Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats of th simpl harmonic oscillator and th classical stats in which a pndulum swings back and forth is not compltly clar Whil it is tru that th high n limit of th x-spac probability distribution in th nrgy ignstat n> approachs th tim-avragd probability distribution for th classical systm, th probability distribution for th quantum stat is tim-indpndnt: it is a stationary stat On th othr hand, th classical motion obsrvd by anyon looking at a pndulum is manifstly on with a tim dpndnt probability distribution! Our aim in this sction is to construct quantum stats that go in th classical limit to th swinging pndulum stat Evidntly, ths quantum stats, having tim-varying probability distributions, cannot b ignstats of th Hamiltonian Th motion of a classical oscillator is bst undrstood in phas spac For th on-dimnsional oscillator th phas spac is two-dimnsional, with position x along th x-axis, momntum p along th y-axis Th swinging pndulum dscribs a circl (in suitably scald units) cntrd at th origin in th (x,p) plan, with constant angular vlocity ω Th (dimnsionlss) quantum variabls corrsponding to (x, p) ar (ξ, π ), whr ξ = mω/ x, π = bp/ =id / dξ (s th arlir lctur on th simpl harmonic oscillator) Now, th annihilation oprator a is dfind by a ( ξ iπ) / = +, so th classical circular motion in th(x,p) plan vidntly corrsponds to th xpctation valu of a rotating at frquncy ω in th complx plan In fact, this is prcisly th bhavior of th oprator a in th Hisnbrg rprsntation! Thrfor, to find a stat approximating th classical bhavior, w nd to xamin ignstats of a Som Exponntial Oprator Algbra As a prliminary task, w shall stablish som oprator idntitis that prov usful both in undrstanding th ignstats of a and in latr work Suppos that th commutator of two oprators A, B, = c, whr c commuts with A and B, usually it s just a numbr, for instanc or i Thn 3 3 A, = A, + + ( λ /!) B + ( λ /3!) B + = λc ( ) ( ) 3 = λc+ λ /! Bc+ λ /3! 3B c+
That is to say, th commutator of A with is proportional to itslf That is rminiscnt of th simpl harmonic oscillator commutation rlation [ Ha, ] = ωa which ld dirctly to th laddr of ignvalus of H sparatd by ω Will thr b a similar laddr of ignstats of A in gnral? Assuming A (which is a gnral oprator) has an ignstat a > with ignvalu a, A a>= a a> Applying[ A, ] = λc to th ignstat a > : A a>= A a>+ λc a>= ( a+ λc) a> Thrfor, unlss it is idntically zro, a> is also an ignstat of A, with ignvalu a + λc W conclud that instad of a laddr of ignstats, w can apparntly gnrat a whol continuum of ignstats, sinc λ can b st arbitrarily! W shall soon rturn to this puzzling rsult, with an xampl To find mor oprator idntitis, prmultiply [ A, ] = λc by to find: = A+ λc [, ] A = A+ λ A B This idntity is only tru for oprators A, B whos commutator c is a numbr (Wll, c could b an oprator, providd it still commuts with both A and B) Our nxt task is to stablish th following vry handy idntity, which is also only tru if [A,B] commuts with A and B: = A+ B A B Th proof (du to Glaubr, givn in Mssiah) is as follows:, Ax Bx Tak f ( x) =, Ax Bx Ax Bx df / dx = A + B Bx Bx ( )( ) ( )( [ ] ) = f x A + B = f x A+ x A, B + B It is asy to chck that th solution to this first-ordr diffrntial quation qual to on at x = 0 is
3 ( ) ( + ) x [ A, B] x A B f x = so taking x = givs th rquird idntity, A + B A B =, It also follows that B A A B [ A, B ] = providd as always that [A,B] commuts with A and B Eignstats of th Annihilation Oprator Rcall that th annihilation oprator a applid to a simpl harmonic oscillator nrgy ignstat movs down on stp of th laddr, and sinc th nrgy ignvalus cannot b ngativ, â annihilats th lowst stat, a ˆ 0>= 0 It is worth rcalling th wav function rprsntation of this rsult, with ˆ ξ = mω/ xˆ, ˆ π = bpˆ / =id / dξ aˆ = ˆ ξ + i ˆ π /,, ( ) and th solution is ( ) 4 mω d aˆ 0>= ξ + f ( ξ) = 0 π dξ f ξ / ξ =, suitably normalizd What about ignstats of a with nonzro ignvalus? Lt us writ 4 4 mω d mω ξ + f ξ = λ π dξ π ( ) f ( ξ ) so w v absorbd th normalization factor into th ignvalu, and w can now cancl it from both sids, to giv df ( ξ ) = ( λ ξ) f ( ξ) dξ asily intgratd to giv f ( ξ ) ( λξ) = C / (Strictly spaking, I should not quat th vctor a ˆ 0 with a function of ξ, which is not a vctor I should hav writtn x a ˆ 0 on th lft, and it should b undrstood that th function of ξ is to b translatd into a function of x ) Th ignstats of th annihilation oprator aˆ λ >= λ λ >
4 with nonzro ignvalus hav th sam wav function as th ground stat, but cntrd somwhr ls at ξ = λ, in fact, whr λ is th ignvalu of a But not that ths ignstats ar not dlta functions, lik th ignstats of th position oprator thy ar, lik th ground stat of th oscillator, stats with th last possibl uncrtainty, p x=, as compact as possibl in th (x,p) phas spac Ths stats ar also not orthogonal to ach othr, in fact any two of thm hav nonzro ovrlap That dos not contradict arlir rsults on Hrmitian oprators, bcaus a isn t Hrmitian Thr is on big diffrnc btwn ths stats λ and th ground stat 0 : th ground stat is also an ignstat of th Hamiltonian, so th only tim dpndnc is in th ovrall phas factor That is mphatically not th cas for stats with nonzro λ: thy corrspond to th pndulum having bn pulld to on sid, and it will bgin to swing This is th motion w ar going to analyz using th quantum formalism Th Translation Oprator W hav stablishd that if A a a a >= >and [ A, B] = c, a numbr, thn a> is an ignstat of A with ignvalu a + λc How dos this rlat to th continuum of ignstats of th annihilation oprator w hav just found? In othr words, if w tak A to b th annihilation oprator â, is thr an oprator corrsponding to B so that translats th cntr of th wav function a distanc λ in x-spac? Th answr is ys, and it is asily found from th Taylor sris: In an obvious notation: d η d f ( ξ + η) = f ( ξ) + η f ( ξ) f ( ξ) dξ +! dξ + η d ξ ( ξ + η d ) = ( ξ ) f f so th appropriat oprator B is just d/ dξ Now w v discovrd th oprator that shifts a wav function along th x-axis by a prscribd amount, th so-calld translation oprator, w can us it to transform th stat 0 to th stat λ : d λ d ξ λ >= 0 > Now, from th introductory sction abov (or th prvious lctur), d a a = iπ = dξ so th translation oprator can b writtn in trms of th annihilation and cration oprators: ( aa ) λa a / λ >= 0>= 0 >
5 whr in th scond stp w hav usd =, Sinc a 0> = 0, a 0> =, so A+ B A B / λa λ >= 0 > From this, w can xprss λ as a sum ovr th ignstats of th Hamiltonian Expanding th xponntial, ( λa ) / λ >= + λa + + 0 >! and rcalling that th normalizd nrgy ignstats ar w find ( a ) n >= 0 > n! 3 / λ λ λ >= λ 0 >+ λ >+ >+ 3>+! 3! n It is asy to chck that this stat is corrctly normalizd Tim Dvlopmnt of an Eignstat of a Now that w hav xprssd th ignstat λ> as a sum ovr th ignstats n> of th niωt Hamiltonian, finding its tim dvlopmnt is straightforward: sinc nt ( ) >= n>, iωt 3 3iωt / iωt λ λ λ( t) >= 0 >+ λ >+ >+ 3 >+,! 3! which can b writtn iωt / λ a λ( t) >= 0 > Having stablishd th tim dpndnc, w can now go back to ξ-spac, using a = ξ iπ / = ξ d / dξ / to find ( ) ( ) () iωt i t / ( d/ d )/ iωt ω / / ( / ) d/ d iωt λ λ ξ ξ λ λξ λ ξ /4 λ t >= 0>= 0>
6 whr w hav usd A+ B A B [ A, B] = in th scond stp Th Taylor sris, and hnc th shift oprator, will still b valid for a shift by a complx amount, so th wavfunction is: 4 i t m iωt ω ω ( / ) / i t / / ξ λ ω λξ /4 ψ λ ( ξ, t) = π