CHAPTER10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS


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1 Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER0 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 spcecoordites 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
2 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 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 selfdoit opertors Properties of Hermiti (or selfdoit) opertors:  Opertors ssocited to me vlues re Hermiti (or selfdoit)  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
3 Refereces: Feym Lectures Vol. III; Chpter 6, 0 Clude CoheToudi, 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
4 CHAPTER0 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 sptilcoordites bsis d the mometumcoordites 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
5 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 sptilcoordites 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 oedimesio 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 weightigfctor 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
6 (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. wvefuctio 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 spcecoordites bsis. = x [ x ] 6
7 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
8 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 bsesttes 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 spcecoordites 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
9 ( x )= F(k) e ikx dk Fourier trsform Bsefuctio 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 rewrite 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
10 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 spcecoordite 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
11 The sttes described i () costitute bse (the ustifictio comes from the Fourier trsform theory), { p,  < p < } cotiuum bse of mometumcoordites (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)
12 Summry bout the mometum coordites: A rbitrry wvefuctio c be expressed s lier combitio of mometumcoordites 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 sptilcoordites bse { x, x }; d the liermometum bse { p,  < p < }.
13 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
14 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
15 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
16 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 righthd 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
17 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 63. 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 crosssectio 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
18 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 wvefuctios. 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
19 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 complexvrible 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
20 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 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 oormlizble 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
21 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
22 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
23 ) 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
24 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
25 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 mevlue 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 complexvrible sigle wvefuctio ( x, y, z, t). Whe mkig mesuremets o ech of the N ideticlly prepred systems of the esemble (see leftside of the figure below) let s ssume we get series of results (see rightside of the figure below) like this: 5
26 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
27 ... 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
28 0.4.A Observbles, eigevlues d eigesttes 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
29 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 selfcosistet 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
30 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 selfcosistet 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
31 Notice (47) is selfcosistet 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 selfcosistet 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
32 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
33 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 ~ 37 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 mometumcoordites d sptilcoordites 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
34 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
35 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
36 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
37 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
38 [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 sptilcoordites 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
39 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 sptilcoordites 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 sptilcoordites; i.e. expressed i wy redy to be pplied whe the sttes re expressed i the sptilcoordites. 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
40 = 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
41 = * 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
42 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
43 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
44 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
45 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 sptilcoordites bsis or i the mometum coordite bses. Here we ddress the represettio of the Hmiltoi opertor i the sptilcoordites 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
46 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
47 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
48 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
49 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
50 ~ 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 reexpressed 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
51 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 bsestte, d s the bse stte m, we obti, (, ~ m ) = ( m, ~ ) * Hece their mtrix represettio re relted through, [ ~ m = [ ~ m * (89) 5
52 (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
53 Tht is, D ~ =  D ~ (9) 0.5.C Hermiti or selfdoit 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 selfdoit 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 selfdoit) opertors Opertors ssocited to me vlues re Hermiti (or selfdoit) 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
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