied i () bove. Expressios (8), () d (4) re equivlet. 0
The sttes described i () costitute bse (the ustifictio comes from the Fourier trsform theory), { p, - < p < } cotiuum bse of mometum-coordites (5) Exercise: Prove tht p p = ( p p ) Now we formlly c ustify tht i expressio (4): p = (p) I effect, multiplyig (4) with br p, oe obtis, p = p (p ) p d p = (p ) p p dp usig p p = ( p p ) p = (p) (6)
Summry bout the mometum coordites: A rbitrry wvefuctio c be expressed s lier combitio of mometum-coordites p, = where (p) = p (p) d p = p p dp -i(p/ )x e ( x' ) ' p =(p) is the Fourier trsform of = (x) (7) p (p) Amplitude probbility tht prticle i the stte c be foud, upo mkig mesuremet, i the stte p. p dp ( p) dp Probbility tht prticle i the stte be foud with mometum withi the itervl ( p, p+ d p). < p > = p (p) dp = p p dp Averge lier mometum of esemble of systems i the stte Expressios (4) d (7) summrizes the obective of this sectio, showig expsio of the stte i two differet bsis, the cotiuum sptil-coordites bse { x, x }; d the lier-mometum bse { p, - < p < }.
SUMMARY Coordites bsis = = x x x (x) ; Mometum bsis = = p p dp p (p) d p (p) is the Fourier trsform of = (x) p is such tht, x p p ( x ) = π e i ( p / )x 0..C Tesor Product of Stte Spces Cosider the stte spce comprised of wvefuctios describig the sttes of give system ( electro, for exmple). We use the idex- to differetite it from the stte spce comprised of wvefuctios describig the stte of other system (other electro, for exmple) which is iitilly locted fr wy from the system. The systems (the electros) my evetully get closer, iterct, d the go fr wy gi. How to describe the spce stte of the globl system? The cocept of tesor product is itroduced to llows such descriptio Let { u i (), i,, 3,...} be bsis i the spce ε, d { v i (), i,, 3,...} be bsis i the spce The tesor product of of ε d ε, deoted by ε ε ε, is defied s spce whose bsis is formed by elemets of the type, { u i () v () } ε 3
Tht is, elemet of ε ε is lier combitio of the form, The tesor product i, c i u i () v () is defied with the followig properties, [ () ] () [ () () ] () [ () ] [ () () ] [ () () ] () [ () () () [ () () ] [ () ] + [ () () ] + [ () () ] () ] Two importt cses my rise. Cse : The wvefuctio is the tesor product of the type, () () Tht is c be expressed s the tesor product of stte from ε d stte from ε. i, i i b i u i () b v () ui () v () Cse : The wvefuctio cot be expressed s the tesor product betwee stte purely from the spce ε d stte purely from the spce ε. I this cse, the wvefutio tkes the form, i, c i ui () v () Let s cosider, for exmple the cse i which ech spce ε d ε hs dimesio. i, c i u i() v () i c i u v() ci u v() i() i () i c u () v() c u () v() 4
c u() v() c u() v() Notice, cot be expressed i the form () () To mke the cse eve simpler, let s ssume c = c = 0, c u () v () + c u () v () etglemet cot be expressed i the form of sigle term of the form () () 0. The Schrödiger Equtio s postulte 0..A The Hmiltoi equtios expressed i the cotiuum sptil coordites. The Schrödiger Equtio. 3 I chpter 8 we obtied the geerl Hmiltoi equtios tht describe the time evolutio of the wvefuctio (t) = A ( t ), da H ( t) A (8) dt i Chpter 9 described the prticulr cse of electro movig i lttice (the ltter costituted by toms seprted distce b. Whe we took the limit b 0 the Hmiltoi equtios i (5) took the form ( x,t) ( x,t) i V ( x,t) ( x,t) (9) t m x Let s cosider ow rbitrry geerl cse. eff How does the Hmiltoi equtios (8) look like whe expressed i the i the cotiuum spce coordites { x, x }? Let s fid out such geerl forml expressio (oe tht is more geerl th expressio (9) ). First otice tht the mplitudes system, = A A i (8) ccout for the stte describig the qutum Sice A c lso be writte s A =, Eq. (5) c lso be expressed s, d i = H dt 5
Let s lso recll, from Chpter 7, tht the coefficiets H opertor H ~ (specific to the problem beig solved.) Tht is, deped o t. d i = H ~ dt I the cotiuum spce coordites we should expect, d i x = dt x H ~ x x d x (x) ( x ) re obtied from the Hmiltoi H H ~. I geerl, H ~ d d i ( x ) = dt H( x, x )( x ) d x (0) where we hve defied H( x, x ) x H ~ x Quotig Feym, 4 Accordig to (0), the rte of chge of t x would deped o the vlue of t ll other poits x. x H ~ x is the mplitude per uit time tht the electro will ump from x to x. It turs out i ture, however, tht this mplitude is zero except for poits x very close to x. This mes (s we sw i the exmple of the chi of toms) tht the right-hd side of Eq. (0) c be expressed completely i terms of d the sptil derivtives of, ll evluted t x. The correct lw of physics is d H( x, x ) ( x ) d x = ( x ) + V ( x) ( x ) Postulte () m Where did we get tht from? Nowhere. It cme out of the mid of Schrodiger, iveted i his struggle to fid uderstdig of the experimetl observtio of the experimetl world. Usig () i (0) oe obtis, 6
i t V ( x, t) x m Schrodiger Equtio This equtio mrked historic momet costitutig the birth of the qutum mechicl descriptio of mtter. The gret historicl momet mrkig the birth of the qutum mechicl descriptio of mtter occurred whe Schrodiger first wrote dow his equtio i 96. For my yers the iterl tomic structure of the mtter hd bee gret mystery. No oe hd bee ble to uderstd wht held mtter together, why there ws chemicl bidig, d especilly how it could be tht toms could be stble. (Although Bohr hd bee ble to give descriptio of the iterl motio of electro i hydroge tom which seemed to expli the observed spectrum of light emitted by this tom, the reso tht electros moved this wy remied mystery.) Schrodiger s discovery of the proper equtios of motio for electros o tomic scle provided theory from which tomic pheome could be clculted qutittively, ccurtely d i detil. Feym s Lectures, Vol III, pge 6-3. Although the result () is kid of postulte, we do hve some clues bout how to iterpret it, bsed o the prticulr cse of the dymics of electro i crystl lttice, studied i Chpter 9. () 0..B Iterprettio of the Wvefuctio Eistei s view o the grulrity ture of the electromgetic rditio I Chpter 5, hrmoic fuctio ws used to describe the motio of free prticle i logy to the existet formlism to describe electromgetic wves, π ε ( x, t) ε o Cos [ x νt] electromgetic wve λ where, the electromgetic itesity I (eergy per uit time crossig uit cross-sectio re perpediculr to the directio of rditio propgtio) is proportiol to ε ( x, t). Eistei (i the cotext of tryig to expli the results from the photoelectric effect) itroduced the grulrity iterprettio of the electromgetic wves (lter clled photos), bdoig the more clssicl cotiuum iterprettio. I Eistei s view, the itesity is iterpreted s sttisticl vrible I c o ε N h. Here N costitutes the verge umber of photos per secod crossig uit re perpediculr to the directio of rditio propgtio; ε ~ N Averge vlues re used i this iterprettio becuse the emissio process of photos by give source is sttisticl i ture. The exct umber of photos crossig uit re per uit time fluctutes roud verge vlue N. 7
Mx Bor s Probbilistic Iterprettio of the wvefuctio I logy to Eistei s view of rditio, Mx Bor proposed similr view to iterpret the prticle s wve-fuctios. I Mx Bor s view, ( x, t) plys role similr to ε ( x, t), ( x, t) is mesure of the probbility of fidig the prticle roud give plce x d t give time t. This iterprettio ws itroduced yers fter Schrodiger (96) hd developed forml qutum mechics descriptio. More specificlly, ( x, y,z, t) plys the role of probbility desity. Pictorilly, the prticle is more likely to be t loctios where the wvefutio hs pprecible vlue. Determiistic evolutio of the wvefuctio The predictios of qutum mechics re sttisticl. I order to kow the stte of motio of prticle, we must mke mesuremet But mesuremet ecessrily disturbs the system i wy tht cot be completely determied. However, otice tht, beig the solutio of differetil equtio (the Schrodiger equtio), vries with time i wy tht is completely determiistic. Tht is, if were kow t t=0, the Schrodiger equtio determies precisely its form t y future time. Tht is, QM mkes determiistic predictio of mplitude probbility wve. However, the ltter does ot covey to determiistic outcomes. There is oe further poit to cosider. How to determie the wvefuctio t t=0? How do the experimetl mesuremets led to the recostructio of the wvefuctio? Or, how to prepre system i defiitely uique stte? If we could ot recostruct wvefuctio, wht would be the beefit of hvig theoreticl formultio tht describes determiistic evolutio of somethig we do ot kow? As it turs out, despite the fct tht mesuremet i QM i geerl ffect the stte of system, such recostructio is possible i some cses (thik of system tht re i sttiory stes.) But i lrger cotext, to beefit of the QM determiistic formultio wht we eed is to prepre system (or my systems) i defiite stte; for the theory could the be used to mke predictios bout the evolutio of tht prticulr stte. We will ddress this issue i the ext chpters, fter the itroductio of observbles d eigesttes. We will see tht fidig eigesttes commo to differet observbles rrows the selectio pool of sttes i which the system c be foud. This procedure leds to the cocept of esemble of system costituted by (i this wy) eqully prepred systems, which costitutes the lbortory i which the QM cocept re develop. (A the ed, 8
systems cot be determied with bsolute certity simply becuse set of mesuremets t t=0 t most my led to the determitio of but ot to uiquely defie ). Let s expli the sttisticl iterprettio bit further i the cotext of esemble of ideticlly prepred systems. Esemble 5 Imgie very lrge umber of ideticlly prepred idepedet system (ssumed to be ll of them i the sme stte), ech of them cosistig of sigle prticle movig uder the ifluece of give exterl force. All these systems re ideticlly prepred. The whole esemble is ssumed to be described by complex-vrible sigle wvefuctio ( x, y, z, t), which cotis ll the iformtio tht c be obtied bout them. describes the whole esemble... Esemble is used to mke probbilistic predictio o wht my hppe i prticulr member of the esemble. N It is postulted tht: If mesuremet of the prticle s positio re mde o ech of the N member of the esemble, the frctio of times the prticle will be foud withi the volume elemet d 3 r = dy dz roud the positio r ( x, y, z, t) t the time t is give by (3) * ( x, y, z, t) ( x, y, z, t) d 3 r where * stds for the complex cougte umber. Notice tht this is othig but the lguge of probbility; i this cse, positio probbility desity P. P * ( x, y, z,t) ( x, y, z,t) ( x, y, z,t) ( x, y, z, t ) 9
Cutio: For coveiece, we shll ofte spek of the wvefuctio of prticulr system, BUT it must lwys be uderstood tht this is shorthd for the wvefuctio ssocited with esemble of ideticl d ideticlly prepred systems, s required by the sttisticl ture of the theory. 6 0..C Normliztio coditio for the wvefuctio The probbilistic iterprettio of the wvefuctio implies, therefore, the followig requiremet: * 3 ( x, y, z,t) ( x, y, z,t) d r (4) All spce becuse give prticle, the likelihood to fid it ywhere should be oe. Iheret to this requiremet is tht, ( r, t) 0 (5) r Notice tht if is solutio of the Schrodiger equtio, the fuctio c (c beig costt) is lso solutio. The multiplictive fctor c therefore hs to be chose such tht the fuctio c stisfies the coditio (4). This process is clled ormlizig the wvefuctio. I geerl, there will be solutios to the Schrodiger equtio () whose solutio ted to ifiite vlue. This mes they re o-ormlizble d therefore c ot represet prticle probbility desity. Such fuctios must be reected o the grouds of Bor s probbility iterprettio. Qutum mechics sttes re represeted by squre- itegrble fuctios tht stisfy the Schrodiger equtio. The prticulr subset of squre itegrble fuctios form vector spce clled the Hilbert spce. QUESTION: Suppose tht is ormlized t t 0. As the time evolves, will chge. How do we kow if it will remi ormlized? Here we show tht the Schrodiger equtio hs the remrkble property tht it utomticlly preserves the ormliztio of the wvefuctio: 0
If stisfies the Schrodiger equtio The if the potetil is rel dt Proof: Let s strt with d dt ( x, t) = 0 (6) d ( x,t) ( x,t) (7) t We provide below grphic ustifictio of (7). fx, t ) fx, t +) x x t O the other hd, fx, t +) - fx, t ) = t [fx, t + ) - fx, t) ] t ) t t t (8) We use the Schrodiger equtio (9) to clculte the time derivtives, i V ( x,t) t m x i i V ( x,t) m x i i V ( x,t) t m x Tkig the complex cougte, d ssumig tht the potetil is rel, i i V( x,t) t m x Addig the lst two expressios, t t i m x x
i m Replcig (9) i (8) we obti, Accordigly, d dt t i m x x x x x x ( x,t) ( x,t) t i m x x x i m x x The expressio o the right is zero becuse x 0 x (9) 0..D The Philosophy of Qutum Theory There hs bee cotroversy over the Qutum Theory s philosophic foudtios. Neils Bohr hs bee the pricipl rchitect of wht is kow s the Copehge iterprettio (sttisticl iterprettio) Eistei ws the pricipl critic of Bohr s iterprettio. His sttemet God does ot ply dice with the uiverse, refers to the bdomet of strict cuslity d idividul evets by qutum theory. Heiseberg coutercts rguig: We hve ot ssumed tht the qutum theory (s opposed to the clssicl theory) is sttisticl theory, i the sese tht oly sttisticl coclusios c be drw from exct dt. I the formultio of the cusl lw, mely, if we kow the preset exctly, we c predict the future it is ot the coclusio, but rther the premise which is flse. We cot kow, s mtter of priciple, the preset i ll its detils. Louis de Broglie, o the other hd, rgues tht tht limited kowledge of the preset my be rther limittio of the curret mesuremet methods beig used. He recogizes tht
) it is certi tht the methods of mesuremet do ot llow us to determie simulteously ll the mgitude which would be ecessry to obti picture of the clssicl type, d tht b) perturbtios itroduced by the mesuremet, which re impossible to elimite, prevet us i geerl from predictig precisely the results which it will produce d llow oly sttisticl predictios. The costructio of purely probbilistic formule ws thus completely ustified. But, the ssertio tht i) The ucerti d icomplete chrcter of the kowledge tht experimet t its preset stge gives us bout wht relly hppes i microphysics, is the result of ii) rel idetermicy of the physicl sttes d of their evolutio, costitutes extrpoltio tht does ot pper i y wy to be ustified. De Broglie cosiders possible tht lookig ito the future we will be ble to iterpret the lws of probbility d qutum physics s beig the sttisticl results of the developmet of completely determied vlues of vribles which re t preset hidde from us. Louis de Broglie s view give bove highlights the obectio to qutum mechics philosophic idetermiism. Accordig to Eistei: The belief of exterl world idepedet of the perceivig subect is the bsis of ll turl sciece. Qutum mechics, however, regrds the iterctio betwee obect d observer s the ultimte relity; reects s meigless d useless the otio tht behid the uiverse of our perceptio there lies hidde obective world ruled by cuslity; cofies itself to the descriptio of the reltios mog perceptios 7 Physics hs give up o the problem of tryig predictig exctly wht will hppe i defiite circumstce. 0.3 Expecttio vlues 0.3.A Expecttio (or me) vlue of prticle s positio Let s ssume we hve system cosistig of box cotiig sigle prticle, which is (we ssume) i stte. The expecttio vlue of the prticle s positio is defied by, x - x ( x) (30) But wht does this itegrl exctly me? It is worth to emphsize first wht type of iterprettio should be voided. 8 3
Expressio (30) does ot imply tht if you mesure the positio of the prticle over d over gi the - x ( x) would be the verge of the results. I fct, if repeted mesuremets were to be mde o the sme prticle, the first mesuremet (whose outcome is upredictble) will mke the wvefuctio to collpse to stte of correspodig prticle s positio x (let s sy x 0 ); subsequet mesuremets (if they re performed quickly) will simply repet tht sme result x 0. O the cotrry, x - x ( x) mes the verge obtied from mesuremets performed o my systems, ll i the sme stte. Tht is, A esemble of systems is prepred, ech i the sme stte, d mesuremet of the positio is performed i ll of them. x is the verge from such mesuremet. describes the whole esemble... Esemble is used to mke probbilistic predictio o wht my hppe i prticulr member of the esemble. N Fig. 0. Esemble of ideticlly prepred systems. Whe we sy tht system is i the stte, we re ctully referrig to esemble of systems ll of them i the sme stte. Thus, represets the whole esemble. d x 0.3.B Clcultio of m dt As time goes o, the expecttio vlue x my chge with time, sice the wvefuctio evolves with time. Let s clculte its rte of chge. d x dt d dt x ( x, t) x t t x ( x, t) t x t For the cse where the potetil is rel, we obtied i expressio (9) tht, 4
t t i m x x d x dt i x m x x i x m x x x After itegrtig by prts, it gives d x dt i m x x Itegrtig oe more time by prts (ust the secod term o the right side,) d x i dt m x (3) We would be tempted to postulte tht the expecttio (or me) vlue of the lier d x mometum is equl to p m. I tht cse, expressio (3) would give, dt d x p m dt = (3) i x i x However, look t the fuctios iside the itegrl. How to uderstd tht term like x would led to the verge vlue of the lier mometum? This ppers bit strge, to sy the lest. (We will more sese of this lst result i the sectios below, whe the cocept of qutum mechics opertors is itroduced). 0.3.C Expecttio (verge) vlues re clculted i esemble of ideticlly prepred systems I geerl, the me-vlue of give physicl property f (mely, eergy, lier mometum, positio, etc.), more geericlly clled observble, is obtied by mkig mesuremet i ech of the N eqully prepred systems of esemble (d ot by vergig repeted mesuremets o sigle system.) The whole esemble is ssumed to be described by complex-vrible sigle wvefuctio ( x, y, z, t). Whe mkig mesuremets o ech of the N ideticlly prepred systems of the esemble (see left-side of the figure below) let s ssume we get series of results (see rightside of the figure below) like this: 5
N systems re foud to hve vlue of f equl to f, from which we deduce tht the prticulr system collpsed to the stte f right fter the mesuremet N systems re foud to hve vlue of f equl to f, from which we deduce tht the prticulr system collpsed to the stte f right fter the mesuremet etc. Accordigly, the verge vlue of f is clculted s follows, f v N f N f N N Usul procedure to clculte verge vlues (33) N Notice tht whe the totl umber of mesuremet N is very lrge umber, the rtio N is othig but the probbility of fidig the system i the prticulr stte f. QM postultes tht the vlue of the vlue f, f ψ should be iterpreted s the probbility to obti f ψ = N Qutum Mechics postulte (34) N Represetig the esemble ψ Hece, expressio (33) c be writte s, f v f f ψ (35) 6
... Esemble... N Before the mesuremets The vlue of f i ech system is ukow. N After the mesuremets The vlue of f hs bee mesured i ech system; fterwrds we proceed to clculte the verge vlue < f >. 0.4 Opertors ssocited to Observbles Qutities such s positio, mometum, or eergy (which re mesured experimetlly) re clled observbles. I clssicl physics, observbles re represeted by ordiry vribles (E, p, for exmple). I qutum mechics observbles re represeted by opertors (qutities tht operte o fuctio to give ew fuctio.) Whe system i stte eters some pprtus, like, for exmple, mgetic field i the Ster Gerlch experimet, or mser resot cvity, it my leve i differet stte. Tht is, s result of its iterctio with the pprtus, the stte of the system is modified. Symboliclly, the pprtus c be represeted by correspodig opertor F ~ such tht, = F ~ (36) Note: We will distiguish the opertors (from other qutities) by puttig smll ht ~ o top of its correspodig symbol. We show below how to ssocite qutum mechics opertor F ~ to give physicl qutity f. 7
0.4.A Observbles, eigevlues d eige-sttes Let s cosider physicl qutity or observble f ( f could be the prticle s mometum for exmple) tht chrcterizes the stte of qutum system. gulr I qutum mechics, the differet vlues tht give physicl qutity f (observble) c tke, re clled its eigevlues; f, f, f 3, (37) The set of these qutum eigevlues is referred to s the spectrum of eigevlues of the correspodig qutity f. For simplicity, let ssume for the momet tht the spectrum of eigevlues is discrete. will deote the stte where the qutity f hs the vlue f ; These sttes will be clled eigesttes (38) We will ssume tht these eigestes stisfy, m = ( x) m ( x) = m (39) Let s ssume lso tht the eigesttes ssocited to the observble f costitute bsis, { ; =,, 3, } Bsis of eigesttes A rbitrry stte c the be represeted by the expsio, = A =. (40) where A = = ( x) ( x) Sice the stte must be ormlized stte, the A = = Accordig to expressio (35), the me vlue of f, whe the system is i the stte, is give by, f = v f A = f (4) Expressio (4) is still very geerl, sice we do ot kow yet the eigesttes. (We will describe below how to figure out these sttes). 8
0.4.B Defiitio of the QM opertor F ~ to be ssocited with the observble physicl qutity f. 9,0 The opertor F ~ to be ssocited with the observble f is such tht, whe ctig o rbitrry stte, stisfies the followig: F ~ f v Defiitio of the Opertor F ~ ssocited to the observble f (4) (the verge vlue o the right side is clculted over the esemble represeted by the stte ) But, how to obti explicit expressio for such opertor F ~? If the observble qutity were, for exmple, the lier mometum, how to build its correspodig qutum mechics opertor? Before buildig the opertors explicitly, we first we derive i this sectio geerl selfcosistet expressio tht shows how opertor, ssocited to give observble, should look like (see expressio (47) below). Subsequetly, sectios 0.4.C d 0.4.D will provide specific procedure o how to costruct the positio d mometum opertors. Derivig self-cosistet expressio of QM opertor F ~ If the system is i the stte, = A =, the verge vlue of the qutity f is give by (4), f = v f A = f The requiremet to build the opertor F ~ is, F ~ = f = = v f A = f f * f (43) 9
Notice, we c obti more compct expressio for the opertor F ~ if we use the ottio, P ~ pro, Proectio opertor (44) Expressio (44) describes opertor tht whe ctig o stte gives the proectio of tht stte log the stte ; tht is, P ~ pro, = = (45) Accordigly, (43) c be expressed s, F ~ = [ f ] (46) stte umber opertor where deote the stte where the qutity f hs the vlue f ; This ottio trick llows us to express the opertor i more compct form, F ~ = f (47) eigevlues Proectio opertorp ~ pro, built out of eigesttes ssocited to the physicl qutity f. This is self-cosistet expressio for the qutum opertor F ~ tht will be ssocited with the clssicl qutity f. It is expressio tht is comptible with the ~ requiremet tht Ψ F Ψ f. Note: Although the opertor defied i (47) would pper to deped o the prticulr bsis composed by eigesttes, the opertor should remi the sme if we chged the bsis. This clim is supported by the fct tht the vlue of f, which itervees i the defiitio of F ~, should be idepedet of the bsis chose. v v 30
Notice (47) is self-cosistet with its defiitio. By pplyig the opertor eigesttes, it gives, F ~ to oe of the Tht is, oce the opertor F ~ F ~ = f (48) (ssocited to the physicl qutity f ) is kow, the eigefuctios of tht give physicl qutity re the solutios of the equtio F ~ = where is costt. Still, otice tht (47) d (48) give ust self-cosistet expressios for the opertor F ~. It is expressio tht is comptible with the requiremet tht F ~ f v, but it is defied i terms of eigesttes tht, for give physics qutity f, we do ot kow yet but would like to fid out. Accordigly, F ~ is ot completely kow yet. But, prphrsig Ldu, Although the opertor F ~ is defied by (48), which itself cotis the eigefuctios, o further coclusios c be drw from the results we hve obtied. However, s we shll see below, the form of the opertors for vrious physicl qutities c be determied from direct physicl cosidertios, which subsequetly, usig the bove properties of the opertors, will eble us to fid the eigefuctios d eigevlues by solvig the equtio F ~ = Expressios (4) d (47) will guide us to build the qutum opertors. 0.4.C Defiitio of the Positio Opertor X ~ We re lookig for opertor X ~ such tht X ~ gives us the me vlue of the positio whe the system is i the stte, which requires, X ~ *( x) [ ( X ~ )( x) ] x - x x ( x), X ~ =? (49) [ *( x) ] x [ ( x) ] ( X ~ ) ( x ) = x ( x ) (50) Notice, we cot sy X ~ = x; tht would be icorrect. (For istce, wht vlue of x would you choose to mke the expressio X ~ = x vlid?). More pproprite is first to defie the idetity fuctio I, which stisfies, 3
I(x) = x, The (50) could the be writte s, ( X ~ ) ( x) = (I ) ( x) becuse tht would give [ X ~ ]( x) = [I ] ( x) = I( x) ( x) = x ( x). Tht is, X ~ = I Aswer (5) I summry, The Positio Opertor X ~ X ~ is the opertor ssocited to x X ~ = I where I is the idetity fuctio; I (x) = x ( X ~ ) x= x x (5) X ~ = [ * x ] [ X ~ x ] = [ * x ] x [x] x Notice, it is strightforwrd to relize tht X is the opertor ssocited to ~ x, ~ 3 X is the opertor ssocited to x 3, etc. More geerl, X ~ is the opertor ssocited to Averge vlue of physicl qutity x (53) Qutum mechics opertor x v X ~ ~ x X v x v ~ X 3
if h(x) is polyomil or coverget series, (54) the h( X ~ ) will be the opertor ssocited to h ( x) For exmple: If physicl qutity chges s, h(x) = 3 x - 7 x 3 the the correspodig opertor will be 3 X ~ 3-7 X ~, tht is, the lst expressio is obtied by evlutig h( X ~ ). Averge vlue Qutum mechics of physicl qutity opertor h (x) h( X ~ ) v For exmple, if h(x) were clssicl potetil tht depeds oly o positio x, the h( X ~ ) would be the the correspodig qutum mechics potetil opertor. Tke the cse of the hrmoic oscilltor, where V(x) = (/)k x the correspodig QM opertor is ~ V ~ = (/)k X. 0.4.D Defiitio of the Lier Mometum Opertor Here we costruct the qutum Lier Mometum Opertor, d give its represettio i both the mometum-coordites d sptil-coordites bsis. This exmple will help to illustrte tht opertors re geerl mthemticl cocepts whose represettio depeds o the bse sttes beig used. The tsk of defiig the mometum opertor is fcilitted by the fct tht the eigefuctios d eigevlues re lredy kow. Thus Sectios 0.4.D d 0.4.D my pper to be trivil; still it helps to illustrte the coectio betwee opertor d the me vlue of the correspodig observble. 0.4.D The Lier Mometum Opertor P ~ expressed i the mometum sttes bsis { p } I this cse, the observble f refers to the lier mometum p. Before buildig the lier mometum opertor let s summrize wht we kow bout the mometum sttes. Accordig to de Broglie, for give p we ssocite mometum stte p give by expressio () bove, 33
p x p ( x ) = x π e i ( p / ) x (55) Represettio of the mometum stte p i the spcecoordite bsis { x } x p p ( x ) = e i ( p / ) x π Sometimes, for coveiece, we will use the ottio mometum p or p isted of p. A rbitrry stte c be expressed s lier combitio of the mometum sttes, where = (p) = p = p p dp = p (p) d p (56) -i(p/ )x e ( x' ) ' Now we wt to build the Lier Mometum Opertor tht will be ssocited to the physicl qutity p. Accordig to expressio (47), F ~ = p, but with the summtio exteded to itegrl over cotiuum vrible, the mometum opertor should hve the form, P ~ = p p p dp Lier Mometum Opertor (57) Applyig the mometum opertor P ~ to rbitrry stte gives, P ~ = p p p dp, Accordig to (56) p =(p), which gives P ~ = p (p) p dp (58) ~ P Ψ is lier combitio of sttes p. 34
Let verify if the opertor P ~ defied i (57) stisfies the requiremet tht P ~ is equl to the verge mometum by ). Sice = p (p) d p, we hve, = p (the verge tke over esemble chrcterized p (p ) d p (59) Similr to the delt Dirc obtied i expressio (6) bove, it c be demostrted tht, p p = ( p p ) (60) From (58) d (59), P ~ = [ p (p ) d p ] [ p (p) p dp ] = d p p (p) (p ) p p dp = d p p (p) (p ) ( p p ) dp = d p p (p ) (p ) Usig p isted of p = p (p) d p Sice =(p) is the Fourier trsform of = ( x ), the term o the right side of the lst expressio is the the verge lier mometum of system i the stte. (See the summry fter expressio (7) bove). P ~ = p (p) d p = p Averge mometum (6) Hece, the opertor P ~ defied i (57), therefore, fulfills the geerl requiremet (4) impossed to qutum mechics opertors. 35
Mtrix represettio of the Lier Mometum Opertor P ~ i the mometum coordites bsis { p, - < p < } Usig (60), otice i (57) tht pplyig P ~ to prticulr stte p gives, P ~ p = p p (6) Usig gi (60) we obti ~ P = p P ~ p = p ( p p ) (63) pp' Elemets of the mtrix represettio of the opertor i the mometum bsis Summry The Mometum Opertor P ~ P ~ = p p p dp (64) The remiig clcultios re reltively strightforwrd becuse of the fct tht we ssume rbitrry stte c expressed s lier combitio of moemetum sttes = p p dp = p (p) d p where = (p) is the Fourier trsform of (x) Accordigly, d P ~ = P ~ = = p p p dp = p (p) p dp [ * x ] [P ~ x ] p (p) d p p 36
0.4.D The lier mometum opertor P ~ expressed i the sptil coordites bsis { x, x } Wht bout if the expsio of i the mometum bsis { p, - < p < } is ot kow, d, isted, its expsio i the sptil coordite is vilble? (i.e. (x) is kow for every vlue of x, but its Fourier trsform of (p), is ot redy vilble). Wht to do to fid P ~ (without hvig to go through the trouble of expressig i terms of the mometum bsis s expressio (64) requires)? It is show below tht ltertive wy (ltertive to (64) ) to express the lier mometum does exist. Applyig (57), P ~ = we hve = P ~ = p p p dp, to the stte fuctio, where p p dp = p p p dp = p (p ) dp p (p) p dp x P ~ = p (p) x p dp, [P ~ x= p (p) p x dp Usig (53) x p p ( x ) = π e i ( p / ) x [P ~ x= p (p) π e i ( p / ) x dp, Otherwise, if the Fourier tyrsform (p) of (x) were kow, the P ~ = [ p p p dp ] = p p p dp = p (p) p dp 37
[P ~ x = (p) p π e i ( p / ) x dp, = (p) i d π e i ( p / ) x dp = i d (p) π e i (p / ) x dp = i d (p) p xdp, Hece, P ~ = = i i d d x (65) The Mometum Opertor P ~ (i the sptil-coordites spce) P ~ = i d (66) The remiig clcultio is fcilitted if rbitrry stte is expressed s lier combitio of sptil coordites, so the derivtive c be strightforrdly evluted. P ~ = which leds to i d p v = P ~ d = i 38
I pssig, otice tht the lst expressio is ideticl to (3) where we clculted the rte of d x chge of the verge positio of prticle m. At tht time, the presece of sptilderivtive ppered to mke o sese i beig ivolved i wht could be iterpreted s dt d i the velocity of prticle. Now we see tht such sptil derivtive ppers becuse we re usig the represettio of the mometum opertor i the sptil coordites (tht sptil derivtive does ot pper whe workig i the mometum bsis). 0.4.D3 Systemtic wy to express the opertors P ~, bsis ~ P, ~ 3 P, etc., i the sptil-coordites We lredy kow how to express the opertors P ~, P, P, i the mometum coordite bsis; we ust eed to pply 57) repetitive wys. Tht expressio will give us the opertors redy to pply o sttes whose expsio i the p sttes is redy vilble. But wht we wt ow is to hve those opertors i the sptil-coordites; i.e. expressed i wy redy to be pplied whe the sttes re expressed i the sptil-coordites. Here we show systemtic wy to obti such expressios. ~ ~ 3 Costructio of the Lier MometumP ~ i the sptil- coordites bse We lredy kow how to express P ~ i the mometum bsis, P ~ = p p p dp. Which, whe opertig i esemble of systems chrcterized by the wvefuctio = p p dp = p (p ) dp, stisfies, p v = p p p dp (67) First, let s work out the fctor p p iside the itegrl i expressio (67). Sice p = (p) is the Fourier trsform of x, we hve p p = p (p) = p -i(p/ )x e ' ( x' ) 39
= -i(p/ )x p e ( x' ) ' = i [ d e -i(p/ )x ] ( x' ) ' Itegrtig by prts = p p = p i Replcig (68) i (67), p v = p p i [e -i(p/ )x ] i = p p i d ( x') ' d (68) d dp (69) d dp. p v = i Wht is this? Notice, if we wted to expd the stte i the mometum bse we would write, i d = p p i d d i This is exctly wht we hve i the expressio bove. Accordigly, d (70) 40
= * x,t i d x,t ] If we defie the opertor P ~ = The lst expressio becomes, Tht is, P ~ = i d i p v = d (7) * x,t P ~ x,t ] (7) stisfies the requiremet for beig the lier mometum opertor. Costructio of the opertor ssocited to By defiitio: p v p = v p p p dp (73) The systemtic procedure cosists i evlutig first the fctor p p iside the itegrl. p p = p -i(p/ )x e ' ( x' ) = p -i(p/ )x e ( x' ) ' Cotiuig with similr procedure s bove (itegrtig by prts), we obti, p p = p Replcig (74) i (73) leds to, i d (74) p = v p p i d dp 4
Fctorig out = p p i d dp The itegrl term is the expsio of the ket i d i the mometum bse p = v i d = * x,t i d (75) x,t ] If we defie the opertor ~ P = The lst expressio becomes, i d (76) Tht is, ~ P = i d ssocited with p. p = v * x,t ~ P x,t ] (77) stisfies the requiremet for beig the qutum opertor to be More geerl, for positive iteger, Tht is, ~ P = i p = v d * x, t i d x,t ] (78) is the qutum mechics opertor ssocited to p Averge vlue of physicl qutity Qutum mechics opertor 4
p v p v p v i i i d d d If g(p, t) is polyomil, or bsolutely coverget series, i the clssicl mometum vrible p, oe obtis, g ( p, t) v = * x, t d g(, t ) i x,t ] (79) Averge vlue of physicl qutity v Qutum mechics opertor d g i g ( p, t) (, t) A exmple of (79) costitutes the QM opertor ssocited to the kietic eergy. m p v ~ P m = = m i d d (80) m 0.4.E The Hmiltoi opertor 0.4.E. Me eergy i terms of the Hmiltoi opertor Oe of the virtues of usig QM opertors is tht me vlues c be expressed idepedet of prticulr bsis. For exmple, recll tht < x > = X ~ d <p> = P ~. We re goig to do somethig similr here for the verge eergy <E>. I Chpter 8, the Hmiltoi Opertor H ~ ws recogized through its mtrix represettio [H], the ltter beig iterpreted s eergy mtrix due to the fct tht, whe workig with 43
sttiory sttes, the compoets of the mtrix were the eergy of the correspodig sttiory sttes. I the lguge of eigevlues used i this chpter we c write, H ~ E = E E (8) where { E, =,, 3, } is the bsis costituted by eergy sttiory sttes For system i rbitrry stte let s clculte the expecttio (or me) vlue of eergy. = E E (8) Its me eergy is give by, E E v E ψ = E * E E = E E E = H ~ E E This fctor c be expressed i terms of the Hmiltoi opertor From (74): H ~ E = E E = H ~ E E = E = H ~ (83) v Thus, we hve foud gi (s we did for the lier mometum d the positio opertors) elegt wy to express me vlue (i this cse for the eergy) tht does ot mke referece to the prticulr bse sttes. Whe explicit ottio of the bsis sttes is required to clculte sttiory sttes bsis { E, =,, 3, } E v, we my use the E E v E But, i some cses it my be coveiet to use differet bse. For geerl bsis { for =,,3, } we will hve, ψ 44
E = H ~ v Expressig d i the { } bsis = [ i i* i ] H ~ [ ] = i i H ~ (84) i, 0.4.E. Represettio of the Hmiltoi Opertor i the sptil coordite bsis Similr to the cse of the lier mometum opertor ddressed i the previous sectios, the Hmiltoi opertor c dopt differet shpes depedig o whether we re workig i the sptil-coordites bsis or i the mometum coordite bses. Here we ddress the represettio of the Hmiltoi opertor i the sptil-coordites I (83), let s tke the cse of sptil coordite bsis Defiig E E v v = H ~ = [ - = [ - = - x * x ] H ~ x * x ] [ - x * [ - x x ] H ~ x x ] x H ~ x x ] - H( x, x ) x H ~ x (85) (see lso expressio (0) bove). E = v - = - x * ( x) * - - H( x, x ) x H( x, x ) ( x ) But Schrodiger estblished tht (see expressio () bove) - H( x, x ) ( x ) d x = ( x ) + V ( x)( x ) d m 45
Accordigly E v = - ( x) * [ d m If we defie the potetil opertor V ~, ~ V Ψ V Ψ ( x ) + V ( x)( x )] (86) E = v - ( x) * d [ m + V ~ ] (x) H ~ E = H ~ v ~ Represettio of the Hmiltoi d H Opertor i the sptil coordites + V ~ (87) m bsis ~ Notice, ccordig to expressio (80) P m = d m, therefore we c write, ~ H ~ P m ~ V Summry Observble Me Opertor Opertor i the sptil vlue coordites represettio positio X ~ X ~ x x = X ~ X ~ = x mometum P ~ d i p p = P ~ P ~ = i d 46
Eergy ~ d H + V ~ m E E = H ~ 0.5 Properties of Opertors 0.5.A Correspodece betwee brs d kets Br Ket χ χ m m m m b b m * m m m m m b m b m bφ bφ Φ b * Φ b * Φ * Φ Φ Φ * Util ow we hve defied the ctio of lier opertor A ~ o kets the ctio of opertor o brs. χ. We wt to defie 0.5.B The Adoit opertor Cosider opertor A ~. We picture opertor A ~ tht whe ctig o rbitrry d wvefuctio the result is 3, or, or I, etc. For give A ~, we will defie its 47
correspodet doit opertor o wvefuctio ; tht is is, A ~. Through the defiitio we will fid out wht does A ~ do if if ~ A ~ = 3 the we eed to fid A =?; or A ~ d = etc. the we eed to fid ~ A =? I dditio, we would like to kow lso how to express the ctio of opertor oto wvefuctios i the lguge of brs d kets. For opertor ctig o ket, wht does A ~ χ me? For opertor ctig o br, wht does A ~ me? First, by defiitio, A ~ χ A ~ χ A ~ χ mes the opertor A ~ ctig o the ket χ. Let be give br, d A ~ give opertor. For ech rbitrry ket χ oe c ssocited complex umber ( A ~ χ ) ( A ~ ~ χ ) Give d A Notice tht through this ssocitio, the br is ssocited with ew br u, which stisfies, u ( A ~ χ ) (i) Let s defie the opertor A ~ such tht, A ~ u (ii) From (i) d (ii) ~ A ~ A (88) 48
A ~ is clled the doit opertor of the lier opertor A ~. Exercise: Show tht the doit opertor A ~ is lier. Exmple: Show tht Let s pply (88), ( (B ~ ) = B ~ ~ ~ A A, to the opertor A ~ = ~ B ) ~ B B ~ ~ [ B *] * ~ [ B ] * ~ [ B ] * B ~ Hece, (B ~ ) = B ~ (iii) Notice: Accordig to (iii) the defiitio (88) c lso be stted s, 49
~ A ~ A (iv) Exercise. Give the opertor ~ D ( D ~, ) D ~ = = d, wht is the opertor D ~? d x * x) [ d ψ * x) x d x Itegrtig by prts = - d φ x) x = d x d ]x d x [ - D ~ x) ]* x This implies, = ( - D ~, ) D ~ = - D ~ (v) Expressio (iv) bove, ~ ~ A A, is very ppelig to be re-expressed s A ~ ( A ~ ) A ~ A ~ (vi) ~ which prompts the followig iterprettio, A ~ A ~ = A ~ A ~ A ~ A ~ A ~ ssigs to the br the br A ~ 50
Similrly, ~ A ~ A ~ = A ~ ~ ( A ) ~ A A ~ = A ~ From the expressio A ~ = A ~, we obti, ( A ~ ) = A ~ Usig the defiitio (88) = A ~ = ( A ~ ) Tht is, A ~ = ( A ~ ) = ( A ~ ) = A ~ (vii) We hve rrived to the result tht it is uecessry to put the prethesis. We emphsize lso the results show bove i the shded blocks: A ~ = A ~ A ~ = A ~ (viii) A ~ = A ~ Mtrix Represettio of the doit opertor I the expressio (88) bove, if we tke s the bse-stte, d s the bse stte m, we obti, (, ~ m ) = ( m, ~ ) * Hece their mtrix represettio re relted through, [ ~ m = [ ~ m * (89) 5
(Notice the order of the idexes is reversed) ~ is lso clled the Hermiti doit opertor of ~. Properties: ~ ~ ( A B) = B ~ A ~ (90) Exmple: Wht is the Hermiti cougte opertor of the positio opertor X ~? ( X ~, ) (, X ~ ) = * x) [ X ~ ]x = * x) x x = x * x) x = [x x) ] * x = [ X ~ x) ] * x ( X ~, ) = ( X ~ ) Sice this expressio is vlid for y rbitrry sttes d, the X ~ = X ~ Tht is, the Hermiti cougte of the positio opertor is itself. (9) Exmple: Give the opertor ~ D D ~ =? ( D ~, ) D ~ = = d, wht is the Hermiti cougte opertor? d x * x) [ d ψ * x) x d x Itegrtig by prts = - = ( - D ~, ) d φ x) x = d x d ]x d x [ - D ~ x) ]* x 5
Tht is, D ~ = - D ~ (9) 0.5.C Hermiti or self-doit opertors My importt opertors of qutum mechics hve the specil property tht whe you tke the Hermiti doit you get bck the sme opertor. ~ = ~. (93) Such opertors re clled the self-doit or Hermiti opertors. Exmple: The positio opertor X ~ is self doit opertor becuse the exmple i the previous sectio (see expressio 9). Exmple: Let s see if the lier mometum opertor P ~ is self doit ( P ~, ) (,P ~ ) = * x) [ i = i * x) d ]x d x d ψ x d x Itegrtig by prts = - i d φ x) x d x X ~ = X ~, s show i Tht is, = [ i d φ d x = ( P ~, ) x) ]* x P ~ = P ~ (94) Properties of Hermiti (or self-doit) opertors Opertors ssocited to me vlues re Hermiti (or self-doit) I sectio 0.4 bove we defied qutum mechics opertors ssocited to clssicl observble qutity. The defiitio ivolved the clcultio of me vlues of observbles. From the fct tht me vlues re rel, we c drw some coclusios cocerig the properties of those opertors. f v = F ~ 53
Tht is, Sice this qutity is rel, it will be equl t its complex cougte = F ~ * = F ~ F ~ = F ~ Usig defiitio (87) F ~ F ~, F ~ = F ~ which implies, F ~ = F ~ Opertors correspodig to observbles (95) (i.e. opertors obtied through the requiremet f = F ~ ) v must be hermitis (self-doit). The eigevlues of Hermiti (self-doit) opertor re rel. Let be d eigevlue of F ~ d the correspodig eigevector Sice F ~ is Hermiti (self-doit), F ~ ) = F ~, I prticulr, for =, Usig (96) F ~ = (96) F ~ = F ~, for y stte we will hve, F ~ = F ~ which implies = * = = ( for eigevlues of Hermiti opertors) (97) Eigefuctios of Hermiti opertor correspodig to differet eigevlues re orthogol Let, 54
F ~ = where k F ~ Sice k d F ~ k = k = k, d F ~ k = k F ~ = F ~, d s well s k re rel, we obti, F ~ k This implies, = k k k d F ~ k = k k k = k k, ( - k ) k = 0 Thus, k implies k = 0 ( for Hermiti opertors) (98) 0.5.D Observble Opertors Whe workig i spce of fiite dimesio, it c be demostrted tht it is lwys possible to form bsis with the eige-vectors of Hermiti opertor. But, whe the spce is ifiite dimesiol, this is ot ecessrily the cse. Tht is the reso why it is useful to itroduce the cocept of observble opertor. A Hermiti (or self-doit) opertor ˆ is observble (99) opertor if its orthoorml eige-sttes form bsis... 0.5.E Opertors o ssocited to me vlues I the previous sectio we ddressed opertors ssocited to physicl qutities f(x) d g(p) tht depeded respectively o positio-x oly, or mometum-p oly, which tur out to be hermiti (self-doit) opertors. However, for qutities tht deped o the product of x d p, this my ot be the cse. Exmple : Clculte *( x ) [ X ~ P ~ ] ( x ) Nively, let s cll tht qutity xp v (i. e. ssumig tht the clcultio of the itegrl will give s positive umber d thus reflectig qutity ssocited to clssicl mesuremet of xp. we will see below tht this ssumptio is icorrect; still the clcultio to be performed below is correct). 55
xp v *( x ) [ X ~ P ~ ] ( x ) *(x) [ x d (x i ) ] Rerrgig the order of the term, x *(x) [ d (x i ) ] - - dv Itegrtig by prts d [x *(x) ] } [ ) i (x ] [ *(x) ] } [ ) i (x ] - [x d *(x) ] } [ ) i (x ] i - [x d *(x)] [ ) i (x ] * i + [x i d *(x) ] (x) i + [x i d (x)] * (x) i + [ X ~ P ~ ] * (x) (x) 56
xp v i + i + [ * ( x ) ~ ~ [ X P ] (x) ] * * xp v (00) This result idictes tht xp v is ot rel qutity. If we write (00) more explicitly, we obti, *(x) [ ~ P ~ X ] ( x ) = i + [ Or, equivletly, ~ ~ [ X P ] (x) ] * * ( x ) X ~ P ~ = i + X ~ P ~ ] * X ~ P ~ = i + X ~ P ~ (0) ~ ~ ( X P ) = i + X ~ P ~ (0) Expressio (0) shows more explicitly tht the opertor X ~ P ~ is ot hermiti; tht is, ~ ~ ~ ~ X P X P. Followig similr procedure, oe c obti the followig result, P ~ X ~ ~ ~ ~ = + X P (03) i where ~ ~ is the idetity opertor; 0.6 The commuttor I geerl, two opertor do ot commute; tht is, of X ~ d P ~, for exmple). A ~ B ~ is differet th B ~ A ~ (tht is the cse The commuttor betwee two opertors is defied s, ~ ~ ~ ~ ~ [ A, B ~ ] AB B A (06) 0.6.A Expressio for the geerlized ucertity priciple For two give observble opertors A ~ d B ~, let s defie the correspodig stdrd devitio (the sttistics is tke from esemble chrcterized by the wvefuctio, 57
A ~ ~ ( A A ) (07) B ~ ~ ( B B ) To simplify the ottio, let s work with the Hermiti opertors ~ d b ~ defied s, ~ ~ A A ~ d b ~ B ~ B ~ (08) Notice, ~ ~ ~ [ ~, b] [ A, B ] (09) A d B c the be expressed s, σ ψ ~ ψ ~ ψ ~ ψ (0) A σ B ~ bψ ~ bψ ψ ~ b ψ Cosider the ot Hermiti opertor C ~, ~ ~ C ~ i λ b where is rel costt () Notice: ~ C ~ ~ i λ b, d ~ ~ C C ~ C ~ C 0 ψ ~ ~ ~ ~ ( i λ b )( i λ b ) ψ 0 ( ~ ~ b ~ i [,b ~ ] ) 0 Usig (0), σ A i ~ ~ [ A,B ] 0 () σ B ~ ~ Notice tht the term [ A, B] must be purely imgiry umber. The fuctio f = f () defied s, f () σ A stisfies, ccordig to (), f ( ) 0 ~ ~ i [ A,B ] (3) σ B 58
I dditio f " ( ) A 0. Therefore the vlue of t which f ' ( ) 0 is miimum; such vlue is, mi i σ B ψ ~ ~ [ A,B] ψ The vlue of f t mi is, ~ ~ ~ ~ f ( mi ) A [ A,B ] [ A, B] 4 A 4 σ B B ~ ~ [ A, B ] σ B Accordig to () this vlue must be greter or equl to zero. A 4 B ~ ~ [ A,B ] 0 where A B ~ [ A ~, B] Geerlized ucertity priciple (4) 4 ~ [ A ~, B], ccordig to (3), is purely imgiry umber. 0.6.B Cougte observbles Stdrd devitio of two cougte observbles Two observble opertors A ~ d B ~ re clled cougte observbles if their commuttor is equl to i. [ A ~, B ~ ] i defiitio of cougte observbles (5) The result (4) the gives for this type of pir opertors the followig requiremet, A B ucertity priciple for cougte observbles (6) I prticulr, the result (03) idictes tht X ~ d P ~ costitutes pir of cougte observbles. Hece, x p (7) 0.6.C Properties of observble opertors tht do commute Some of the theorems listed below re vlid eve if the opertors re ot observbles. But we will ssume the ltter, sice the mi obective of this sectio is to show tht it is possible to build bsis out of eigefuctios commo to both observbles. 59
Let s strt with the ssumptio tht for the opertor A ~ ll its eigevlues {,,, } d eigefuctios {,, } re kow. Theorem-: Let A ~ d B ~ two observble opertor such tht [ A ~, B ~ ] 0 If the is eigefuctio of A ~ (8) ~ B is lso eigefuctio of A ~, with the sme eigevlue. Proof: A ~ B ~ A ~ = = B ~ Sice A ~ d B ~ commute ~ A, ~ A, ( ) B ~ A ~ B ~ = Thus, the theorem is prove. is the spce geerted by the eigefuctios of A ~ tht hve eigevlue ( ) (9) is the spce geerted by the eigefuctios of A ~ tht hve eigevlue ~ B ~ B The opertor B ~ ctig o eigefuctio i the spce of A ~ gives stte tht remis i the spce. Two cses rise: i) The eigevlue is o-degeerte. ( i.e. the eigevlue-spce ssocited to is oe dimesiol. ) B ~ is therefore proportiol to ; tht is, B ~ = b (0) 60
We fid tht ii) The eigevlue is Tht is, there exists A ~ = is the eigefuctio commo to both A ~ d B ~. g -fold degeerte. g idepedet eigevectors ssocited to the sme eigevlue. for =,,, g The result (9) idictes tht the followig wvefuctios B ~ for =,,, g re lso eigefuctios of A ~ d with the sme eigevlue. But we cot stte, i geerl, tht the re lso eigefuctios of B ~. All we c sy t the momet is tht they belog to the spce, B ~. Let s work out explicitly the simple cse of for =,,, g () g =. Our tsk is to fid two eigesttes commo to A ~ d B ~. Below we preset two wys to proceed. Method-. The dimesio of the spce is. Sice B ~ is observble, it must hve two orthoorml eigesttes i. Let s cll them b d b, B ~ b = b b d B ~ b = b b () (we cot esure whether b b or b b ). O the other hd, sice b d b belog to, the they re lier combitio of d, b = b b = b + b + b Applyig the opertor A ~ of these two eigesttes, A ~ b = b A ~ + b A ~ = b + b = [ b + b ] 6
Similrly, A ~ b = b (3) A ~ b = b (4) From (), (3), d (4), we relize tht b d b B ~. re commo eigesttes to A ~ d Method-. I the method- bove we hd to ssume tht b d b were kow. Let s cosider tht tht such eigesttes re ot kow t the begiig. Tht is, let s sty restricted to the fct tht oly the eigevlue d the eigefuctios d re kow. The tsk is the how to fid the two eigesttes of B ~ (oce we fid them, we kow, by the procedure of method- bove, tht they will be lso eigesttes of A ~ ). d B ~ Let s express B ~ re kow. d B ~ re kow. d B ~ s lier combitios of B ~ B ~ = = + + d (5) The coefficiets k re kow. So we post ourselves the followig problem: To fid p, q d such tht, b = p + q is eigestte of the opertor B ~ Solutio: Usig (6) i (7), ( p d q re ukow) (6) B ~ b = b ( is ukow) (7) B ~ ( p + q Sice B ~ is lier opertor, ) = ( p + q ) (8) p B ~ ( p + q B ~ = p + q 6
p( + ) + + q ( + ) = p + q ( p + q ) + This implies, + ( p + q ) = p + q p + q = p p + q = q (9) which c be rewritte s, p p q q (30) The coefficiets k re kow This is eigevlue problem. We expect the to obti couple of idepedet solutios, (p q ) with eigevlue d (3) (p q ) with eigevlue Usig these solutios i expressios (6) d (7) bove, gives couple of eigestes for the opertor B ~. b = p + q b = p + q Eigesttes of B ~ (3) It is strightforwrd to verify tht these two sttes re lso eigesttes of A ~. We hve show the procedure to obti eigesttes commo to A ~ d B ~. Tht is, there exist i eigevectors of B ~. Therefore, for the degeerte cse, it is lso possible to fid (33) i eigevectors tht re commo to A ~ d B ~ 63
I geerl, the eigevlues d will be differet; but it my lso occur tht they re equl. Theorem- Similrly, if { u r } is bse composed of sttes tht re mutul eigefuctio of A ~ d the [ A ~, B ~ ] 0 Proof: For y stte = A ~ ~ = A u c = u c ; B ~ A ~ = Tht is, u ~ B u c = u b c ; c ~ B = ~ B u c = u b c ~ ~ ~ AB = A u b c = u b c B ~ A ~ ~ ~ = AB (34) Hece, from (3) d (4), we hve the followig theorem: ~ B, Theorem-3: Two observbles A ~ d B ~ stisfy [ A ~, B ~ ] 0 If d oly if there exists bsis composed of sttes tht re mutul (35) eigefuctio of A ~ ~ d B. 0.7 How to prepre the iitil qutum sttes I the previous sectios we hve itroduced wve fuctios s mthemticl etities tht represet sttes d opertors s represetig the physicl qutities of systems. We hve described how to predict the probble outcomes of mesuremet oce wve fuctio is kow. We hve lso see the ssocitio betwee opertors d the verge vlues of the physicl qutities. Wht hs ot bee ddressed yet is how to prepre qutum stte esembles upo which mesuremet c be mde. We rised this questio i i Sectio 6.4 of Lecture 6, but we postpoed its discussio util we were rmed with better mthemticl tools d expded cocepts (like QM opertors, observbles, d, i prticulr, commuttio properties). It is time the to retke d expli those questios. Let s strt metioig, 64
A system, clssiclly described s oe of degrees of freedom, is completely specified qutum mechiclly by ormlized wve fuctio ( q, q,, q ) which cotis rbitrry fctor of modulus. All possible iformtio bout the system c be derived from this wve fuctio. (36) How to build QM wve fuctio? It turs out, How to use the clssicl mesuremets of the physicl properties of system to build QM wve fuctio? the stte of system, specified by wve fuctio i the Hilbert spce, represets theoreticl bstrctio. Oe cot mesure the stte directly i y wy. Wht oe does do is to mesure certi physicl qutities such s eergies, momet, etc., which Dirc referred to s observbles. From these observtios oe the hs to ifer the stte of the system. We hve itroduced wvefuctios s represetig sttes d opertors s represetig the physicl observbles of the system. We must ow see how these qutities re relted to ctul mesuremets. Give system i defiite stte, described by give wvefutio, how c we predict its physicl properties, which re to be compred with experimets? Give? How to predict eergy lier mometum gulr mometum of the system Coversely, how do we icorporte iformtio bout system, gied s result of mesuremet, ito the wvefuctio? How, for exmple, do we determie the iitil stte 0 system upo which we c predict, usig the Schrodiger, the stte t lter time t? F. Mdl, Qutum Mechics, d Ed. Butter Worth Scietific Publictios. 65
How to ifer the stte of the system?? Hvig mesured Eergies Lier momet Agulr mometum 0.7.A Kowig, wht c we predict bout evetul outcomes from the mesuremet? Cosider F ~ to be observble opertor ssocited to the observble f, which hs complete orthoorml set of eigefuctios {,, } d correspodig eigevlues { f, f, } If system is i eigestte of F ~, the mesurig f gives defiite result f. I geerl, system will be i stte, which is ot eigestte of F ~. But it c be expded i the form = c. Mesurig f o system i stte o loger leds to defiite result; rther there exists probbility distributio to fid yoe of the vlues f, f, P ( f ) = c Postulte (37) The most importt implictio of this postulte is tht, i cotrst to clssicl mechics, system with the mximl specifictio of its stte (mely with give wvefuctio ) still shows dispersio; mesurig f does ot ecessrily leds to oe defiite result. This mes, if we mesured f o lrge umber of systems (the esemble chrcterized by ) the the differet possible results f, f, would occur with frequecy give by (37). 0.7.B After mesuremet, wht c we sy bout the stte of system? We come ow to the coverse questio of how iformtio obtied bout system through mesuremets is icorported i the wvefuctio. Let s ssume the observble f is mesured o system i the (38) stte, d the vlue f ( eigevlue of F ~ ) is observed. If this sttemet is to me ythig, it must imply tht, If oe repets the mesuremet of f sufficietly quickly oe (39) ecessrily gi fids the sme vlue f. [I geerl, this oly holds for sufficietly short itervl of time betwee the two mesuremets (for if the itervl is too log the system will hve chged pprecibly, ccordig to the Schrodiger equtio.] 66
If the secod mesuremet is to hve the defiite result f, the the wvefuctio ' (represetig the system immeditely fter the first mesuremet) must be eigefuctio of F ~ belogig to the eigevlue f. Let s mke some predictios bout ' : Cse: The stte before the first mesuremet is ukow If f is o-degeerte, the the stte fter the mesuremets ' is. If f is -fold degeerte (i.e. r for r =,, hve the sme eigevlue f ), the we oly kow tht ' lies i -dimesiol sub-spce, The mplitudes r r r ' c ( c ukow) (40) r c r r c re i geerl ukow. Cse: The stte before the first mesuremet is kow r We shll postulte tht the wvefuctio ' is give by (40), d tht the mplitudes c re kow d give by, r r c ( c kow) (4) r I other words, the correspodig mplitudes ' re the sme: r r c i the expsios of d Tht is, before the mesuremet, we hve r r k c k c (4) k r c k k k r r c r r r If fter mesuremet o tht mesuremet reders the eigevlue f, the: ' r c r c r r r r r r r r r c (43) r 67
I the prticulr cse tht before the mesuremet we kew tht is eigefuctio belogig to the eigevlue f the, the bove simply mes tht, ' = (44) Tht is, the wvefuctio is uchged by the mesuremet. We hve ust described how to icorporte iformtio bout system, obtied s result of experimets, ito the wvefuctio describig tht system. I geerl, before the mesuremet, there my be vriety of possible results. But oce the experimets hve bee crried out d oe prticulr result obtied, we c discrd most of the wvefuctio (correspodig to ll the results which were ot observed) d reti oly tht prt which o immedite mesuremet would led to the sme result. We re thus prtilly ble to determie the stte of qutum system. This determitio is complete oly if the eigevlue mesured is o-degeerte. I geerl it will be degeerte d further mesuremets re required to determie the stte completely. (The ltter is chieved with simulteous mesuremets of severl observbles, which we ddress i the sectios below). 0.7.C Simulteous mesuremet of observbles So fr, we hve delt with the mesuremets of oly oe observble F ~. This does ot i geerl specify system completely; we must cosider the mesuremets of severl observbles. Accordig to clssicl mechics, the stte of system with degrees of freedom c be determied, t the time t, by mesurig the positio d mometum coordites q i (t), p i (t) ; i=,,,. We c thik of these mesuremets s performed simulteously, or s crried out i very rpid successio so tht the coordites hve ot vried pprecible (o ccouts of their developmet ccordig to the equtios of motio ) durig the time required for ll the mesuremet. But it is iheret i the cocepts of clssicl physics tht, - The mesuremets do ot ffect the vlues of the vribles describig the system (i.e. the mesuremets do ot ffect the stte of the system). - Tht the order of mesurig the coordites is immteril. - By kowig the coordites, ll coceivble observbles c be clculted. I qutum mechics the situtio is etirely differet. - It is o loger possible to specify the simulteous vlues of ll observbles of system. - I geerl, observbles re mutully icomptible. Mesurig oe of them prtly or completely destroys y kowledge bout the others. 68
Cosider two observbles A ~ d B ~ which possess mutul eigefuctio, i.e A ~ = d B ~ = b. If system were i such stte the, ccordig to the postulte (48), successive mesuremets with A ~ d B ~ (performed sufficietly quickly for the time vritio to be uimportt) would give the results d b. However, if A ~ d B ~ hve o commo eigefuctios, successive ltertive mesuremets of A ~ d B ~, which ecessrily leve the system i eigesttes of A ~ d B ~ respectively, destroy the iformtio obtied erlier. The icomptibility of opertors is relted to the ucertity priciple I wht follows, we will cosider first, uder wht coditios oe c scribe simulteous eigevlues to severl observbles 0.7.C Defiitio of comptible (or simulteously mesurble) opertors If A ~ d B ~ re mesured o system (i quick successio) with the results r d b s, d immedite remesuremet of A ~ or B ~ ecessrily reproduces these results, whtever the iitil stte (45) of the system, the we sy tht A ~ d B ~ re comptible or simulteously mesurble. 0.7.C Coditio for observbles A ~ d B ~ to be comptible It follows theorem of fudmetl importce: Proof: A ecessry d sufficiet coditio for A ~ d B ~ to be comptible is for them to possess complete orthoorml set of (46) simulteous eigefuctios. i) Necessry coditio: If A ~ d B ~ re comptible the there exists such set. Let s ssume the system is iitilly i stte. After mesuremet of A ~ with result r, the system is i eigestte Expdig r i terms of the eigefuctios of B ~ gives, r. r = s s c s, (47) where B ~ k = b k k. A mesuremet of B ~ o the system i the stte (47) (i.e. mesurig B ~ immeditely fter we hve mesured A ~ ) with the result b s, leves the system i the eigestte opertor B ~. s of the 69
A secod mesuremet of A ~ will give gi the sme result r (becuse A ~ d B ~ re comptible opertors), which implies tht s is lso eigestte of A ~. Hece, we relize tht i (47), y eigestte of A ~ c be expressed s lier combitio of sttes tht re simulteous eigesttes of A ~ d B ~. Sice y rbitrry stte is lier combitio of eigesttes { r } of A ~, d ech r i this set is lier combitio of sttes s tht re simulteous eigesttes of A ~ d B ~, the y rbitrry stte is lier combitio of sttes tht re simulteous eigesttes of A ~ d B ~. Tht is, if A ~ d B ~ re comptible observbles, such complete set of sttes tht re simulteous eigesttes of A ~ d B ~ does exists. ii) Sufficiet coditio: If there exist complete set of simulteous eigefuctios of A ~ d B ~, the such observbles re comptible. Let { u r } be complete set of simulteous eigefuctios of A ~ d B ~, Let s ssume the system is iitilly i stte. After mesuremet of A ~ with result r, the system is i eigestte A subsequet mesuremet of B ~ will give the result b r d leve the system i the eigestte u. r u r. Repetig the mesuremets with A ~ d B ~ will give the sme results r d b r. Puttig together the results (35) d (46): Theorem: Ay oe of these coditios imply the other oe; () A ~ d B ~ re comptible () A ~ d B ~ possess bsis composed of simulteous eigevlues. (48) (3) A ~ d B ~ commute 0.7.C3 Complete set of commutig opertors The previous sectios serve to idicte tht: - We cot icrese our kowledge bout the stte of system by successive mesuremets of rbitrry observbles. - Rther, hvig mesured A ~, we my mesure secod observble B ~ oly if it commutes with A ~, - d so o. If the mesuremet of A ~ gives the vlue tht is g ) fold degeerte, it mes: r ( r 70
- The wvefuctio lies i the g ( r ) dimesiol spce. r If fter subsequet immedite mesuremet of observble B ~ (tht commutes with the observble A ~ ) the result is b s, the: - The wvefuctio of the system lies i subspce r,b whose s dimesiolity g ( r, br ) g ( r ) - r,b comprises ll simulteous eigefuctios of A ~ d B ~ with eigevlues s r d b respectively s I this wy we cotiue mesurig more d more commutig observbles A ~, B ~, C ~, D,. At ech step: - We reduce the dimesiolity of the spce withi which the wvefuctio my lie. - The degeercy of the stte is reduced Filly we hve set of commutig opertors { A ~, B ~,,Q ~ } such tht to y set of eigevlues { r, b s,, q z } there exists exctly oe eigefuctio; tht is g ( r,br,..., qz ) This stte cot be resolved further by more mesuremets. We cll such observtio mximl or complete, d the correspodig set of observbles complete set of commutig observbles. Exmple. For the cse of prticle i cetrl field, H ~, commutig observbles. L ~ d L ~ z costitute complete set of Notice, fter observtio with complete set of commutig observbles { A ~, B ~,,Q ~ } we do ot kow the vlues of ll possible observbles. - We cot scribe defiite vlue to observble W ~ tht does ot commute with ll the observbles i the set { A ~, B ~,,Q ~ } - Oly if W ~ commutes with { A ~, B ~,,Q ~ } c we scribe defiite vlue to it. However, fter observtio with complete set of commutig observbles { A ~, B ~,, Q ~ } we do of course kow the exct wvefuctio of the system. Hece. - We c clculte the sttistics (me vlues, probbility distributios, etc.) of y observble W ~. It should be cler tht there is ot oe uique complete set of commutig observbles. 7
- If we hd tke observble W ~ for the first mesuremet, d W ~ does ot commute with every opertor of the set { A ~, B ~,,Q ~ }, we could still obti differet complete set of commutig observbles. - Exmple. For the cse of prticle i cetrl field, { H ~, commutig observbles. Sice with L ~ x or L ~ z. But { H ~, L ~, observbles. L ~, L ~ z } form complete set of L ~ y commute with L ~ z, we cot determie their vlues simulteously L ~ x } or { H ~, L ~, L ~ y } would be eqully cceptble complete set of commutig --- APPENDIX Altertive procedure: Derivig self-cosistet expressio of QM opertor F ~ If the system is i the stte, = A =, the verge vlue of the qutity f is give by (4), f = v f A = f The requiremet to build the opertor F ~ is, F ~ = f v f A = f (43) We wt to mipulte the expressio o the right side of the equlity such tht it tkes form like the followig, Opertor = ( x) [Opertor ( x). 7
= = = f A * A = f f * We re ot there yet. We eed itegrl to show up i the lst expressio. Oe optio is to express i its itegrl form = F ~ = ( x) ( x) ( x). Thus, oe obtis, f ( x) ( x) Iterchgig the order of the summtio d itegrl, f ( x) The left side bove hs the form F ~ ( x) [ F ~ ( x), ( x) [ F ~ ( x) = ( x) The lst expressio implies: Or, more geerl, [ F ~ ]( x) = [ F ~ = = f ( x) This guy hs to be f ]( x) f [ F ~ ]x f (44) umber stte umber Notice, we could obti more compct expressio for the opertor F ~, if we give to the fctor the coottio of beig the result of opertor ctig o the stte. For tht purpose, we use the ottio, 73
P ~ pro, Proectio opertor (45) to describe opertor, which whe ctig o geerl stte gives the proectio of tht stte log the stte ; tht is, P ~ pro, = (45) Accordigly, (44) c be expressed s, F ~ = [ f ] (46) stte umber opertor where deote the stte where the qutity f hs the vlue f ; This ottio trick llows us to express the opertor i more compct form, F ~ = f (47) eigevlues Proectio opertorp ~ pro, built out of eigesttes ssocited to the physicl qutity f. This is self-cosistet expressio for the qutum opertor F ~ tht will be ssocited with the clssicl qutity f. It is expressio tht is comptible with the requiremet tht F ~ I pssig, otice tht pplyig the opertor F ~ to oe of the eigesttes, expressio (47) gives, F ~ = f (48) f v Appedix- Accordigly, ech stte B ~ c be expded i terms of the bse { } 74
B ~ = g () Notice, this gives the mtrix represettio [ ] of the opertor B ~ i the { } bsis. Sice B ~ is observble, we c clim tht the wvefuctios B ~ re lierly idepedet. Tht is to sy, the mtrix obtied from () should be of rk- g [ Ideed, sice B ~ is observble, we should be ble to express y eigefuctio from i terms of its eigefuctios B ~ u = b u. u, =,, g, where There will be trsformtio of coordites from the { = T u which gives, B ~ g g = T We c thus idetify v k g = T b k B ~ g u = v B ~ u k T b u } to the { u } bse: s elemets of bse. It reltes to { u } through, { v } is bse becuse the trsformtio of coordites mtrix [T][b] stisfies, k det [T][b] = det [T] det [b] 0. (Here [b] stds for the digol mtrix tht represets B ~ i its eigesttes bsis) Hece, the vectors v B ~ i (4) re lierly idepedet ] k k Sice the vectors B ~ i () re lierly idepedet, the oe c implemet process of digoliztio. Tht is, there exist i eigevectors of B ~. Therefore, for the degeerte cse, it is lso possible to fid (3) i eigevectors tht re commo to A ~ d B ~ REFERENCES Mdl, Qutum Mechics, Butterworths Scietific Publictios 75
3 4 5 6 7 8 9 0 See Sectio 6-4 i Feym Lectures: Vol III (See Fig 6- i this referece for illustrtio of the Delt Dirc). See Feym Lectures, Vol III, pge 6-9 d 6-0. See Feym Lectures, Vol III, Sectio 6-5. See Feym Lectures, Vol III, pge 6- It is iterestig to observe ew strtegies beyod the esemble pproch: B. L. Altshuler, JETP Lett. 4, 648 (985). P. A. Lee d A. D. Stoe, Phys. Rev. Lett. 55, 6 (985). See lso the itroductio rticle by Igor V. Lerer, So Smll Yey Still Git, Sciece 36, 63 (007). B. H. Brsde d C. J. Jochi, Qutum Mechics, d Editio, pge 56. Eisberg, Resick, Qutum Physics, pge 80, d Editio Wiley (985). D. Griffiths, Itroductio to Qutum Mechics, Secod Editio,; Perso Pretice Hll (005), pge 5. L. D. Ldu d E.M. Lifshitz, Qutum Mechics, No-Reltivistic Theory, Pergmo Press, 965; Chpter, Sectio 3. He itroduces the cocept of Qutum opertors i the cotext of fidig me vlues of observbles. Feym Lectures, Vol III, Sectio 0-4 d Sectio 0-5. Feym lso itroduces the cocept of Qutum opertors i the cotext of fidig me vlues of observbles. L. D. Ldu d E. M. Lifshitz, Qutum Mechics, No-Reltivistic Theory, Pergmo Press, 965; Chpter, Sectio 3. This sectio follows closely the book by Mdl, Qutum mechics 76