CHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS

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1 Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER- WAVEFUNCTIONS, OBSERVABLES d OPERATORS. Represettos the sptl d mometum spces..a Represetto of the wvefucto the sptl coordtes bss { posto x }..A The Delt Drc..A Comptblty betwee the physcl cocept of mpltude probblty d the otto used for the er product...b Represetto of the wvefucto the mometum coordtes bss { mometum p }..B Represetto of the mometum p stte spce-coordtes bss { posto x }..B Idetfyg the mpltude probblty mometum p s the Fourer trsform of the fucto ( x)..c Tesor Product of Stte Spces. The Schrödger equto s postulte..a The Hmlto equtos expressed the cotuum sptl coordtes. The Schrodger Equto...B Iterpretto of the wvefucto Este s vew o the grulrty ture of the electromgetc rdto. Mx Bor s probblstc terpretto of the wvefucto. Determstc evoluto of the wvefucto Esemble..C Normlzto codto of the wvefucto Hlbert spce Coservto of probblty..d The Phlosophy of Qutum Theory.3 Expectto vlues.3.a Expectto vlue of prtcle s posto.3.b Expectto vlue of the prtcle s mometum.3.c Expectto (verge) vlues re clculted esemble of detclly prepred systems

2 .4 Opertors ssocted to observbles.4.a Observbles, egevlues d egesttes.4.b Defto of the qutum mechcs opertor F to be ssocted wth the observble physcl qutty f.4.c Defto of the Posto Opertor X.4.D Defto of the Ler Mometum Opertor P.4.D. Represetto of the ler mometum opertor P the mometum bss { mometum p }.4.D. Represetto of the ler mometum opertor P the sptl coordtes bss { posto x }.4.D3 Costructo of the opertors P,.4.E The Hmlto opertor P,.4.E. Evluto of the me eergy terms of the Hmlto opertor.4.e. Represetto of the Hmlto opertor the sptl coordte bss.5 Propertes of Opertors.5.A Correspodece betwee brs d kets.5.b Adjot opertors.5.c Hermt or self-djot opertors Propertes of Hermt (or self-djot) opertors: - Opertors ssocted to me vlues re Hermt (or self-djot) - Egevlues re rel - Egevectors wth dfferet egevlues re orthogol.5.d Observble Opertors.5.E Opertors tht re ot ssocted to me vlues 3 P.6 The commuttor.6.a The Heseberg ucertty relto.6.b Cojugte observbles Stdrd devto of two cojugte observbles.6.c Propertes of observble opertors tht do commute.6.d How to uquely detfy bss of egefuctos? Complete set of commutg observbles.7 Smulteous mesuremet of observbles.7.a Defto of comptble (or smulteously mesurble) opertors.7.b Codto for observbles A d B to be comptble.8 How to prepre the tl qutum sttes.8.a Kowg wht c we predct bout evetul outcomes from mesuremet?

3 .8.B After mesuremet, wht c we sy bout the stte? Refereces: Feym Lectures Vol. III; Chpter 6, Clude Cohe-Toudj, B. Du, F. Lloe, Qutum Mechcs, Wley. "Itroducto to Qutum Mechcs" by Dvd Grffths; Chpter 3. B. H. Brsde & C. J. Joch, Qutum Mechcs, Pretce Hll, d Ed. 3

4 CHAPTER- WAVEFUNCTIONS, OBSERVABLES d OPERATORS Qutum theory s bsed o two mthemtcl tems: wvefuctos d opertors. The stte of system s represeted by wvefucto. A exct kowledge of the wvefucto s the mxmum formto oe c hve of the system: ll possble formto bout the system c be clculted from ths wvefucto. Quttes such s posto, mometum, or eergy, whch oe mesures expermetlly, re clled observbles. I clsscl physcs, observbles re represeted by ordry vrbles. I qutum mechcs observbles re represeted by opertors;.e. by quttes tht upo operto o wvefucto gvg ew wvefucto. Ths chpter presets three m sectos: The frst cludes descrpto of the sptl-coordtes bss d the mometum-coordtes bss, whch re typclly used to represet qutum stte. The ext descrbes how to buld the qutum mechcs opertor correspodg to gve observble. The fl secto ddresses how to buld (mthemtcl) qutum stte from gve set of expermetl results. The key mthemtcl cocept used here s the complete set of commutg opertors. Represetto of the wvefuctos the sptl d mometum spces A rbtrry stte c be expded terms of bse sttes tht coveetly ft the prtculr problem uder study. For geerl descrptos, two bses re frequetly used: the sptl coordte bss d the ler mometum bss. These two bss re ddressed ths secto...a Represetto of the wvefucto the sptl coordtes bss { x, x } Chpter 9 helped to provde some clues o the proper terpretto of the wvefucto (the solutos of the Schrodger equto.) Ths cme through the lyss of the prtculr cse of electro movg cross dscrete lttce: the wvefucto s pctured s wve of mpltude probbltes (complex umbers whose mgtude s terpreted s probbltes). Notce, however, tht whe tkg the lmtg cse of the lttce spcg tedg to zero, oe 4

5 eds up wth stuto whch the electro s propgtg through cotuum le spce. Thus, ths lmtg cse tkes us to the study of prtcle movg cotuum spce. I logy to the dscrete lttce, where the locto of the toms guded the selecto of the stte bss { }, the cotuum spce we cosder the followg cotuum set, { x, x } Cotuum sptl-coordtes bse () I logy to the dscrete cse, x stds for stte whch prtcle s locted roud the coordte x. x For every vlue x log the le oe coceves correspodg stte. If oe cludes ll the pots o the le, complete bss set results s dcted (), whch wll be used to descrbe geerl qutum stte d, hece, to descrbe the oe-dmeso moto of prtcle. A gve stte specfes the prtculr wy whch the mpltude probblty of prtcle s dstrbuted log le. Oe wy of specfyg ths stte s by specfyg ll the mpltudeprobbltes tht the prtcle wll be foud t ech bse stte x; we wrte ech of these mpltudes s x. We must gve fte set of mpltudes, oe for ech vlue of x. Thus, = x x dx Represetto of the wvefucto () the sptl coordtes bss x s the mpltude probblty tht system the stte be foud the stte x. About the x otto We could use ltertve ottos, lke, for exmple, x (x ) = x x dx = x A Ψ (x ) dx ; s to mke ths expso to resemble the expso of vector terms of bsevectors x, d the correspodg coeffcets A Ψ (x ) plyg the role of weghtg-fctor coeffcets. Sometmes the followg otto s preferred:x ( x ) = x x dx = A Ψ x (x) dx ; () 5

6 x = (x) s the mpltude probblty tht system the stte be foud the stte x. Multplyg the expresso () bove by prtculr br x, we should obt, x = ( x ) (Ths result wll be justfed bt more rgorously the ext secto below; see the delt Drc secto). Tht s, ( x) x Ampltude probblty tht the prtcle tlly the stte be foud (mmedtely fter mesuremet) t the stte x. (3) Thus, we wll use dstctly the followg otto = x x dx.= Represetto of the x xdx. wve-fucto the (4) sptl coordtes bss umber Number Cuto: x does ot me [x]* (x) dx. Workg wth the sptl coordtes bse { x } my costtute the oly occso whch the otto x becomes cofusg wth the defto of the sclr product. Note: I Chpter 8 we used the otto (t) = A (t), where the mpltudes A (t) = were determed by the Hmlto equtos d = H j dt j j. I ths chpter, we re usg sted cotuum bse { x, x }; ccordgly the mpltudes wll be expressed s x, whch s lso wrtte s x. Soo we hve to ddress the how the Hmlto equtos look lke whe usg cotuum bse. The Delt Drc A subproduct of the mthemtcl mpulto expressg stte gve bss s the closure reltoshp tht the compoets of the bss set must comply: the Delt Drc reltoshp. Ths s llustrted for the cse of the spce-coordtes bss. 6

7 x x = = x = x [ x ] dx x [ x ] dx x x [ x ] dx x = x x [ x ] dx ( x ) ( x ) Or, equvletly x = f x x x [ x ] dx O the other hd, t s ccepted tht for rbtrry fucto f, the delt fucto s defed s ( x - x ) f x dx The lst two expressos re cosstet f, x x = ( x x ). Thus, we hve the followg result: Cse of dscrete sttes Delt Kroeecker Cse of cotuum sttes Delt Drc (x -x ) (5) m = m x x = ( x - x ) Comptblty betwee the physcl cocept of mpltude probblty d the otto used for the er product * x) x dx We kow tht gve stte, the mpltude probbltesx tht pper the expso = x x dx re determed by the Hmlto equtos. 7

8 Suppose we hve prtcle the stte d we wt to kow the mpltude probblty (fter gve mesuremet process) to fd the prtcle t the stte. Tht s, we wt to evlute. There re my pth wy for the stte to trst to stte. It could do t by pssg frst through y of the bse-sttes x. Sce ech stte x geertes pth, d ll these pths hve the sme tl d fl stte, the, ccordg to the rules estblshed Chpter 7, the totl mpltude probblty wll be gve by, x = ll x x x Frst, the sum over smll rego of wdth dx would be xx multpled by dx. The, sce x vres from to the expresso bove c be wrtte s, = x x dx (6) For the terms sde the sde the tegrl, let s use the otto troduced expresso (3) bove, x = x. Also, sce x = x * the x = x * =*(x), oe obts, = * x) x dx (7) The mpltude probblty s equl to the er product betwee the fuctos d..b Represetto of the wvefucto the mometum coordtes bss { p } I the prevous secto, rbtrry stte ws expressed terms of the compoets of the spce-coordtes bss { x, x }. Here we preset ltertve bss to express the stte, whch s the cotuum bse of mometum coordtes. We wll relze tht we re lredy fmlr wth the sttes comprsg ths ew bss. Recll tht Chpter 5 we troduced the Fourer trsform F= F(k) of fucto (x). It volved the troducto of the complex hrmoc fuctos e k, where e k (x) = e kx for k 8

9 ( x )= F(k) e kx dk Fourer trsform Bse-fucto e k evluted t x where the weght coeffcets F(k), referred to s the Fourer trsform of the fucto, re gve by, F(k) = e - k x' ( x' ) dx' However, t s coveet to express the lst two expressos terms of the vrble p = k, ( x )= (p) e ( p / ) x d p (8) where (p) = e Fourer trsform of -(p/ )x ( x' ) dx' (9) I expressos (8) d (9), the expresso terpretto, p( x ) = e ( p / ) x π e ( p / x π hve cler physcs Accordg to de Brogle, p represets ple wve of defte ler mometum p. () We c re-wrte expresso (8) terms of fuctos, 9

10 ( x )= ( p) Fourer trsform of ( x) e ( p / ) x d p p = ( p) p dp Fucto Sum of fuctos () Expressos (8) d () re equvlet. Plcg p brcket otto Frst, tkg to ccout expresso (4), = x xdx, Let s pply t to the fucto p, where p (x) = p π x p ( x ) dx e ( p / ) x ; we obt, The bove serves to defe the mometum sttes bss: { p, - < p < } p = x p ( x ) dx x e ( p / ) x dx π Represetto of the mometum stte p the spce-coordte bss { x } () x p p ( x ) = π e ( p / )x Ampltude probblty tht prtcle, stte of mometum p, be foud t the coordte x. (3) Ths s the de Brogle hypothess the lguge of mpltude probbltes. Plcg expresso () brcket otto

11 I expresso () we c mke the the followg ssocto, Hece, = (p) p dp p = (p) p dp (4) Fourer trsform ( x ) Expresso (4) gves s ler combto of the mometum sttes p defed () bove. Expressos (8), () d (4) re equvlet. The sttes defed () costtute bse (the justfcto comes from the Fourer trsform theory), { p, - < p < } cotuum bse of mometum-coordtes (5) Exercse: Prove tht p p = ( p p ) Now we formlly c justfy tht expresso (4): p = (p) I effect, multplyg (4) wth br p, oe obts, p = p (p ) p d p = (p ) p p dp usg p p = ( p p ) p = (p) (6)

12 Summry bout the mometum coordtes: A rbtrry wvefucto c be expressed s ler combto of mometum-coordtes p, = where (p) = p (p) d p = p p dp -(p/ )x e ( x' ) dx' p =(p) s the Fourer trsform of = (x) (7) p (p) Ampltude probblty tht prtcle the stte c be foud, upo mkg mesuremet, the stte p. p dp ( p) dp Probblty tht prtcle the stte be foud wth mometum wth the tervl ( p, p+ d p). < p > = p (p) dp = p p dp Averge ler mometum of esemble of systems the stte Expressos (4) d (7) summrzes the objectve of ths secto, showg expso of the stte two dfferet bss, the cotuum sptl-coordtes bse { x, x }; d the ler-mometum bse { p, - < p < }.

13 SUMMARY Coordtes bss = = x x dx x (x) dx ; Mometum bss = = p p dp p (p) d p (p) s the Fourer trsform of = (x) p s such tht, x p p ( x ) = π e ( p / )x..c Tesor Product of Stte Spces Cosder the stte spce comprsed of wvefuctos descrbg the sttes of gve system ( electro, for exmple). We use the dex- to dfferette t from the stte spce comprsed of wvefuctos descrbg the stte of other system (other electro, for exmple) whch s tlly locted fr wy from the system. The systems (the electros) my evetully get closer, terct, d the go fr wy g. How to descrbe the spce stte of the globl system? The cocept of tesor product s troduced to llows such descrpto Let { u (),,, 3,...} be bss the spce ε, d { v (),,, 3,...} be bss the spce The tesor product of of ε d ε, deoted by ε ε ε, s defed s spce whose bss s formed by elemets of the type, { u () v j () } ε 3

14 Tht s, elemet of ε ε s ler combto of the form, The tesor product,j c j u () v j () s defed wth the followg propertes, [ () ] () [ () () ] () [ () ] [ () () ] [ () () ] () [ () () () [ () () ] [ () ] + [ () () ] + [ () () ] () ] Two mportt cses my rse. Cse : The wvefucto s the tesor product of the type, () () Tht s c be expressed s the tesor product of stte from ε d stte from ε.,j b u () b j v j () j j u () v j () Cse : The wvefucto cot be expressed s the tesor product betwee stte purely from the spce ε d stte purely from the spce ε. I ths cse, the wvefuto tkes the form,, j c j u () v j () Let s cosder, for exmple the cse whch ech spce ε d ε hs dmeso., j c j u () v j () c u v() c u v() () () c u v() c u v() () () 4

15 c u v() c u v() () () Notce, cot be expressed the form () () To mke the cse eve smpler, let s ssume c = c =, c u () + c u () etglemet () v () cot be expressed the form of sgle term of the form () () v. The Schrödger Equto s postulte..a The Hmlto equtos expressed the cotuum sptl coordtes. The Schrödger Equto. 3 I chpter 8 we obted the geerl Hmlto equtos tht descrbe the tme evoluto of the wvefucto (t) = A ( t ), da H j ( t) Aj (8) dt j Chpter 9 descrbed the prtculr cse of electro movg lttce (the ltter costtuted by toms seprted dstce b. Whe we took the lmt b the Hmlto equtos (5) took the form ( x,t) ( x,t) V ( x,t) ( x,t) (9) t m x Let s cosder ow rbtrry geerl cse. eff How does the Hmlto equtos (8) look lke whe expressed the the cotuum spce coordtes { x, x }? Let s fd out such geerl forml expresso (oe tht s more geerl th expresso (9) ). Frst otce tht the mpltudes system, = j j A j A j (8) ccout for the stte descrbg the qutum Sce A j c lso be wrtte s A j = j, Eq. (5) c lso be expressed s, d = H j j dt j 5

16 Let s lso recll, from Chpter 7, tht the coeffcets H j opertor H (specfc to the problem beg solved.) Tht s, deped o t. d = H j j dt j I the cotuum spce coordtes we should expect, d x = dt x H x x d x (x) ( x ) re obted from the Hmlto H j H j. I geerl, H d d ( x ) = dt H( x, x )( x ) d x () where we hve defed H( x, x ) x H x Quotg Feym, 4 Accordg to (), the rte of chge of t x would deped o the vlue of t ll other pots x. x H x s the mpltude per ut tme tht the electro wll jump from x to x. It turs out ture, however, tht ths mpltude s zero except for pots x very close to x. Ths mes (s we sw the exmple of the ch of toms) tht the rght-hd sde of Eq. () c be expressed completely terms of d the sptl dervtves of, ll evluted t x. The correct lw of physcs s d H( x, x ) ( x ) d x = ( x ) + V ( x) ( x ) Postulte () m dx Where dd we get tht from? Nowhere. It cme out of the md of Schrodger, veted hs struggle to fd uderstdg of the expermetl observto of the expermetl world. Usg () () oe obts, 6

17 t V ( x, t) x m Schrodger Equto Ths equto mrked hstorc momet costtutg the brth of the qutum mechcl descrpto of mtter. The gret hstorcl momet mrkg the brth of the qutum mechcl descrpto of mtter occurred whe Schrodger frst wrote dow hs equto 96. For my yers the terl tomc structure of the mtter hd bee gret mystery. No oe hd bee ble to uderstd wht held mtter together, why there ws chemcl bdg, d especlly how t could be tht toms could be stble. (Although Bohr hd bee ble to gve descrpto of the terl moto of electro hydroge tom whch seemed to expl the observed spectrum of lght emtted by ths tom, the reso tht electros moved ths wy remed mystery.) Schrodger s dscovery of the proper equtos of moto for electros o tomc scle provded theory from whch tomc pheome could be clculted qutttvely, ccurtely d detl. Feym s Lectures, Vol III, pge 6-3. Although the result () s kd of postulte, we do hve some clues bout how to terpret t, bsed o the prtculr cse of the dymcs of electro crystl lttce, studed Chpter 9. ()..B Iterpretto of the Wvefucto Este s vew o the grulrty ture of the electromgetc rdto I Chpter 5, hrmoc fucto ws used to descrbe the moto of free prtcle logy to the exstet formlsm to descrbe electromgetc wves, π ε ( x, t) ε o Cos [ x νt] electromgetc wve λ where, the electromgetc testy I (eergy per ut tme crossg ut cross-secto re perpedculr to the drecto of rdto propgto) s proportol to ε ( x, t). Este ( the cotext of tryg to expl the results from the photoelectrc effect) troduced the grulrty terpretto of the electromgetc wves (lter clled photos), bdog the more clsscl cotuum terpretto. I Este s vew, the testy s terpreted s sttstcl vrble I ocε N h. Here N costtutes the verge umber of photos per secod crossg ut re perpedculr to the drecto of rdto propgto; ε N Averge vlues re used ths terpretto becuse the emsso process of photos by gve source s sttstcl ture. The exct umber of photos crossg ut re per ut tme fluctutes roud verge vlue N. 7

18 Mx Bor s Probblstc Iterpretto of the wvefucto I logy to Este s vew of rdto, Mx Bor proposed smlr vew to terpret the prtcle s wve-fuctos. I Mx Bor s vew, ( x, t) plys role smlr to ε ( x, t), ( x, t) s mesure of the probblty of fdg the prtcle roud gve plce x d t gve tme t. Ths terpretto ws troduced yers fter Schrodger (96) hd developed forml qutum mechcs descrpto. More specfclly, ( x, y,z, t) plys the role of probblty desty. Pctorlly, the prtcle s more lkely to be t loctos where the wvefuto hs pprecble vlue. Determstc evoluto of the wvefucto The predctos of qutum mechcs re sttstcl. I order to kow the stte of moto of prtcle, we must mke mesuremet But mesuremet ecessrly dsturbs the system wy tht cot be completely determed. However, otce tht, beg the soluto of dfferetl equto (the Schrodger equto), vres wth tme wy tht s completely determstc. Tht s, f were kow t t=, the Schrodger equto determes precsely ts form t y future tme. Tht s, QM mkes determstc predcto of mpltude probblty wve. However, the ltter does ot covey to determstc outcomes. There s oe further pot to cosder. How to determe the wvefucto t t=? How do the expermetl mesuremets led to the recostructo of the wvefucto? Or, how to prepre system deftely uque stte? If we could ot recostruct wvefucto, wht would be the beeft of hvg theoretcl formulto tht descrbes determstc evoluto of somethg we do ot kow? As t turs out, despte the fct tht mesuremet QM geerl ffect the stte of system, such recostructo s possble some cses (thk of system tht re sttory stes.) But lrger cotext, to beeft of the QM determstc formulto wht we eed s to prepre system (or my systems) defte stte; for the theory could the be used to mke predctos bout the evoluto of tht prtculr stte. We wll ddress ths ssue the ext chpters, fter the troducto of observbles d egesttes. We wll see tht fdg egesttes commo to dfferet observbles rrows the selecto pool of sttes whch the system c be foud. Ths procedure leds to the cocept of esemble of system costtuted by ( ths wy) eqully prepred systems, whch costtutes the lbortory whch the QM cocept re develop. (A the ed, 8

19 systems cot be determed wth bsolute certty smply becuse set of mesuremets t t= t most my led to the determto of but ot to uquely defe ). Let s expl the sttstcl terpretto bt further the cotext of esemble of detclly prepred systems. Esemble 5 Imge very lrge umber of detclly prepred depedet system (ssumed to be ll of them the sme stte), ech of them cosstg of sgle prtcle movg uder the fluece of gve exterl force. All these systems re detclly prepred. The whole esemble s ssumed to be descrbed by complex-vrble sgle wvefucto ( x, y, z, t), whch cots ll the formto tht c be obted bout them. descrbes the whole esemble... Esemble s used to mke probblstc predcto o wht my hppe prtculr member of the esemble. N It s postulted tht: If mesuremet of the prtcle s posto re mde o ech of the N member of the esemble, the frcto of tmes the prtcle wll be foud wth the volume elemet d 3 r =dx dy dz roud the posto r ( x, y, z, t) t the tme t s gve by (3) * ( x, y, z, t) ( x, y, z, t) d 3 r where * stds for the complex cojugte umber. Notce tht ths s othg but the lguge of probblty; ths cse, posto probblty desty P. P * ( x, y, z,t) ( x, y, z,t) ( x, y, z,t) ( x, y, z, t ) 9

20 Cuto: For coveece, we shll ofte spek of the wvefucto of prtculr system, BUT t must lwys be uderstood tht ths s shorthd for the wvefucto ssocted wth esemble of detcl d detclly prepred systems, s requred by the sttstcl ture of the theory. 6..C Normlzto codto for the wvefucto The probblstc terpretto of the wvefucto mples, therefore, the followg requremet: * 3 ( x, y, z,t) ( x, y, z,t) d r (4) All spce becuse gve prtcle, the lkelhood to fd t ywhere should be oe. Iheret to ths requremet s tht, ( r, t) (5) r Notce tht f s soluto of the Schrodger equto, the fucto c (c beg costt) s lso soluto. The multplctve fctor c therefore hs to be chose such tht the fucto c stsfes the codto (4). Ths process s clled ormlzg the wvefucto. I geerl, there wll be solutos to the Schrodger equto () whose soluto ted to fte vlue. Ths mes they re o-ormlzble d therefore c ot represet prtcle probblty desty. Such fuctos must be rejected o the grouds of Bor s probblty terpretto. Qutum mechcs sttes re represeted by squre- tegrble fuctos tht stsfy the Schrodger equto. The prtculr subset of squre tegrble fuctos form vector spce clled the Hlbert spce. QUESTION: Suppose tht s ormlzed t t. As the tme evolves, wll chge. How do we kow f t wll rem ormlzed? Here we show tht the Schrodger equto hs the remrkble property tht t utomtclly preserves the ormlzto of the wvefucto:

21 If stsfes the Schrodger equto The f the potetl s rel dt Proof: Let s strt wth d dt ( x, t) dx = (6) d ( x,t) dx ( x,t) dx (7) t We provde below grphc justfcto of (7). fx, t ) fx, t +) x x t fx, t +)dx - fx, t ) dx = t [fx, t + ) - fx, t) ] O the other hd, t ) t t t (8) We use the Schrodger equto (9) to clculte the tme dervtves, V ( x,t) t m x V ( x,t) m x V ( x,t) t m x Tkg the complex cojugte, d ssumg tht the potetl s rel, V( x,t) t m x Addg the lst two expressos, t t m x x

22 m Replcg (9) (8) we obt, Accordgly, d dt t m x x x x x x ( x,t) dx ( x,t) t m dx dx x x x m x x The expresso o the rght s zero becuse x x (9)..D The Phlosophy of Qutum Theory There hs bee cotroversy over the Qutum Theory s phlosophc foudtos. Nels Bohr hs bee the prcpl rchtect of wht s kow s the Copehge terpretto (sttstcl terpretto) Este ws the prcpl crtc of Bohr s terpretto. Hs sttemet God does ot ply dce wth the uverse, refers to the bdomet of strct cuslty d dvdul evets by qutum theory. Heseberg coutercts rgug: We hve ot ssumed tht the qutum theory (s opposed to the clsscl theory) s sttstcl theory, the sese tht oly sttstcl coclusos c be drw from exct dt. I the formulto of the cusl lw, mely, f we kow the preset exctly, we c predct the future t s ot the cocluso, but rther the premse whch s flse. We cot kow, s mtter of prcple, the preset ll ts detls. Lous de Brogle, o the other hd, rgues tht tht lmted kowledge of the preset my be rther lmtto of the curret mesuremet methods beg used. He recogzes tht

23 ) t s cert tht the methods of mesuremet do ot llow us to determe smulteously ll the mgtude whch would be ecessry to obt pcture of the clsscl type, d tht b) perturbtos troduced by the mesuremet, whch re mpossble to elmte, prevet us geerl from predctg precsely the results whch t wll produce d llow oly sttstcl predctos. The costructo of purely probblstc formule ws thus completely justfed. But, the sserto tht ) The ucert d complete chrcter of the kowledge tht expermet t ts preset stge gves us bout wht relly hppes mcrophyscs, s the result of ) rel determcy of the physcl sttes d of ther evoluto, costtutes extrpolto tht does ot pper y wy to be justfed. De Brogle cosders possble tht lookg to the future we wll be ble to terpret the lws of probblty d qutum physcs s beg the sttstcl results of the developmet of completely determed vlues of vrbles whch re t preset hdde from us. Lous de Brogle s vew gve bove hghlghts the objecto to qutum mechcs phlosophc determsm. Accordg to Este: The belef of exterl world depedet of the percevg subject s the bss of ll turl scece. Qutum mechcs, however, regrds the tercto betwee object d observer s the ultmte relty; rejects s megless d useless the oto tht behd the uverse of our percepto there les hdde objectve world ruled by cuslty; cofes tself to the descrpto of the reltos mog perceptos 7 Physcs hs gve up o the problem of tryg predctg exctly wht wll hppe defte crcumstce..3 Expectto vlues.3.a Expectto (or me) vlue of prtcle s posto Let s ssume we hve system cosstg of box cotg sgle prtcle, whch s (we ssume) stte. Sce (x) s terpreted s probblty, the expectto vlue of the prtcle s posto s defed by, x - But wht does ths tegrl exctly me? x ( x) dx (3) 3

24 It s worth to emphsze frst wht type of terpretto should be voded. 8 Expresso (3) does ot mply tht f you mesure the posto of the prtcle over d over g the - x ( x) dx would be the verge of the results. I fct, f repeted mesuremets were to be mde o the sme prtcle, the frst mesuremet (whose outcome s upredctble) wll mke the wvefucto to collpse to stte of correspodg prtcle s posto x (let s sy x ); subsequet mesuremets (f they re performed quckly) wll smply repet tht sme result x. O the cotrry, x - x ( x) dx mes the verge obted from mesuremets performed o my systems, ll the sme stte. Tht s, A esemble of systems s prepred, ech the sme stte, d mesuremet of the posto s performed ll of them. x s the verge from such mesuremet. descrbes the whole esemble... Esemble s used to mke probblstc predcto o wht my hppe prtculr member of the esemble. N Fg.. Esemble of detclly prepred systems. Whe we sy tht system s the stte, we re ctully referrg to esemble of systems ll of them the sme stte. Thus, represets the whole esemble. d x.3.b Clculto of m dt As tme goes o, the expectto vlue x my chge wth tme, sce the wvefucto evolves wth tme. Let s clculte ts rte of chge. d x dt d dt x ( x, t) dx x t ( x, t) dx x dx t 4

25 5 dx t t x For the cse where the potetl s rel, we obted expresso (9) tht, x x m t t x x m x dt x d x x x m x After tegrtg by prts, t gves x x m dt x d Itegrtg oe more tme by prts (just the secod term o the rght sde,) dx x m dt x d (3) We would be tempted to postulte tht the expectto (or me) vlue of the ler mometum s equl to dt x d m p. I tht cse, expresso (3) would gve, dt x d m p dx x = x (3) However, look t the fuctos sde the tegrl. How to uderstd tht term lke x would led to the verge vlue of the ler mometum? Ths ppers bt strge, to sy the lest. (Ths result wll mke more sese the sectos below, whe the cocept of qutum mechcs opertors s troduced)..3.c Expectto (verge) vlues re clculted esemble of detclly prepred systems I geerl, the me-vlue of gve physcl property f (mely, eergy, ler mometum, posto, etc.), more geerclly clled observble, s obted by mkg mesuremet ech of the N eqully prepred systems of esemble (d ot by vergg

26 repeted mesuremets o sgle system.) The whole esemble s ssumed to be descrbed by complex-vrble sgle wvefucto ( x, y, z, t). Whe mkg mesuremets o ech of the N detclly prepred systems of the esemble (see left-sde of the fgure below) let s ssume we get seres of results (see rghtsde of the fgure below) lke ths: N systems re foud to hve vlue of f equl to f, from whch we deduce tht the prtculr system collpsed to the stte f rght fter the mesuremet N systems re foud to hve vlue of f equl to f, from whch we deduce tht the prtculr system collpsed to the stte f rght fter the mesuremet etc. Accordgly, the verge vlue of f s clculted s follows, f v N f N f N N Usul procedure to clculte verge vlues (33) N Notce tht whe the totl umber of mesuremet N s very lrge umber, the rto N s othg but the probblty of fdg the system the prtculr stte f. QM postultes tht the vlue of the vlue f, f ψ should be terpreted s the probblty to obt f ψ = N Qutum Mechcs postulte (34) N Represetg the esemble ψ Hece, expresso (33) c be wrtte s, f v f f ψ (35) 6

27 ... Esemble... N Before the mesuremets The vlue of f ech system s ukow. N After the mesuremets The vlue of f hs bee mesured ech system; fterwrds we proceed to clculte the verge vlue < f >..4 Opertors ssocted to Observbles Quttes such s posto, mometum, or eergy (whch re mesured expermetlly) re clled observbles. I clsscl physcs, observbles re represeted by ordry vrbles (E, p, for exmple). I qutum mechcs observbles re represeted by opertors (quttes tht operte o fucto to gve ew fucto.) Whe system stte eters some pprtus, lke, for exmple, mgetc feld the Ster Gerlch expermet, or mser resot cvty, t my leve dfferet stte. Tht s, s result of ts tercto wth the pprtus, the stte of the system s modfed. Symbolclly, the pprtus c be represeted by correspodg opertor F such tht, = F (36) Note: We wll dstgush the opertors (from other quttes) by puttg smll ht o top of ts correspodg symbol. We show below how to ssocte qutum mechcs opertor F to gve physcl qutty f..4.a Observbles, egevlues d ege-sttes 7

28 Let s cosder physcl qutty or observble f ( f could be the prtcle s gulr mometum for exmple) tht chrcterzes the stte of qutum system. I qutum mechcs, the dfferet vlues tht gve physcl qutty f (observble) c tke, re clled ts egevlues; f, f, f 3, (37) The set of these qutum egevlues s referred to s the spectrum of egevlues of the correspodg qutty f. For smplcty, let ssume for the momet tht the spectrum of egevlues s dscrete. wll deote the stte where the qutty f hs the vlue f ; These sttes wll be clled egesttes (38) We wll ssume tht these egestes stsfy, m = ( x) m ( x) dx = m (39) Let s ssume lso tht the egesttes ssocted to the observble f costtute bss, { ; =,, 3, } Bss of egesttes A rbtrry stte c the be represeted by the expso, = A =. (4) where A = = ( x) ( x) dx. Sce the stte must be ormlzed stte, the A = = Accordg to expresso (35), the me vlue of f, whe the system s the stte, s gve by, f = v f A = f (4) Expresso (4) s stll very geerl, sce we do ot kow yet the egesttes. We descrbe ext how to fgure out those sttes. 8

29 .4.B Defto of the QM opertor F to be ssocted wth the observble physcl qutty f. 9, The opertor F to be ssocted wth the observble f s such tht, whe ctg o rbtrry stte, stsfes the followg: F f v Defto of the Opertor F ssocted to the observble f (4) (the verge vlue o the rght sde s clculted over the esemble represeted by the stte ) But, how to obt explct expresso for such opertor F? If the observble qutty f were, for exmple, the ler mometum, how to buld ts correspodg qutum mechcs opertor? Before buldg the opertors explctly, we frst we derve ths secto geerl selfcosstet expresso tht shows how opertor, ssocted to gve observble, should look lke (see expresso (47) below). Subsequetly, sectos.4.c d.4.d wll provde specfc procedure o how to costruct the posto d mometum opertors. Dervg self-cosstet expresso of QM opertor F If the system s the stte, = A =, the verge vlue of the quttyc f s gve by (4), f = v f A = f The requremet to buld the opertor F s, F = f = = v f A = f f * f (43) 9

30 Notce, we c obt more compct expresso for the opertor F f we use the otto, P proj, Projecto opertor (44) Expresso (44) descrbes opertor tht whe ctg o stte gves the projecto of tht stte log the stte ; tht s, P proj, = (45) Accordgly, (43) c be expressed s, F = [ f ] (46) stte umber opertor where deote the stte where the qutty f hs the vlue f ; Ths otto trck llows us to express the opertor more compct form, F = f (47) egevlues Projecto opertorp proj, bult out of egesttes ssocted to the physcl qutty f. Ths s self-cosstet expresso for the qutum opertor F tht wll be ssocted wth the clsscl qutty f. It s expresso tht s comptble wth the requremet tht Ψ F Ψ f. Note: The opertor defed (47) should rem the sme f we chged the bss to express the. Ths clm s supported by the fct tht the vlue of f, whch tervees the defto of F, should be depedet of the bss chose. v v Notce (47) s self-cosstet wth ts defto. By pplyg the opertor egesttes, t gves, F to oe of the F j = f j (48) j 3

31 Tht s, oce the opertor F (ssocted to the physcl qutty f ) s kow, the egefuctos of tht gve physcl qutty re the solutos of the equto F = where s costt. Stll, otce tht (47) d (48) gve just self-cosstet expressos for the opertor F. It s expresso tht s comptble wth the requremet tht F f v, but t s defed terms of egesttes j tht, for gve physcs qutty f, we do ot kow yet but would lke to fd out. Accordgly, F s ot completely kow yet. But, prphrsg Ldu, Although the opertor F s defed by (48), whch tself cots the egefuctos, o further coclusos c be drw from the results we hve obted. However, s we shll see below, the form of the opertors for vrous physcl quttes c be determed from drect physcl cosdertos, whch subsequetly, usg the bove propertes of the opertors, wll eble us to fd the egefuctos d egevlues by solvg the equto F = Expressos (4) d (47) wll gude us to buld the qutum opertors..4.c Defto of the Posto Opertor X We re lookg for opertor X such tht X gves us the me vlue of the posto whe the system s the stte, whch requres, X *( x) [ ( X )( x) ] dx x - x x ( x) dx, X =? (49) [ *( x) ] x [ ( x) ] dx ( X ) ( x ) = x ( x ) (5) Notce, we cot sy X = x; tht would be correct. (For stce, wht vlue of x would you choose to mke the expresso X = x vld?). More pproprte s frst to defe the detty fucto I, whch stsfes, I(x) = x, The (5) c be wrtte s, ( X ) ( x) = (I ) ( x) 3

32 becuse tht would gve [ X ]( x) = [I ] ( x) = I( x) ( x) = x ( x). Tht s, X = I Aswer (5) I summry, The Posto Opertor X X s the opertor ssocted to x X = I where I s the detty fucto; I (x) = x ( X ) x= x x (5) X = [ * x ] [ X x ] dx = [ * x ] x [x] x Notce, t s strghtforwrd to relze tht X s the opertor ssocted to 3 X s the opertor ssocted to x 3, etc. I geerl, x, d tht More geerl, X s the opertor ssocted to Averge vlue of physcl qutty x (53) Qutum mechcs opertor x v X x X v x v X f h(x) s polyoml or coverget seres, (54) the h( X ) wll be the opertor ssocted to h ( x) 3

33 For exmple: If physcl qutty chges s, h(x) = 3 x - 7 x 3 the the correspodg opertor wll be 3 X 3-7 X, tht s, the lst expresso s obted by evlutg h( X ). Averge vlue Qutum mechcs of physcl qutty opertor h (x) h( X ) v For exmple, f h(x) were clsscl potetl tht depeds oly o posto x, the h( X ) would be the the correspodg qutum mechcs potetl opertor. Tke the cse of the hrmoc osclltor, where V(x) = (/)k x the correspodg QM opertor s V = (/)k X..4.D Defto of the Ler Mometum Opertor Here we costruct the qutum Ler Mometum Opertor, d gve ts represetto both the mometum-coordtes d sptl-coordtes bss. Ths exmple wll help to llustrte tht opertors re geerl mthemtcl cocepts depedet of the bse used, but whose represetto depeds o the prtculr bse sttes beg selected..4.d The Ler Mometum Opertor P expressed the mometum sttes bss { p } I ths cse, the observble f refers to the ler mometum p. Before buldg the ler mometum opertor let s summrze wht we kow bout the mometum sttes. Accordg to de Brogle, for gve p we ssocte mometum stte p gve by expresso () bove, p x p ( x ) dx = x π e ( p / ) x dx (55) Represetto of the mometum stte p the spce-coordte bss { x } x p p ( x ) = π e ( p / ) x 33

34 Sometmes, for coveece, we wll use the otto mometum p or p sted of p. From the Fourer trsform theory we lered tht rbtrry stte c be expressed s ler combto of the mometum sttes, where = (p) = p = p p dp = p (p) d p (56) -(p/ )x e ( x' ) dx' Now we wt to buld the Ler Mometum Opertor tht wll be ssocted to the physcl qutty p. We re lookg for the opertor P such tht P gves us the me vlue of the mometum whe the system s the stte. P x, P =? But frst, f system s the stte = Wht s the verge mometum p? Aswer: p Hece, = p (p) d p We re lookg for the opertor P such tht P = p = p p dp = p (p) d p; P =? p (p) d p, p = = p (p) d p (d p) p (p ) (p) = (d p) (p ) p(p). 34

35 = (d p) (p ) p (p ) ( p p ) dp Usg p p = ( p p ) = ( p p ) = p p = (d p) (p ) p (p ) p p dp = [ p (p) d p] [ p (p ) p dp ] p = [ p (p ) p dp ] (57) Wth the ht tke from expresso (47), we relze the followg pertor, P = p p p dp Ler Mometum Opertor (58) would do the job I effect, pplyg P to rbtrry stte gves, P = [ = p p p dp], p p p dp, Accordg to (56) p =(p), whch gves P = p p (p) dp (59) Replcg (59) (57) oe obts, p = P (6) Ths cofrms the oprtor defed (58) s deed the mometum opertor. 35

36 Mtrx represetto of the Ler Mometum Opertor P the mometum coordtes bss { p, - < p < } Notce (57) tht pplyg P to prtculr stte p gves, Summry P p = p p (6) p P p = p p p pp' = p ( p p ) P = p P p = p ( p p ) (6) The Mometum Opertor P P = Elemets of the mtrx represetto of the opertor the mometum bss p p p dp = p p dp = p (p) d p where = (p) s the Fourer trsform of (x) P = p p p dp = p (p) p dp P = p (p) d p = p.4.d The ler mometum opertor P expressed the sptl coordtes bss { x, x } Expresso (59) dctes tht f the system were the stte d we wted to evlute P the we hve to frst expd the mometum bss = p (p)dp 36

37 Tht wy, oce ll the (p) vlues re kow, the we would be ble to evlute P = p p (p) dp Wht bout f the expso of the mometum bss { p, - < p < } s ot hdy vlble, d, sted, ts expso the sptl coordte s vlble? (.e. (x) s kow for every vlue of x, but ts Fourer trsform (p) s ot redy vlble). Wht to do to fd P (wthout hvg to go through the trouble of expressg terms of the mometum bss s expresso (59) requres)? Here ltertve to (59) s preseted. Applyg (58), P = oe obts, P = [ p p p dp, to the stte fucto = = p p p dp] = p p (p) dp p p p dp p (p ) dp x P = p (p) x p dp, [P x= p (p) p x dp Usg x p p ( x ) = π e ( p / ) x [P x= p (p) π e ( p / ) x dp, [P x = (p) p π e ( p / ) x dp, 37

38 = (p) d dx π e ( p / ) x dp = d dx (p) π e (p / ) x dp = d dx (p) p xdp, Hece, P = = d dx d dx x (63) The Mometum Opertor P ( the sptl-coordtes spce) P d = (64) dx (ltertve to expresso (58)) P = d dx whch leds to p v = P d = dx dx I pssg, otce tht the lst expresso s detcl to (3) where we clculted the rte of d x chge of the verge posto of prtcle m. At tht tme, the presece of sptldervtve ppered to mke o sese beg volved wht could be terpreted s dt d dx the velocty of prtcle. Now we see tht such sptl dervtve ppers becuse we re usg the represetto of the mometum opertor the sptl coordtes (tht sptl dervtve does ot pper whe workg the mometum bss). 38

39 .4.D3 Systemtc wy to express the opertors P, bss Costructo of the opertor ssocted to By defto: p = v = p v P, 3 P, etc., the sptl-coordtes p (p) d p (65) p p p dp The systemtc procedure cossts evlutg frst the fctor p p sde the tegrl. p p = p -(p/ )x e ' ( x' ) dx = p -(p/ )x e ( x' ) dx' Cotug wth smlr procedure s bove (tegrtg by prts), we obt, p p = p Replcg (66) (65) leds to, p = v d (66) dx p p Fctorg out = p p d dp dx d dp dx The tegrl term s the expso of the ket d the mometum bse dx p = v d dx 39

40 = * x,t d dx x,t ] dx If we defe the opertor P = The lst expresso becomes, d dx (67) Tht s, P = d dx ssocted wth p. p = v * x,t P x,t ] dx (68) stsfes the requremet for beg the qutum opertor to be More geerl, for postve teger, Tht s, P = p = v d dx * x, t d dx x,t ] dx (69) s the qutum mechcs opertor ssocted to p Averge vlue of physcl qutty p v p v p v Qutum mechcs opertor d dx d dx d dx If g(p, t) s polyoml, or bsolutely coverget seres, the clsscl mometum vrble p, oe obts, 4

41 g ( p, t) v = * x, t d g(, t) dx x,t ] dx (7) Averge vlue of physcl qutty v Qutum mechcs opertor d g dx g ( p, t) (, t) A exmple of (7) costtutes the QM opertor ssocted to the ketc eergy. m p v P m = = m d dx d m dx.4.e The Hmlto opertor.4.e. Me eergy terms of the Hmlto opertor Oe of the vrtues of usg QM opertors s tht me vlues c be expressed depedet of prtculr bss. For exmple, recll tht < x > = X d <p> = P. We re gog to do somethg smlr here for the verge eergy <E>. I Chpter 8, the Hmlto Opertor H ws recogzed through ts mtrx represetto [H], the ltter beg terpreted s eergy mtrx due to the fct tht, whe workg wth sttory sttes, the compoets of the mtrx were the eergy of the correspodg sttory sttes. I the lguge of egevlues used ths chpter we c wrte, H E = E E where { E, =,, 3, } s the bss costtuted by eergy sttory sttes For system rbtrry stte let s clculte the expectto (or me) vlue of eergy. = E E (7) Its me eergy s gve by, (7) E E v E ψ = E * E E 4

42 = E E E = H E E Ths fctor c be expressed terms of the Hmlto opertor From (7): H E = E E = H E E = E = H (73) v Thus, we hve foud g (s we dd for the ler mometum d the posto opertors) elegt wy to express me vlue ( ths cse for the eergy) tht does ot mke referece to the prtculr bse sttes. Whe explct otto of the bss sttes s requred to clculte sttory sttes bss { E, =,, 3, } E v, we my use the v E E E ψ (74) But, some cses t my be coveet to use dfferet bse. For geerl bss { j for j =,,3, } we wll hve, E = H v Expressg d the { j } bss = [ * ] H [ j j j ] = H j j (75), j.4.e. Represetto of the Hmlto Opertor the sptl coordte bss Smlr to the cse of the ler mometum opertor ddressed the prevous sectos, the Hmlto opertor c dopt dfferet shpes depedg o whether we re workg the sptl-coordtes bss or the mometum coordte bses. Here we ddress the represetto of the Hmlto opertor the sptl-coordtes 4

43 I (73), let s tke the cse of sptl coordte bss Defg E E v v = H = [ - = [ - = - x * x dx ] H x * x dx ] [ - x * [ - x x dx ] H x x dx ] x H x x dx dx ] - H( x, x ) x H x (76) where x H x s the mpltude per ut tme tht the electro wll jump from x to x. (see lso expresso () bove). E = v - = - x * ( x) * - - H( x, x ) x dx dx H( x, x ) ( x ) dx dx Accordgly E v = - But Schrodger estblshed tht (see expresso () bove) - H( x, x ) ( x ) d x = ( x ) + V ( x)( x ) d m dx d m dx ( x) * [ ( x ) + V ( x)( x )] dx If we defe the potetl opertor V, V Ψ V Ψ (77) E = v - ( x) * d [ m + V ] (x) dx dx H 43

44 E = H v Represetto of the Hmlto d H Opertor the sptl coordtes + V (78) m dx bss Notce, ccordg to expresso (67) P m = d m dx, therefore we c wrte, H P m V Summry Observble Me Opertor Opertor the sptl vlue coordtes represetto posto X X x x = X X = x mometum P d dx p p = P P d = dx d H m + V dx Eergy E E = H.5 Propertes of Opertors.5.A Correspodece betwee brs d kets Br Ket χ χ 44

45 m m m m b b m * m m m m m b m b m (79) bφ bφ Φ b * Φ b * Φ * Φ Φ Φ * Utl ow we hve defed the cto of ler opertor A o kets the cto of opertor o brs. χ. We wt to defe.5.b The Adjot opertor Cosder opertor A. We pcture opertor A tht whe ctg o rbtrry d wvefucto the result s, for exmple, 3 ; or ; or I, etc. For gve A, we dx wll defe ts correspodet djot opertor A. Through the defto we wll fd out wht does A do o wvefucto ; tht s s, f f A = 3 the we wll fd out A =?; or A d = dx etc. the we wll fd out A =? I ddto, we would lke to kow lso how to express the cto of opertor oto wvefuctos the lguge of brs d kets. Let A be opertor. For opertor ctg o ket, wht does A χ me? For opertor ctg o br, wht does A me? A s the djot opertor of A 45

46 f t whch fulflls the followg er product for y rbtrry wvefuctos d, A,, A (8) Iterpretto of the djot opertor brcket otto. Frst, strght forwrd trslto of expresso (8) s to wrte s s, A A (8) Tkg the cojugte vlues, A A We would lke to kow lso how to tke the opertors out of the br or the ket. Frst, A A Tht s, we defe A A (8) Secod, let s compre the ctos of opertor o the brs d kets A A I other words, Exmple: Show tht A A Tht s, A s defed through the followg expresso A ppled o the ket gves the ket A A ppled o the br (A ) = A Applyg the defto of djot opertor to A, A A (83) gves the br A 46

47 A A Applyg the defto of djot opertor to ( A ) A A, A * Usg the defto of djot * A (A ) A Ths mples, Notce, ) (A A (84) A A ) (A A Tht s, (A ) A (85) Usg (84) A A (86) Summry A A. A A. (87) A A Questo: How to terpret the followg qutty A? 47

48 O oe hd, Does t me ( A ) or ( A )? A = ( A ) A = ( A ) (88) O the other hd, A = A From (88) d (89) = ( A ) (89) ( A ) = ( A ) (9) Tht s, t uecessry to plce the prethess. Tht s, A, oe obts the sme result by mkg A to ct frst o the br (d the multply by ), or by mkg A to ct frst o the ket (d the multply by ). I short, A = ( A ) = ( A ) = A A = A = A (9) Exercse: Show tht the djot opertor A s ler. Exmple: Show tht (B ) = B 48

49 Let s pply the defto (8), A A, to the opertor A = ( B ) B B Hece, (B ) = B Exercse. Gve the opertor [ B *] * [ B ] * [ B ] * D B ( D, ) D = = d, wht s the opertor D? d x * x) [ d ψ * x) x dx d x Itegrtg by prts = - Ths mples, D = - D = ( - D, ) d φ x) x dx = d x Mtrx Represetto of the djot opertor (, d ]x dx d x [ m ) = ( m, ) * Hece ther mtrx represetto re relted through, - D x) ]* x dx [ m = [ m * (9) 49

50 (Notce the order of the dexes s reversed) Exercse: Show tht ( A B) = B A (93) Exmple: Wht s the djot opertor of the posto opertor X? ( X, ) (, X ) = * x) [ X ]x dx = * x) x x dx = x * x) x dx = [x x) ] * x dx = [ X x) ] * x dx ( X, ) = ( X ) Sce ths expresso s vld for y rbtrry sttes d, the X = X Tht s, the djot of the posto opertor s tself. (94).5.C Hermt or self-djot opertors My mportt opertors of qutum mechcs hve the specl property tht whe you tke the Hermt djot you get bck the sme opertor. =. (95) Such opertors re clled the self-djot or Hermt opertors. Exmple: The posto opertor X s self djot opertor becuse the exmple the prevous secto (see expresso 9). Exmple: Let s see f the ler mometum opertor P s self djot ( P, ) (,P ) = * x) [ = * x) d ]x dx d x d ψ x dx d x Itegrtg by prts X = X, s show 5

51 = - d φ x) x dx d x = [ d φ d x x) ]* x dx Tht s, P = P = ( P, ) (96) Propertes of Hermt (or self-djot) opertors Opertors ssocted to me vlues re Hermt (or self-djot) I secto.4 bove we defed qutum mechcs opertors ssocted to clsscl observble qutty. The defto volved the clculto of me vlues of observbles. From the fct tht me vlues re rel, we c drw some coclusos cocerg the propertes of those opertors. Tht s, f v = F Sce ths qutty s rel, t wll be equl t ts complex cojugte = F * = F F = F Usg defto of djot opertor F = F whch mples, F = F Opertors correspodg to observbles (97) (.e. opertors obted through the requremet f = F ) v must be hermts (self-djot). The egevlues of Hermt (self-djot) opertor re rel. Let j be d egevlue of F d j the correspodg egevector F j = j j (98) 5

52 Sce F s Hermt (self-djot), F = F, for y stte we wll hve, F ) = F, I prtculr, for =j, Usg (98) F j j = j F j j j j j j j whch mples = j * j = j j j = j j j ( for egevlues of Hermt opertors) (99) Egefuctos of Hermt opertor correspodg to dfferet egevlues re orthogol Let, F j = where j k F Sce k j j d F k = k j = k, j j d F k j = k F = F, d j s well s k re rel, we obt, F k j = Ths mples, k k j k j d F k j = k k j k j = k k, j j j Thus, ( j - k ) k j = j k mples k j = ( for Hermt opertors) ().5.D Observble Opertors Whe workg spce of fte dmeso, t c be demostrted tht t s lwys possble to form bss wth the ege-vectors of Hermt opertor. But, whe the spce s fte dmesol, ths s ot ecessrly the cse. Tht s the reso why t s useful to troduce the cocept of observble opertor. A Hermt (or self-djot) opertor ˆ s observble () opertor f ts orthoorml ege-sttes form bss... 5

53 .5.E Opertors tht re ot ssocted to me vlues I the prevous secto we ddressed opertors ssocted to ) clsscl physcl quttes f(x) tht depeds o the posto x, d ) physcl quttes g(p) tht deped o the mometum tur out to be hermt (self-djot) opertors. Here we explore buldg opertors ssocted to clsscl physcl quttes lke xp. Let s clculte Nvely, let s cll tht qutty *( x ) [ X P ] ( x ) dx xp v. e. oe would expect tht the clculto of the tegrl wll gve postve umber d thus reflect qutty ssocted to clsscl mesuremet of xp. We wll see below tht ths ssumpto s correct. As mtter of fct we wll relze tht X P s ot Hermt. xp v *( x ) [ X P ] ( x ) dx *(x) [ x d (x dx ) ] dx Rerrgg the order of the terms, x *(x) [ d (x dx ) ]dx - dx - dv Itegrtg by prts d [x *(x) ] } [ ) (x ]dx [ *(x) ] } [ ) (x ]dx - [x d *(x) ] } [ ) dx (x ]dx 53

54 - [x d *(x)] [ ) dx (x ]dx * + [x d *(x) ] (x) dx dx + [x d (x)] * (x) dx dx + [ X P ] * (x) (x) dx xp v + + [ * ( x ) [ X P ] (x) dx ] * * xp v () Ths result dctes tht xp v s ot rel qutty. Let s put the result () more explctly terms of opertors. Expresso () c be wrtte more explctly, *(x) [ P X ] ( x )dx = + [ [ X P ] (x) dx ] * * ( x ) X P = + X P ] * X P = + X P X P = + ( X P ) Although ths hs bee demostrted for detcl br d ket, the procedure wth dfferet br d ket would gve smlr result. Tht s, X P = + ( X P ) (3) whch shows more explctly tht the opertor X P s ot hermt 54

55 Also, sce ( A B) = B A X P = + X P = + P X (4) P X More properly, ths result should be expressed s, X P = I + P X where I s the detty opertor; Wht to do the f qutty lke xp s preset the clsscl Hmlto? How to buld the correspodg qutum Opertor? The result (4), tht X P P X, dctes tht dfferet qutum mechcs opertors my correspod to equvlet clsscl quttes For exmple x p, p x, (/)( x p + p x ) re equvlet clssclly. Stll, three dfferet opertors X P, P X, (/)( X P + P X ) c be ssocted to the sme clsscl qutty. The gudce to select the proper opertor s to mpulte the clsscl qutty such tht the resultg opertor result Hermt opertor. I ths prtculr cse of x p, t turs out tht, (/) ( X P + P X ) (5) fulflls.6 The commuttor I geerl, two opertor do ot commute; tht s, of X d P, for exmple). A B s dfferet th B A (tht s the cse The commuttor betwee two opertors s defed s, [ A, B ] AB B A (6) Accordg to (4), [ X, P ] (7).6.A The Heseberg ucertty relto I ths secto we wll be usg terms lke A ( A A ). Wht does ths term me? To get fmlr frst wth tht termology, let s work out explctly couple of terms. (You my skp the followg prgrph d go drectly to expresso (9)). 55

56 The meg of ( P - < p > ) Frst, let s expd the bsed formed by ts egesttes = p p dp () (Note: To dcte the mometum stte p sometmes I m usg the otto p. Tht s p = p. ) P = P = ( P p ) p dp ( p p ) p dp () Sce < p > s costt, the < p > = < p > p p dp () () () P - < p > = [ P - < p > ] = [ P - < p > ] = ( p - < p > ) p p dp ( p - < p > ) p p dp ( p - < p > ) p p dp We relze ths s the procedure to clculte stdrd devtos. I effect, [ P - < p > I ] = = = ( p - < p > ) p p dp ( p - < p > ) p p dp ( p - < p > ) p dp (8) Probblty wth whch the term ( p - < p > ) occurs For two gve observble opertors A d B, let s defe the correspodg stdrd devto (the sttstcs s tke from esemble chrcterzed by the wvefucto, ( A A ) (9) A 56

57 B ( B B ) To smplfy the otto, let s work wth the Hermt opertors d b defed s, A A d b B B () Notce, [, b] [ A, B ] () A d B c the be expressed s, σ ψ ψ ψ ψ () A σ B bψ bψ ψ b ψ Cosder the ot Hermt opertor C, C λ b where s rel costt (3) Notce: C λ b, d C C C C (4) ψ ( λ b )( λ b ) ψ ( b [,b ] ) Usg (), σ A [ A,B ] (5) σ B Notce tht the term [ A, B] must be purely mgry umber. The fucto f = f () defed s, f () σ A stsfes, ccordg to (5), f ( ) [ A,B ] (6) σ B I ddto f " ( ) B. Therefore the vlue of t whch f ' ( ) s mmum; such vlue s, 57