3/21/2017. Commuting and Non-commuting Operators Chapter 17. A a

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1 Commutig ad No-commutig Operators Chapter 17 Postulate 3. I ay measuremet of the observable associated with a operator A the oly values that will ever be observed are the eige values, a, which satisfy the eige value equatio; A a e. g. H E. If the system is a eigestate of A with eigevalue a, the ay measuremet of the observable quatity A of that state will yield a value a, oly. I quatum mechaics two observables A ad B of a quatum system ca be predicted (foud) exactly oly if the outcomes of the measuremets of the two observables are idepedet of the order i which they are determied. Classical mechaics Ucertaity i measuremets is oly limited by capabilities of techique, measurig istrumet precisio ad the ivestigator skill. Further, i quatum systems there exists a limitatio o the ucertaity associated with some simultaeous measuremets that ca be made, regardless of the methodology employed. Classical mechaics allows predictio of the iformatio with o limits o the amout of iformatio obtaiable. 1

2 Quatum mechaics. Two observables ca be kow simultaeously with high accuracy (i some cases). But i other cases the two observables measured would have a iate ucertaity. This ucertaity caot be removed by the improvemet of the techique of measuremet. If operators A ad B are associated with observed values (measuremets) of ad, they ca be measured simultaeously by experimet oly if the measuremet process does ot chage the system. If the first measuremet chages the system, the secod measuremet will ot be performed o the same system dealt at the first measuremet. The positio (x) of a electro i a atom, if measured by usig a probe, such as a light ray will chage the velocity of the electro by electro iteractig with the light ray. So if the velocity (v), of the electro is measured after measurig the positio, it is the velocity of a differet system!! Quatum systems are described by a wave fuctio, say (x); assume they are eigefuctios of operators A ad B. The outcome of the measuremet of property A followed by measuremet of property B would the be as follows; i stages; # B [ A ( x )] i.e. B [ A B[ B The secod measuremet (property B) would ot chage the wave fuctio (i.e. the state of the system) oly if is a eige fuctio of B. B[ A Now, Reversig the order of operatios o yields; A [ B ( x )] A [ ( x )] A ( x ) ( x ) i.e. x ( ) B [ A B Makig AB [ BA [

3 AB [ BA [ The above, is made possible because the operators did ot chage the wavefuctio., i.e. the state of the system. A a B b The secod measuremet will chage the wave fuctio (state) if the wave fuctio is NOT a eige fuctio of BOTH operators. The requiremet for beig able to be simultaeously measure two observables is that the two operators must be eige operators of the same wave fuctio. *Test is: AB [ BA [ A[ Bf B [ Af ] ABf BA f [ AB BA ] f [ A, B ] f Commutator of A ad B. If [ AB BA ] f [ A, B ] f 0 A ad B commutes. A[ Bf B [ Af ABf BAf Note; aother form If ABf [ BAf [ ] [ ABf, ] 0 the value of the commutator is ot equal to zero, values for properties A ad B caot be measured simultaeously ad exactly (with high accuracy), A ad B are o-commutig operators. [ A, B ] [ B, A] Study Example Problem

4 No-commutig operators - example R z R y R z Ster-Gerlach Experimet: A dh oly 0 dz R y R z R y R z R y R y R z R z R y R y R z -R z R y 0 [R y,r z ] 0 A = measurig the z compoets. Ag atom has a sigle upaired electro mag. momet. Atoms oriet oly i two directios w.r.t. field. Measurig i z directio has oly spots, eige fuctios, i.e two states. The two states were described by eige fuctios, ad. The umber of spi states for silver atom are oly, ad the complete set has oly wave fuctios ad. Equal itesity beams The iitial acceptable wavefuctio describig silver atom ca be writte as; Equal itesity of beams; 1 c c 1 A = measurig the z compoets. c1 c 1/ 4

5 A B = measurig the x compoets. B A B A ad B do ot commute. i.e. operators for compoets of magetic momets do ot commute. BA AB A B B A A ad B do ot commute. i.e. operators for compoets of magetic momets do ot commute. 5

6 Heiseberg Ucertaity Priciple. c c c 3 4 A commo statemet of the ucertaity priciple is that the positio ad mometum of a quatum mechaical particle caot be kow exactly ad simultaeously. It is because; [ x, p] 0 A free particle described by ( xt, ) Ae i( kxt ) c c c 3 4 i( kx) Simplified by settig t = 0 ad = 0 Ae Evaluatio of commutator [x,p] AB [ BA [ xp [ px [ d i kx p i Ae kae dx i( kx) x xae ( ) i( kx) For simplicity the property i oe directio cosidered. xp [ px [ d x[ kae ] p[ xae ] kxae i xae dx i( kx) i( kx) i( kx) kxae iae xkae 0 i( kx) i( kx) i( kx) i( kx) 6

7 Heiseberg Ucertaity Priciple. The values determied are ot discrete, hece there are built i ucertaities, x ad p, for example. Ucertaity Priciple; Illustratio of Ucertaity Priciple for 1D Box. Normalized wave fuctios; Alterate defiitio = variace variace = mea of squares square of meas variace (See Example Problem 17.5 for proof) Note: If a system (a state described by a wavefuctio) is a eigestate of the total eergy operator (Hamiltoia, H ) ad if a property P, of which the operator P, does ot commute with H, the P caot be kow accurately for that system. If a operator A does ot commute with aother operator, B, the eigefuctios of A are ot eigefuctios of B ad vice versa. If a operator A commutes with aother operator, B, the eigefuctios of A are eigefuctios of B ad vice versa. The operators A ad B may differ by a multiplicative factor. 7