CHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS

Transcription

1 Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER-0 WAVEFUNCTIONS, OBSERVABLES OPERATORS 0. Represettios i the sptil mometum spces 0..A Represettio of the wvefuctio i the sptil coorites bsis { positio x } 0..A The Delt Dirc 0..A Comptibility betwee the physicl cocept of mplitue probbility the ottio use for the ier prouct. 0..B Represettio of the wvefuctio i the mometum coorites bsis { mometum p } 0..B Represettio of the mometum p stte i spce-coorites bsis { positio x } 0..B Ietifyig the mplitue probbility mometum p s the Fourier trsform of the fuctio ( x) 0. The Schröiger equtio s postulte 0..A The Hmiltoi equtios expresse i the cotiuum sptil coorites. The Schroiger Equtio. 0..B Iterprettio of the wvefuctio Eistei s view o the grulrity ture of the electromgetic ritio. Mx Bor s probbilistic iterprettio of the wvefuctio. Determiistic evolutio of the wvefuctio Esemble 0..C Normliztio coitio 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 clculte i esemble of ieticlly prepre systems

2 0.4 Opertors ssocite to observbles 0.4.A Observbles, eigevlues eigesttes 0.4.B Defiitio of the qutum mechics opertor F ~ to be ssocite 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 coorites 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 coorite bsis 0.5 Properties of Opertors 0.5.A Hermiti cojugte ( or joit ) opertors 0.5.B Hermiti or self-joit opertors Properties of Hermiti (or self-joit) opertors: - Opertors ssocite to me vlues re Hermiti (or self-joit) - Eigevlues re rel - Eigevectors with ifferet eigevlues re orthogol 0.5.C Observble Opertors 0.5.D Opertors o ssocite to me vlues 0.6 The commuttor 0.6.A Expressio for the geerlize ucertity priciple 0.6.B Cojugte observbles Str evitio of two cojugte observbles 0.6.C Properties of opertors tht o commute 0.7 How to prepre the iitil qutum sttes 0.7.A Kowig wht c we preict 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 Refereces: 0.7.C Coitio for observbles A ~ B ~ to be comptible 0.7.C3 Complete set of commutig opertors

3 Feym Lectures Vol. III; Chpter 6, 0 Clue Cohe-Touji, B. Diu, F. Lloe, Qutum Mechics, Wiley. "Itrouctio to Qutum Mechics" by Dvi Griffiths; Chpter 3. B. H. Brse & C. J. Jochi, Qutum Mechics, Pretice Hll, E

4 CHAPTER-0 WAVEFUNCTIONS, OBSERVABLES OPERATORS Qutum theory is bse o two mthemticl items: wvefuctios opertors. The stte of system is represete by wvefuctio. A exct kowlege of the wvefuctio is the mximum iformtio oe c hve of the system: ll possible iformtio bout the system c be clculte from this wvefuctio. Qutities such s positio, mometum, or eergy, which oe mesures experimetlly, re clle observbles. I clssicl physics, observbles re represete by oriry vribles. I qutum mechics observbles re represete by opertors; i.e. by qutities tht upo opertio o wvefuctio givig ew wvefuctio. This chpter presets three mi sectios: The first iclues escriptio of the sptil-coorites bsis the mometumcoorites bsis, which re typiclly use to represet qutum stte. The ext escribes how to buil the qutum mechics opertor correspoig to give observble. The fil sectio resses how to buil buil (mthemticl) qutum stte from give set of experimetl results. The key mthemticl cocept use here is the complete set of commutig opertors 0. Represettio of the wvefuctios i the sptil mometum spces A rbitrry stte c be expe i terms of bse sttes tht coveietly fit the prticulr problem uer stuy. For geerl escriptios, two bses re frequetly use: the sptil coorite bsis the lier mometum bsis. These two bsis re resse i this sectio. 0..A Represettio of the wvefuctio i the sptil coorites bsis { x, x } Chpter 9 helpe to provie some clues o the proper iterprettio of the wvefuctio (the solutios of the Schroiger equtio.) This cme through the lysis of the prticulr cse of electro movig cross iscrete lttice: the wvefuctio is picture s wve of mplitue probbilities (complex umbers whose mgitue is iterprete s probbilities). Notice, however, tht whe tkig the limitig cse of the lttice spcig teig to zero, oe es up with situtio i which the electro is propgtig through cotiuum lie spce. Thus, this limitig cse tkes us to the stuy of prticle movig i cotiuum spce. I logy to the iscrete lttice, where the loctio of the toms guie the selectio of the stte bsis { }, i the cotiuum spce we cosier the followig cotiuum set, { x, x } Cotiuum sptil-coorites bse () 4

5 I logy to the iscrete cse, x sts for stte i which prticle is locte rou the coorite x. x For every vlue x log the lie oe coceives correspoig stte. If oe iclues ll the poits o the lie, complete bsis set results s iicte i (), which will be use to escribe geerl qutum stte, hece, to escribe the oe-imesio motio of prticle. A give stte specifies the prticulr wy i which the mplitue probbility of prticle is istribute log lie. Oe wy of specifyig this stte is by specifyig ll the mplitueprobbilities tht the prticle will be fou t ech bse stte x; we write ech of these mplitues s x. We must give ifiite set of mplitues, oe for ech vlue of x. Thus, = About the x ottio x x Represettio of the wvefuctio () i the sptil coorites bsis We coul use ltertive ottios, like, for exmple, = x A(x) s to mke this expsio to resemble the expsio of vector i terms of bsevectors x with the correspoig coefficiets A(x) plyig the role of weightig-fctor coefficiets. Iste of simply A(x), we coul eve use A Ψ (x ) s to emphsize tht those coefficiets correspo to the stte. = x A Ψ (x ) ; the meig of the A Ψ (x ) ottio becomes clerer but, t the sme time, it my result bit too cumbersome. Hece, the ottio x is frequetly preferre. The followig is other commo ottio for the mplitue probbility, (x) x Amplitue probbility tht the prticle iitilly i the stte be fou (immeitely fter mesuremet) t the stte x. (3) 5

6 Thus, we will use iistictly the followig ottio = x x.= Represettio of the x x. wve-fuctio i the (4) sptil coorites bsis umber Number Cutio: x oes ot me [x]* (x). Workig with the sptil coorites bse { x } my costitute the oly occsio i which the ottio x gets cofuse with the efiitio of the sclr prouct. Note: I Chpter 8 we use the ottio (t) = A (t), where the mplitues A (t) were etermie by the Hmiltoi equtios {, =,, } ws iscrete bse. I this chpter, iste of iscrete bse we re usig cotiuum bse { x, x } ; ccorigly the mplitues re beig expresse s x, or by x. The Delt Dirc A subprouct 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 illustrte for the cse of the spce-coorites bsis. x x = = x = x [ x ] x [ x ] x x [ x ] x = x x [ x ] ( x ) ( x ) Or, equivletly x = x x [ x ] 6

7 f x O the other h, it is ccepte tht for rbitrry fuctio f, the elt fuctio is efie s ( x - x ) f x The lst two expressios re cosistet if, x x = ( x x ). Thus, we hve the followig result: Cse of iscrete sttes Delt Kroeecker Cse of cotiuum sttes Delt Dirc (x -x ) (5) m = m x x = ( x - x ) Comptibility betwee the physicl cocept of mplitue probbility the ottio use for the ier prouct * x) x We kow tht give stte, the mplitue probbilitiesx tht pper i the expsio = x x re etermie by the Hmiltoi equtios. Suppose we hve prticle i the stte we wt to kow the mplitue probbility (fter give mesuremet process) to fi the prticle t the stte. Tht is, we wt to evlute. There re my pth wy for the stte to trsit to stte. It coul o it by pssig first through y of the bse-sttes x. Sice ech stte x geertes pth, ll these pths hve the sme iitil fil stte, the, ccorig to the rules estblishe i Chpter 7, the totl mplitue probbility will be give by, 7

8 x = ll x x x First, the sum over smll regio of with woul be xx multiplie by. The, sice x vries from to the expressio bove c be writte s, = x x (6) We c here itrouce the ottio use bove, mely x = xalso, sice x = x * oe obtis, = * x) x (7) The mplitue probbility is equl to the ier prouct betwee the fuctios 0..B Represettio of the wvefuctio i the mometum coorites bsis { p } I the previous sectio, rbitrry stte ws expresse i terms of the compoets of the spce-coorites bsis { x, x }. Here we preset ltertive bsis wy to express the stte, the cotiuum bse of mometum coorites. We will relize tht we re lrey fmilir with the sttes comprisig this ew bsis. Recll tht i Chpter 5 we itrouce the Fourier trsform F= F(k) of fuctio (x). It ivolve the itrouctio of the complex hrmoic fuctios e k, where e k (x) = ( x )= e ikx for k F(k) e ikx k Fourier trsform Bse-fuctio e k evlute t x where the weight coefficiets F(k), referre to s the Fourier trsform of the fuctio, re give by, 8

9 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 p (8) where (p) = e Fourier trsform of -i(p/ )x ( x' ) ' (9) Expressios (8) (9) hve clerer physics iterprettio, p ( x ) = π e i ( p / ) x Accorig to e Broglie, p represets ple wve of efiite lier mometum p. (0) We c re-write expressio (8) i terms of fuctios, ( x )= ( p) Fourier trsform of ( x) e i ( p / ) x p p = ( p) p p Fuctio Sum of fuctios () Expressios (8) () re equivlet. Plcig expressio () i brcket ottio First, usig s hit expressio (4), = x x, otice tht expressio (0), p (x) = e i ( p / ) x, suggests to efie ket mometum stte p s follows, π 9

10 p = x p ( x ) x e i( p / ) x π Represettio of the mometum stte p i the spce-coorite bsis { x } () x p p ( x ) = π e i ( p / )x Amplitue probbility tht prticle, i stte of mometum p, be fou t the coorite x. (3) This is the e Broglie hypothesis i the lguge of mplitue probbilities. I expressio () we c mke the the followig ssocitio, Hece, = (p) p p p = (p) p p (4) Fourier trsform ( x ) Expressio (4) gives s lier combitio of the mometum sttes p efie i () bove. Expressios (8), () (4) re equivlet. The sttes escribe i () costitute bse (the justifictio comes from the Fourier trsform theory), { p, - < p < } Exercise: Prove tht p p = ( p p ) cotiuum bse of mometum-coorites (5) Now we formlly c justify tht i expressio (4): p = (p) I effect, multiplyig (4) with br p, oe obtis, 0

11 p = p (p ) p p = (p ) p p p usig p p = ( p p ) p = (p) (6) I summry: A rbitrry wvefuctio c be expresse s lier combitio of mometum-coorites p, = p (p) p = p p p where (p) = -i(p/ )x e ( x' ) ' p =(p) is the Fourier trsform of = (x) (7) p (p) Amplitue probbility tht prticle i the stte c be fou, upo mkig mesuremet, i the stte p. p p (p) p Probbility tht prticle i the stte be fou with mometum withi the itervl ( p, p+ p). < p > = p (p) p = p p p Averge lier mometum of system i the stte Expressios (4) (7) summrizes the objective of this sectio, showig expsio of the stte i two ifferet bsis, the cotiuum sptil-coorites bse { x, x }; the lier-mometum bse { p, - < p < }.

12 0. The Schröiger Equtio s postulte 0..A The Hmiltoi equtios expresse i the cotiuum sptil coorites. The Schröiger Equtio. 3 I chpter 8 we obtie the geerl Hmiltoi equtios tht escribe the time evolutio of the wvefuctio (t) = A ( t ), A H j ( t) Aj, (8) t j i Chpter 9 escribe the prticulr cse of electro movig i lttice (the ltter costitute by toms seprte istce 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 cosier ow rbitrry geerl cse. eff How oes the Hmiltoi equtios (8) look like whe expresse i the i the cotiuum spce coorites { x, x }? Let s fi out such geerl forml expressio (oe tht is more geerl th expressio (9) ). First otice tht the mplitues system, = j j A j A j i (8) ccout for the stte escribig the qutum Sice A j c lso be writte s A j = j, Eq. (5) c lso be expresse s, i = H j t j j Let s lso recll, from Chpter 7, tht the coefficiets H j opertor H ~ (specific to the problem beig solve.) Tht is, epe o t. i = H ~ j j t j I the cotiuum spce coorites we shoul expect, re obtie from the Hmiltoi H j H ~ j. I geerl, H ~

13 i x = t x H ~ x x x (x) ( x ) i ( x ) = t H( x, x )( x ) x (0) where we hve efie H( x, x ) x H ~ x Quotig Feym, 4 Accorig to (0), the rte of chge of t x woul epe o the vlue of t ll other poits x. x H ~ x is the mplitue per uit time tht the electro will jump from x to x. It turs out i ture, however, tht this mplitue 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-h sie of Eq. (0) c be expresse completely i terms of the sptil erivtives of, ll evlute t x. The correct lw of physics is H( x, x ) ( x ) x = ( x ) + V ( x) ( x ) Postulte () m Where i we get tht from? Nowhere. It cme out of the mi of Schroiger, ivete i his struggle to fi uerstig of the experimetl observtio of the experimetl worl. Usig () i (0) oe obtis, i t m V ( x, t) x Schroiger Equtio This equtio mrke historic momet costitutig the birth of the qutum mechicl escriptio of mtter. The gret historicl momet mrkig the birth of the qutum mechicl escriptio of mtter occurre whe Schroiger first wrote ow his equtio i 96. For my yers the iterl tomic structure of the mtter h bee gret mystery. No oe h bee ble to uerst wht hel mtter together, why there ws chemicl biig, especilly how it coul be tht toms coul be stble. (Although Bohr h () 3

14 bee ble to give escriptio of the iterl motio of electro i hyroge tom which seeme to expli the observe spectrum of light emitte by this tom, the reso tht electros move this wy remie mystery.) Schroiger s iscovery of the proper equtios of motio for electros o tomic scle provie theory from which tomic pheome coul be clculte qutittively, ccurtely i etil. Feym s Lectures, Vol III, pge 6-3. Although the result () is ki of postulte, we o hve some clues bout how to iterpret it, bse o the prticulr cse of the ymics of electro i crystl lttice, stuie i Chpter B Iterprettio of the Wvefuctio Eistei s view o the grulrity ture of the electromgetic ritio I Chpter 5, hrmoic fuctio ws use to escribe the motio of free prticle i logy to the existet formlism to escribe 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 perpeiculr to the irectio of ritio propgtio) is proportiol to ε ( x, t). Eistei (i the cotext of tryig to expli the results from the photoelectric effect) itrouce the grulrity iterprettio of the electromgetic wves (lter clle photos), boig the more clssicl cotiuum iterprettio. I Eistei s view, the itesity is iterprete s sttisticl vrible I c o ε N h. Here N costitutes the verge umber of photos per seco crossig uit re perpeiculr to the irectio of ritio propgtio; ε ~ N Averge vlues re use 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 rou verge vlue N. Mx Bor s Probbilistic Iterprettio of the wvefuctio I logy to Eistei s view of ritio, Mx Bor propose 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 fiig the prticle rou give plce x t give time t. This iterprettio ws itrouce yers fter Schroiger (96) h evelope forml qutum mechics escriptio. More specificlly, 4

15 ( x, y,z, t) plys the role of probbility esity. Pictorilly, the prticle is more likely to be t loctios where the wvefutio hs pprecible vlue. Determiistic evolutio of the wvefuctio The preictios of qutum mechics re sttisticl. I orer to kow the stte of motio of prticle, we must mke mesuremet But mesuremet ecessrily isturbs the system i wy tht cot be completely etermie. However, otice tht, beig the solutio of ifferetil equtio (the Schroiger equtio), vries with time i wy tht is completely etermiistic. Tht is, if were kow t t=0, the Schroiger equtio etermies precisely its form t y future time. Tht is, QM mkes etermiistic preictio of mplitue probbility wve. However, the ltter oes ot covey to etermiistic outcomes. There is oe further poit to cosier. How to etermie the wvefuctio t t=0? How o the experimetl mesuremets le to the recostructio of the wvefuctio? Or, how to prepre system i efiitely uique stte? If we coul ot recostruct wvefuctio, wht woul be the beefit of hvig theoreticl formultio tht escribes etermiistic evolutio of somethig we o ot kow? As it turs out, espite 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 etermiistic formultio wht we ee is to prepre system (or my systems) i efiite stte; for the theory coul the be use to mke preictios bout the evolutio of tht prticulr stte. We will ress this issue i the ext chpters, fter the itrouctio of observbles eigesttes. We will see tht fiig eigesttes commo to ifferet observbles rrows the selectio pool of sttes i which the system c be fou. This proceure les to the cocept of esemble of system costitute by (i this wy) eqully prepre systems, which costitutes the lbortory i which the QM cocept re evelop. (A the e, systems cot be etermie with bsolute certity simply becuse set of mesuremets t t=0 t most my le to the etermitio of but ot to uiquely efie ). Let s expli the sttisticl iterprettio bit further i the cotext of esemble of ieticlly prepre systems. Esemble 5 Imgie very lrge umber of ieticlly prepre iepeet system (ssume to be ll of them i the sme stte), ech of them cosistig of sigle prticle movig uer the ifluece of give exterl force. 5

16 All these systems re ieticlly prepre. The whole esemble is ssume to be escribe by complex-vrible sigle wvefuctio ( x, y, z, t), which cotis ll the iformtio tht c be obtie bout them. escribes the whole esemble... Esemble is use to mke probbilistic preictio o wht my hppe i prticulr member of the esemble. N It is postulte tht: If mesuremet of the prticle s positio re me o ech of the N member of the esemble, the frctio of times the prticle will be fou withi the volume elemet 3 r = y z rou the positio r ( x, y, z, t) t the time t is give by (3) * ( x, y, z, t) ( x, y, z, t) 3 r where * sts for the complex cojugte umber. Notice tht this is othig but the lguge of probbility; i this cse, positio probbility esity P. P * ( x, y, z,t) ( x, y, z,t) ( x, y, z,t) ( x, y, z, t Cutio: For coveiece, we shll ofte spek of the wvefuctio of prticulr system, BUT it must lwys be uerstoo tht this is shorth for the wvefuctio ssocite with esemble of ieticl ieticlly prepre systems, s require by the sttisticl ture of the theory. 6 ) 0..C Normliztio coitio for the wvefuctio 6

17 The probbilistic iterprettio of the wvefuctio implies, therefore, the followig requiremet: * 3 ( x, y, z,t) ( x, y, z,t) r (4) All spce becuse give prticle, the likelihoo to fi it ywhere shoul be oe. Iheret to this requiremet is tht, ( r, t) 0 (5) r Notice tht if is solutio of the Schroiger 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 coitio (4). This process is clle ormlizig the wvefuctio. I geerl, there will be solutios to the Schroiger equtio () whose solutio te to ifiite vlue. This mes they re o-ormlizble therefore c ot represet prticle probbility esity. Such fuctios must be rejecte o the grous of Bor s probbility iterprettio. Qutum mechics sttes re represete by squre- itegrble fuctios tht stisfy the Schroiger equtio. The prticulr subset of squre itegrble fuctios form vector spce clle the Hilbert spce. QUESTION: Suppose tht is ormlize t t 0. As the time evolves, will chge. How o we kow if it will remi ormlize? Here we show tht the Schroiger equtio hs the remrkble property tht it utomticlly preserves the ormliztio of the wvefuctio: If stisfies the Schroiger equtio The if the potetil is rel t Proof: Let s strt with t ( x, t) = 0 (6) ( x,t) ( x,t) (7) t We provie below grphic justifictio of (7). 7

18 8 fx, t +) - fx, t +) t fx, t ) t [fx, t + ) - fx, t) ] = fx, t ) x x O the other h, t t t t ) (8) We use the Schroiger equtio (9) to clculte the time erivtives, ) ( x,t V x m i t ) ( x,t V i x m i ) ( x,t V i x m i t Tkig the complex cojugte, ssumig tht the potetil is rel, ) V( i m i x,t x t Aig the lst two expressios, m i x x t t x x x m i (9) Replcig (9) i (8) we obti, t x x x m i Accorigly, x,t t x,t t ) ( ) ( x x x m i

19 i m x x The expressio o the right is zero becuse x 0 x 0..D The Philosophy of Qutum Theory There hs bee cotroversy over the Qutum Theory s philosophic foutios. 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 Go oes ot ply ice with the uiverse, refers to the bomet of strict cuslity iiviul evets by qutum theory. Heiseberg coutercts rguig: We hve ot ssume tht the qutum theory (s oppose to the clssicl theory) is sttisticl theory, i the sese tht oly sttisticl coclusios c be rw from exct t. I the formultio of the cusl lw, mely, if we kow the preset exctly, we c preict 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 etils. Louis e Broglie, o the other h, rgues tht tht limite kowlege of the preset my be rther limittio of the curret mesuremet methos beig use. He recogizes tht ) it is certi tht the methos of mesuremet o ot llow us to etermie simulteously ll the mgitue which woul be ecessry to obti picture of the clssicl type, tht b) perturbtios itrouce by the mesuremet, which re impossible to elimite, prevet us i geerl from preictig precisely the results which it will prouce llow oly sttisticl preictios. The costructio of purely probbilistic formule ws thus completely justifie. But, the ssertio tht i) The ucerti icomplete chrcter of the kowlege tht experimet t its preset stge gives us bout wht relly hppes i microphysics, is the result of ii) rel ietermicy of the physicl sttes of their evolutio, costitutes extrpoltio tht oes ot pper i y wy to be justifie. 9

20 De Broglie cosiers possible tht lookig ito the future we will be ble to iterpret the lws of probbility qutum physics s beig the sttisticl results of the evelopmet of completely etermie vlues of vribles which re t preset hie from us. Louis e Broglie s view give bove highlights the objectio to qutum mechics philosophic ietermiism. Accorig to Eistei: The belief of exterl worl iepeet of the perceivig subject is the bsis of ll turl sciece. Qutum mechics, however, regrs the iterctio betwee object observer s the ultimte relity; rejects s meigless useless the otio tht behi the uiverse of our perceptio there lies hie objective worl rule by cuslity; cofies itself to the escriptio of the reltios mog perceptios 7 Physics hs give up o the problem of tryig preictig exctly wht will hppe i efiite 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 efie by, x - But wht oes this itegrl exctly me? x ( x) (30) It is worth to emphsize first wht type of iterprettio shoul be voie. 8 Expressio (30) oes ot imply tht if you mesure the positio of the prticle over over gi the - x ( x) woul be the verge of the results. I fct, if repete mesuremets were to be me o the sme prticle, the first mesuremet (whose outcome is upreictble) will mke the wvefuctio to collpse to stte of correspoig prticle s positio x (let s sy x 0 ); subsequet mesuremets (if they re performe quickly) will simply repet tht sme result x 0. O the cotrry, x - x ( x) mes the verge obtie from mesuremets performe o my systems, ll i the sme stte. Tht is, 0

21 A esemble of systems is prepre, ech i the sme stte, mesuremet of the positio is performe i ll of them. x is the verge from such mesuremet. Esemble... N escribes the whole esemble is use to mke probbilistic preictio o wht my hppe i prticulr member of the esemble. Fig. 0. Esemble of ieticlly prepre 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. 0.3.B Clcultio of t x m 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. t x t x t x t x x t t x ), ( ), ( t t x For the cse where the potetil is rel, we obtie i expressio (9) tht, x x m i t t x x m i x t x x x x m i x After itegrtig by prts, it gives

22 x t i m x x Itegrtig oe more time by prts (just the seco term o the right sie,) x i t m x (3) We woul be tempte to postulte tht the expecttio (or me) vlue of the lier x mometum is equl to p m. I tht cse, expressio (3) woul give, t x m t = (3) i x i x However, look t the fuctios isie the itegrl. How to uerst tht term like x woul le 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 itrouce). 0.3.C Expecttio (verge) vlues re clculte i esemble of ieticlly prepre systems I geerl, the me-vlue of give physicl property f (mely, eergy, lier mometum, positio, etc.), more geericlly clle observble, is obtie by mkig mesuremet i ech of the N eqully prepre systems of esemble ( ot by vergig repete mesuremets o sigle system.) The whole esemble is ssume to be escribe by complex-vrible sigle wvefuctio ( x, y, z, t). Whe mkig mesuremets o ech of the N ieticlly prepre systems of the esemble (see left-sie of the figure below) let s ssume we get series of results (see rightsie of the figure below) like this: N systems re fou to hve vlue of f equl to f, from which we euce tht the prticulr system collpse to the stte f right fter the mesuremet N systems re fou to hve vlue of f equl to f, from which we euce tht the prticulr system collpse to the stte f right fter the mesuremet etc. Accorigly, the verge vlue of f is clculte s follows,

23 f v N f N f N N Usul proceure 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 fiig the system i the prticulr stte f. QM postultes tht the vlue of the vlue f, f ψ = Hece, expressio (33) c be writte s, f v f ψ shoul be iterprete s the probbility to obti N Qutum Mechics postulte (34) N f f ψ (35)... Esemble... N Before the mesuremets The vlue of f i ech system is ukow. N After the mesuremets The vlue of f hs bee mesure i ech system; fterwrs we procee to clculte the verge vlue < f >. 0.4 Opertors Associte to Observbles Qutities such s positio, mometum, or eergy (which re mesure experimetlly) re clle observbles. I clssicl physics, observbles re represete by oriry vribles (E, p, for exmple). I qutum mechics observbles re represete by opertors (qutities tht operte o fuctio to give ew fuctio.) 3

24 Whe system i stte eters some pprtus, like, for exmple, mgetic fiel i the Ster Gerlch experimet, or mser resot cvity, it my leve i ifferet stte. Tht is, s result of its iterctio with the pprtus, the stte of the system is moifie. Symboliclly, the pprtus c be represete by correspoig opertor F ~ such tht, = F ~ (36) Note: We will istiguish the opertors (from other qutities) by puttig smll ht ~ o top of its correspoig symbol. We show below how to ssocite qutum mechics opertor F ~ to give physicl qutity f. 0.4.A Observbles, eigevlues eige-sttes Let s cosier physicl qutity or observble f ( f coul be the prticle s gulr mometum for exmple) tht chrcterizes the stte of qutum system. I qutum mechics, the ifferet vlues tht give physicl qutity f (observble) c tke, re clle its eigevlues; f, f, f 3, (37) The set of these qutum eigevlues is referre to s the spectrum of eigevlues of the correspoig qutity f. For simplicity, let ssume for the momet tht the spectrum of eigevlues is iscrete. will eote the stte where the qutity f hs the vlue f ; These sttes will be clle eigesttes (38) We will ssume tht these eigestes stisfy, m = ( x) m ( x) = m (39) Let s ssume tht the eigesttes ssocite to the observble f costitute bsis, { ; =,, 3, } Bsis of eigesttes A rbitrry stte c the be represete by the expsio, 4

25 = A =. (40) where A = = ( x) ( x) Sice the stte must be ormlize stte, the A = = Accorig to expressio (35), the me vlue of f, whe the system is i the stte, is give by, f = v f A = f (4) Notice, expressio (4) is still very geerl, sice we o ot kow yet the eigesttes. (We will escribe below how to figure out these sttes). 0.4.B Defiitio of the QM opertor F ~ to be ssocite with the observble physicl qutity f. 9,0 The opertor F ~ to be ssocite with the observble f is such tht, whe ctig o rbitrry stte, stisfies the followig: F ~ Defiitio of the Opertor F ~ f (4) v ssocite to the observble f (the verge vlue o the right sie is clculte over the esemble represete by the stte ) But, how to obti explicit expressio for such opertor F ~? If the observble qutity were, for exmple, the lier mometum or the gulr mometum, how to buil their correspoig qutum mechics opertor? Before builig the opertors explicitly, we first we erive i this sectio geerl selfcosistet expressio tht shows how opertor, ssocite to give observble, shoul look like (see expressio (47) below). Subsequetly, sectios 0.4.C 0.4.D will provie specific proceure o how to costruct the positio 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), 5

26 f = v f A = f The requiremet to buil the opertor F ~ is, F ~ = f = = v f A = f f * f (43) Notice, we c obti more compct expressio for the opertor F ~ if we use the ottio, P ~ proj, Projectio opertor (44) Expressio (44) escribes opertor tht whe ctig o stte gives the projectio of tht stte log the stte ; tht is, P ~ proj, = = (45) Accorigly, (43) c be expresse s, F ~ = [ f ] (46) stte umber opertor where eote the stte where the qutity f hs the vlue f ; This ottio trick llows us to express the opertor i more compct form, 6

27 F ~ = f (47) eigevlues Projectio opertorp ~ proj, built out of eigesttes ssocite to the physicl qutity f. This is self-cosistet expressio for the qutum opertor F ~ tht will be ssocite 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 ~ j = f j (48) Tht is, oce the opertor F ~ j f v (ssocite 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) (48) give just self-cosistet expressio for the opertor F ~. It is expressio tht is comptible with the requiremet tht F ~ f v, but it is efie i terms of eigesttes j tht, for give physics qutity f, we o ot kow yet but woul like to fi out. Accorigly, F ~ is ot completely kow yet. But, prphrsig Lu, Although the opertor F ~ is efie by (48), which itself cotis the eigefuctios, o further coclusios c be rw from the results we hve obtie. However, s we shll see below, the form of the opertors for vrious physicl qutities c be etermie from irect physicl cosiertios, which subsequetly, usig the bove properties of the opertors, will eble us to fi the eigefuctios eigevlues by solvig the equtio F ~ = 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, X ~ x - x ( x), X ~ =? (49) 7

28 which requires, *( x) [ ( X ~ )( x) ] x [ *( x) ] x [ ( x) ] ( X ~ ) ( x ) = x ( x ) (50) Notice, we cot sy X ~ = x; tht woul be icorrect. (For istce, wht vlue of x woul you choose to mke the expressio X ~ = x vli?). More pproprite is first to efie the ietity fuctio I, which stisfies, I(x) = x, The (50) coul the be writte s, ( X ~ ) ( x) = (I ) ( x) becuse tht woul 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 ssocite to x X ~ = I (where I is the ietity fuctio) ( X ~ ) x= x x (5) X ~ = [ * x ] [ X ~ x ] = [ * x ] x [x] x Notice, it is strightforwr to relize tht X is the opertor ssocite to ~ x, ~ 3 X is the opertor ssocite to x 3, etc. X ~ is the opertor ssocite to Averge vlue Qutum mechics of physicl qutity opertor x (53) 8

29 x v X ~ x v ~ X More geerl, if h(x) is polyomil or coverget series, (54) the h( X ~ ) will be the opertor ssocite to h ( x) Averge vlue of physicl qutity v Qutum mechics opertor h (x) h( X ~ ) For exmple, if h(x) were clssicl potetil tht epes oly o positio x, the h( X ~ ) woul be the the correspoig qutum potetil opertor. Tke the cse of the hrmoic oscilltor, where V(x) = (/)k x. The correspoig QM opertor is V ~ ~ =(/)k X 0.4.D Defiitio of the Lier Mometum Opertor Here we costruct the qutum Lier Mometum Opertor, give its represettio i both the mometum-coorites sptil-coorites bsis. This exmple will help to illustrte tht opertors re geerl mthemticl cocepts whose represettio epes o the bse sttes beig use. The tsk of efiig the mometum opertor is fcilitte by the fct tht the eigefuctios eigevlues re lrey kow. Thus Sectios 0.4.D 0.4.D my pper to be trivil; still it helps to illustrte the coectio betwee opertor the me vlue of the correspoig observble. 0.4.D The Lier Mometum Opertor mometum sttes bsis { p } P ~ expresse i the I this cse, the observble f refers to the lier mometum p. Before builig the lier mometum opertor let s summrize wht we kow bout the mometum sttes. Accorig to e Broglie, for give p we ssocite mometum stte p give by expressio () bove, p x p ( x ) = x π e i ( p / ) x (55) Represettio of the mometum stte p i the spce-coorite bsis { x } 9

30 x p p ( x ) = e i ( p / ) x π Sometimes, for coveiece, we will use the ottio mometum p or p iste of p. A rbitrry stte c be expresse s lier combitio of the mometum sttes, = p =(p) = p p p = p (p) p (56) -i(p/ )x e ( x' ) ' where Now we wt to buil the Lier Mometum Opertor tht will be ssocite to the physicl qutity p. Accorig to expressio (47), F ~ = f, but with the summtio extee to itegrl over cotiuum vrible, the mometum opertor shoul hve the form, P ~ = p p p p Lier Mometum Opertor (57) Applyig the mometum opertor P ~ to rbitrry stte gives, P ~ = p p p p, Accorig to (56) p =(p), which gives P ~ = p (p) p p ~ P Ψ is lier combitio of sttes p. (58) Let verify if the opertor P ~ efie i (57) stisfies the requiremet tht P ~ is equl to the verge mometum p (the verge tke over esemble chrcterize by ). Sice = p (p) p, = we hve, p (p ) p (59) Similr to the elt Dirc obtie i expressio (6) bove, it c be emostrte tht, 30

31 p p = ( p p ) (60) From (58) (59), P ~ = [ p (p ) p ] [ p (p) p p ] = p p (p) (p ) p p p = p p (p) (p ) ( p p ) p = p p (p ) (p ) Usig p iste of p = p (p) p Sice =(p) is the Fourier trsform of = ( x ), the term o the right sie 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) p Averge mometum (6) The opertor P ~ efie i (57), therefore, fulfills the geerl requiremet (4) imposse to qutum mechics opertors. Mtrix represettio of the Lier Mometum Opertor P ~ i the mometum coorites 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' 3

32 Elemets of the mtrix represettio of the opertor i the mometum bsis Summry The Mometum Opertor P ~ P ~ = p p p p (64) The remiig clcultios re reltively strightforwr becuse of the fct tht we ssume rbitrry stte c expresse s lier combitio of moemetum sttes = p p p = p (p) p where = (p) is the Fourier trsform of (x) Accorigly, P ~ = P ~ = = p p p p = p (p) p p [ * x ] [P ~ x ] p (p) p p 0.4.D The lier mometum opertor P ~ expresse i the sptil coorites bsis { x, x } Wht bout if the expsio of i the mometum bsis { p, - < p < } is ot kow,, iste, its expsio i the sptil coorite is vilble? 3

33 (i.e. (x) is kow for every vlue of x, but its Fourier trsform of (p), is ot rey vilble). Wht to o to fi 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 oes exist. Applyig (57), P ~ = we hve = P ~ = x P ~ = p p p p, to the stte fuctio, where p p 'p = p p p p = p (p) x p p, p (p ) p p (p) p p [P ~ x= p (p) p x p Usig (53) x p p ( x ) = π e i ( p / ) x [P ~ x= p (p) π e i ( p / ) x p, [P ~ x = (p) p π e i ( p / ) x p, If the Fourier tyrsform (p) of (x) were kow, the P ~ = [ p p p p ] = p p p p = p (p) p p 33

34 = (p) i π e i ( p / ) x p = i (p) π e i (p / ) x p = i (p) p xp, Hece, P ~ = = i i x (65) The Mometum Opertor P ~ (i the sptil-coorites spce) P ~ = i (66) The remiig clcultio is fcilitte if rbitrry stte is expresse s lier combitio of sptil coorites, so the erivtive c be strightforrly evlute. P ~ = which les to i p v = P ~ = i I pssig, otice tht the lst expressio is ieticl to (3) where we clculte the rte of x chge of the verge positio of prticle m. At tht time, the presece of sptilerivtive ppere to mke o sese i beig ivolve i wht coul be iterprete s t i the velocity of prticle. Now we see tht such sptil erivtive ppers becuse we re usig 34

35 the represettio of the mometum opertor i the sptil coorites (tht sptil erivtive oes 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-coorites We lrey kow how to express the opertors P ~, P, P, i the mometum coorite bsis; we just ee to pply 57) repetitive wys. Tht expressio will give us the opertors rey to pply o sttes whose expsio i the p sttes is rey vilble. But wht we wt ow is to hve those opertors i the sptil-coorites; i.e. expresse i wy rey to be pplie whe the sttes re expresse i the sptil-coorites. Here we show systemtic wy to obti such expressios. ~ ~ 3 Costructio of the Lier MometumP ~ i the sptil- coorites bse We lrey kow how to express P ~ i the mometum bsis, which stisfies, P ~ = p p p p. p v = p p p p (67) First, let s work out the fctor p p isie the itegrl of (67). Sice p = (p) is the Fourier trsform of x, we hve p p = p -i(p/ )x e ' ( x' ) = -i(p/ )x p e ( x' ) ' = i [ e -i(p/ )x ] ( x' ) ' Itegrtig by prts 35

36 = p p = p i Replcig (68) i (67), p v = p p i [e -i(p/ )x ] i = p p i ( x') ' (68) p (69) p. p v = i Wht is this? Notice, if we wte to exp the stte i the mometum bse we woul write, i = p p i i This is exctly wht we hve i the expressio bove. Accorigly, = * x,t i (70) x,t ] If we efie the opertor P ~ = The lst expressio becomes, i p v = (7) * x,t P ~ x,t ] (7) 36

37 Tht is, P ~ = i stisfies the requiremet for beig the lier mometum opertor. Costructio of the opertor ssocite to By efiitio: p v p = v p p p p (73) The systemtic proceure cosists i evlutig first the fctor p p isie the itegrl. p p = p -i(p/ )x e ' ( x' ) = p -i(p/ )x e ( x' ) ' Cotiuig with similr proceure s bove (itegrtig by prts), we obti, p p = p Replcig (74) i (73) les to, i (74) p = v p p Fctorig out i p = p p i p The itegrl term is the expsio of the ket i i the mometum bse 37

38 p = v i = * x,t i (75) x,t ] If we efie the opertor ~ P = The lst expressio becomes, i (76) Tht is, ~ P = i ssocite 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 * x, t i x,t ] (78) is the qutum mechics opertor ssocite to p Averge vlue of physicl qutity p v p v p v Qutum mechics opertor i i i 38

39 If g(p, t) is polyomil, or bsolutely coverget series, i the clssicl mometum vrible p, oe obtis, g ( p, t) v = * x, t g(, t ) i x,t ] (79) Averge vlue of physicl qutity v Qutum mechics opertor g i g ( p, t) (, t) A exmple of (79) costitutes the QM opertor ssocite to the kietic eergy. m p v ~ P m = = m i (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 expresse iepeet of prticulr bsis. For exmple, recll tht < x > = X ~ <p> = P ~. We re goig to o somethig similr here for the verge eergy <E>. I Chpter 8, the Hmiltoi Opertor H ~ ws recogize through its mtrix represettio [H], the ltter beig iterprete s eergy mtrix ue to the fct tht, whe workig with sttiory sttes, the compoets of the mtrix were the eergy of the correspoig sttiory sttes. I the lguge of eigevlues use i this chpter we c write, H ~ E = E E (8) where { E, =,, 3, } is the bsis costitute 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, 39

40 E E v E ψ = E * E E = E E E = H ~ E E This fctor c be expresse i terms of the Hmiltoi opertor From (74): H ~ E = E E = H ~ E E = E = H ~ (83) v Thus, we hve fou gi (s we i for the lier mometum the positio opertors) elegt wy to express me vlue (i this cse for the eergy) tht oes ot mke referece to the prticulr bse sttes. Whe explicit ottio of the bsis sttes is require 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 ifferet bse. For geerl bsis { j for j =,,3, } we will hve, E = H ~ v Expressig i the { j } bsis = [ i i* i ] H ~ [ j j j ] ψ = i i H ~ j j (84) i, j 40

41 0.4.E. Represettio of the Hmiltoi Opertor i the sptil coorite bsis Similr to the cse of the lier mometum opertor resse i the previous sectios, the Hmiltoi opertor c opt ifferet shpes epeig o whether we re workig i the sptil-coorites bsis or i the mometum coorite bses. Here we ress the represettio of the Hmiltoi opertor i the sptil-coorites I (83), let s tke the cse of sptil coorite 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 Schroiger estblishe tht (see expressio () bove) - Accorigly E H( x, x ) ( x ) x = ( x ) + V ( x)( x ) v = - m m ( x) * [ ( x ) + V ( x)( x )] If we efie the potetil opertor V ~, ~ V Ψ V Ψ (86) 4

42 E = v - ( x) * [ m + V ~ ] (x) H ~ E = H ~ v ~ Represettio of the Hmiltoi H Opertor i the sptil coorites + V ~ (87) m bsis ~ Notice, ccorig to expressio (80) P m = m, therefore we c write, ~ H ~ P m ~ V Summry Observble Me Opertor Opertor i the sptil vlue coorites represettio positio X ~ X ~ x x = X ~ X ~ = x mometum P ~ i p p = P ~ P ~ = i ~ H m + V ~ Eergy E E = H ~ 4

43 0.5 Properties of Opertors 0.5.A Hermiti cojugte (or joit ) opertors ~ the followig reltioship betwee sclr proucts hols, is clle the Hermiti cojugte opertor of the opertor ~ if for y stte ( ~, ) ( ~ ). (88) Here (, ) iicte sclr prouct betwee. Mtrix Represettio of the joit opertor I the expressio (88) bove, if we tke s the bse-stte, s the bse stte m, we obti (, ~ m ) = ( m, ~ ) * Hece their mtrix represettio re relte through, [ ~ m = [ ~ m * (89) (Notice the orer of the iexes is reverse) ~ is lso clle the Hermiti joit opertor of ~. Properties: ~ ~ ( A B) = B ~ A ~ (90) Exmple: Wht is the Hermiti cojugte 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 vli for y rbitrry sttes, the X ~ = X ~ (9) 43

44 Tht is, the Hermiti cojugte of the positio opertor is itself. Exmple: Give the opertor ~ D D ~ =? ( D ~, ) D ~ = =, wht is the Hermiti cojugte opertor? x * x) [ ψ * x) x x Itegrtig by prts = - φ x) x = x ]x x [ - D ~ x) ]* x Tht is, = ( - D ~, ) D ~ = - D ~ (9) 0.5.B Hermiti or self-joit opertors My importt opertors of qutum mechics hve the specil property tht whe you tke the Hermiti joit you get bck the sme opertor. ~ = ~. (93) Such opertors re clle the self-joit or Hermiti opertors. Exmple: The positio opertor X ~ is self joit opertor becuse the exmple i the previous sectio. Exmple: Let s see if the lier mometum opertor P ~ is self joit ( P ~, ) (,P ~ ) = * x) [ i = i * x) ]x x ψ x x Itegrtig by prts X ~ = X ~, s show i 44

45 = - i φ x) x x Tht is, = [ i φ x = ( P ~, ) x) ]* x P ~ = P ~ (94) Properties of Hermiti (or self-joit) opertors: Opertors ssocite to me vlues re Hermiti (or self-joit) I sectio 0.4 bove we efie qutum mechics opertors ssocite to clssicl observble qutity. The efiitio ivolve the clcultio of me vlues of observbles. From the fct tht me vlues re rel, we c rw some coclusios cocerig the properties of those opertors. Tht is, f v = F ~ Sice this qutity is rel, it will be equl t its complex cojugte = F ~ * = F ~ F ~ = F ~ Usig efiitio (87) F ~ F ~, F ~ = F ~ which implies, F ~ = F ~ Opertors correspoig to observbles (95) (i.e. obtie through the requiremet must be hermitis (self-joit). f = F ~ ) v The eigevlues of Hermiti (self-joit) opertor re rel. Let j be eigevlue of F ~ j the correspoig eigevector F ~ j = j j (96) 45

46 Sice F ~ is Hermiti (self-joit), F ~ = F ~, for y stte we will hve, ( F ~, ) = F ~, I prticulr, for =j, Usig (96) ( F ~ j,j ) = (j, F~ j ) ( j j,j ) = (j, j j ) j (j,j ) = j (j,j ) which implies * j = j ( for eigevlues of Hermiti opertors) (97) Eigefuctios of Hermiti opertor correspoig to ifferet eigevlues re orthogol Let, F ~ j = where j k, F ~ Sice ( k j j F ~ k = k j ) = ( k, j k j ) F ~ k, j = ( k F ~ = F ~, j s well s k re rel, we obti, ( F ~ k, j ) = This implies, k, j ( k, j ) ( F ~ k, j = k ( k, j ( k, j ) = k ( k, j ) ( j - k ) k, j ) = 0 j ) j ) Thus, j k implies k, j = 0 ( for Hermiti opertors) (98) 0.5.C Observble Opertors Whe workig i spce of fiite imesio, it c be emostrte tht it is lwys possible to form bsis with the eige-vectors of Hermiti opertor. But, whe the spce is ifiite imesiol, this is ot ecessrily the cse. This is the reso why it is useful to itrouce the cocept of observble opertor. 46

47 A Hermiti (or self-joit) opertor ˆ is observble (99) opertor if its orthoorml eige-sttes form bsis 0.5.D Opertors o ssocite to me vlues I the previous sectio we resse opertors ssocite to physicl qutities f(x) g(p) tht epee respectively o positio-x oly, or mometum-p oly, which tur out to be hermiti (self-joit) opertors. However, for qutities tht epe o the prouct of x p, this my ot be the cse. Exmple : Clculte xp v xp v *( x ) [ X ~ P ~ ] ( x ) *(x) [ x (x i ) ] Rerrgig the orer of the term, x *(x) [ (x i ) ] - - v Itegrtig by prts [x *(x) ] } [ ) i (x ] [ *(x) ] } [ ) i (x ] - [x *(x) ] } [ ) i (x ] i - [x *(x)] [ ) i (x ] 47

48 * i + [x i *(x) ] (x) i + [x i (x)] * (x) i + [ X ~ P ~ ] * (x) (x) xp v i + i + [ * ( x ) ~ ~ [ X P ] (x) ] * * xp v (00) This result iictes 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 proceure, oe c obti the followig result, P ~ X ~ ~ ~ ~ = + X P (03) i where ~ ~ is the ietity opertor; 48

49 0.6 The commuttor I geerl, two opertor o ot commute; tht is, of X ~ P ~, for exmple). A ~ B ~ is ifferet th B A ~ ~ (tht is the cse The commuttor betwee two opertors is efie s, ~ ~ ~ ~ ~ [ A, B ~ ] AB B A (06) 0.6.A Expressio for the geerlize ucertity priciple For two give observble opertors A ~ B ~, let s efie the correspoig str evitio (the sttistics is tke from esemble chrcterize by the wvefuctio, ~ ~ ( A A ) (07) A B ~ ~ ( B B ) To simplify the ottio, let s work with the Hermiti opertors ~ b ~ efie s, ~ ~ A A ~ b ~ B ~ B ~ (08) Notice, ~ ~ ~ [ ~, b] [ A, B ] (09) A B c the be express s, σ ψ ~ ψ ~ ψ ~ ψ (0) A σ B ~ bψ ~ bψ ψ ~ b ψ Cosier the ot Hermiti opertor C ~, ~ ~ C ~ i λ b where is rel costt () Notice: ~ C ~ ~ i λ b, ~ ~ 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 49

50 ~ ~ Notice tht the term [ A, B] must be purely imgiry umber. The fuctio f = f () efie s, f () σ A stisfies, ccorig to (), f ( ) 0 ~ ~ i [ A,B ] (3) σ B I itio 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 Accorig to () this vlue must be greter or equl to zero. A 4 B ~ ~ [ A,B ] 0 where A B ~ [ A ~, B] Geerlize ucertity priciple (4) 4 ~ [ A ~, B], ccorig to (3), is purely imgiry umber. 0.6.B Cojugte observbles Str evitio of two cojugte observbles Two observble opertors A ~ B ~ re clle cojugte observbles if their commuttor is equl to i. [ A ~, B ~ ] i efiitio of cojugte observbles (5) The result (4) the gives for this type of pir opertors the followig requiremet, 50

51 A B ucertity priciple for cojugte observbles (6) I prticulr, the result (03) iictes tht X ~ P ~ costitutes pir of cojugte observbles. Hece, x p (7) 0.6.C Properties of observble opertors tht o commute Some of the theorems liste below re vli eve if the opertors re ot observbles. But we will ssume the ltter, sice the mi objective of this sectio is to show tht it is possible to buil bsis out of eigefuctios commo to both observbles. Let s strt with the ssumptio tht for the opertor A ~ ll its eigevlues {,,, } eigefuctios {,, } re kow. Theorem-: Let A ~ 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 ~ B ~ commute A ~ ( B ~ Thus, the theorem is prove. ) = ( B ~ ) (9) 5

52 ~ A, ~ A, is the spce geerte by the eigefuctios of A ~ tht hve eigevlue is the spce geerte 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-egeerte. (i.e. the eigevlue-spce ssocite to is oe imesiol.) B ~ is therefore proportiol to B ~ = b ; tht is, (0) We fi tht ii) The eigevlue is Tht is, there exists A ~ = is eigefuctio commo to both A ~ B ~. j g -fol egeerte. g iepeet eigevectors ssocite to the sme eigevlue. j for j =,,, g The result (9) iictes tht the followig wvefuctios B ~ j for j =,,, g re lso eigefuctios of A ~ with the sme eigevlue. But we cot stte, i geerl, tht they re lso eigefuctios of B ~. All we c sy t the momet is tht they belog to the spce, B ~ j j. for j =,,, g () 5