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1 Ths cotet has bee dowloaded from IOPscece. Please scroll dow to see the full text. Dowload detals: IP Address: Ths cotet was dowloaded o 4/09/08 at 04:0 Please ote that terms ad codtos aly. You may also be terested : Hgher-Order Lagraga Equatos of Hgher-Order Motve Mechacal System Zhao Hog-Xa, Ma Sha-Ju ad Sh Yog Why ot eergy coservato? Shaw Carlso Dervg relatvstc mometum ad eergy: II Sebastao Soego ad Massmo P Phase-sace study of a sg quatum artcle a costat magetc feld L M Neto The Adabatc Ivarace of the Acto Varable Classcal Dyamcs Clve G Wells ad Stehe T C Slos Aalytcal mechacs ad feld theory: dervato of equatos from eergy coservato N A Vourov A exaso term Hamlto's equatos M. D. Roberts Feld theory wth hgher dervatves-hamltoa structure L M C Coelho de Souza ad P R Rodrgues Hamltoa formulato of the evoluto of movemets over clusters of oteractg artcles M Adrews

2 Part I Fudametals

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4 IOP Publshg A Itroducto to Quatum Theory Jeff Greeste Chater The classcal state I the frst quarter of the 0th cetury t was dscovered that the laws of moto formulated by Galleo, Newto, Lagrage, Hamlto, Maxwell, ad may others were adequate to exla a wde rage of heomea volvg electros, atoms, ad lght. After a great deal of effort, a ew theory (together wth a ew law of moto) emerged 94. That theory s ow as quatum mechacs, ad t s ow the basc framewor for uderstadg atomc, uclear, ad subuclear hyscs, as well as codesed-matter hyscs. The laws of moto (due to Galleo, Newto, ) whch receded quatum theory are referred to as classcal mechacs. Although classcal mechacs s ow regarded as oly a aroxmato to quatum mechacs, t s stll true that much of the structure of the quatum theory s herted from the classcal theory that t relaced. So we beg wth a lghtg revew of classcal mechacs, whose formulato begs (but does ot ed!) wth Newto s law F ma.. Baseball, F ma, ad the rcle of least acto Tae a baseball ad throw t straght u the ar. After a fracto of a secod, or erhas a few secods, the baseball wll retur to your had. Deote the heght of the baseball, as a fucto of tme, as x(t). If we mae a lot of x as a fucto of t, the the curve (we wll call t a trajectory ) the x t lae wll, a uform gravtatoal feld assumg ar resstace s eglgble, have the form of a arabola. There are a fte umber of ossble trajectores. Whch oe of these the baseball actually follows s determed by the mometum of the baseball at the momet t leaves your had. However, f we requre that the baseball returs to your had exactly Δt secods after leavg t, the there s oly oe trajectory that the ball ca follow. For a baseball movg a uform gravtatoal feld t s a smle exercse to determe ths trajectory exactly, but we would le to develo a method whch ca be aled do:0.088/ ch - ª IOP Publshg Ltd 07

5 A Itroducto to Quatum Theory to a artcle movg ay otetal feld V(x). So let us beg wth Newto s law F ma, whch s actually a secod-order dfferetal equato d m x dv (.) It s useful to reexress ths secod-order equato as a ar of frst-order equatos m d dv (.) where m s the mass ad s the mometum of the baseball. We wat to fd the soluto of these equatos such that xt ( 0) X ad xt ( 0 +Δ t) Xf, where X ad X f are, resectvely, the (tal) heght of your had whe the baseball leaves t, ad the (fal) heght of your had whe you catch the ball. Wth the advet of the comuter, t s ofte easer to solve equatos of moto umercally, rather tha struggle to fd a aalytc soluto whch may or may ot exst (artcularly whe the equatos are o-lear). Although the object of ths secto s ot really to develo umercal methods for solvg roblems baseball, we wll, for the momet, roceed as though t were. To mae the roblem sutable for a comuter, dvde the tme terval Δt to N smaller tme tervals of durato ϵ ΔtN /, ad deote, for 0,,, N, t t0 + ϵ, x x( t), ( t), x X, x X 0 N f A aroxmato to a cotuous trajectory x(t) s gve by the set of ots { x } coected by straght les, as show fgure.. We ca lewse aroxmate dervatves by fte dffereces,.e. xt ( + ) xt ( ) x+ x t t ϵ ϵ d t ( + ) t ( ) + t t ϵ ϵ (.3) d x ϵ t t t d t t t t ( x+ x) ( x x) ϵ ϵ ϵ We wll allow these ostos to be dfferet, geeral, sce you mght move your had to aother osto whle the ball s flght. -

6 A Itroducto to Quatum Theory ad tegrals by sums Fgure.. A dscrete aroxmato to the cotuous ath x(t). t0 t0+δt N d tf( t) ϵft ( ) (.4) where f(t) s ay fucto of tme. As we ow from elemetary calculus, the rghthad sde of (.3) ad (.4) equals the left-had sde the lmt that ϵ 0, whch s also ow as the cotuum lmt. We ca ow aroxmate the laws of moto, by relacg tme dervatves (.) by the corresodg fte dffereces, ad fd x x ϵ m d Vx ( ) ϵ (.5) These are teratve equatos. Gve osto x ad mometum at tme t t,we ca use (.5) to fd the osto ad mometum at tme t t +. The fte dfferece aroxmato of course troduces a slght error; x + ad +, comuted from x ad by (.5) wll dffer from ther exact values by a error of order ϵ. Ths error ca be made eglgble, rcle, by tag ϵ suffcetly small. It s the ossble to use the comuter to fd a aroxmato to the trajectory by the shootg method. The method s to mae a guess for the tal mometum 0 P0, ad the use (.) to solve for x,, x,, ad so o, utl xn, N.If xn Xf, the sto; the set { x } s the (aroxmate) trajectory. If ot, mae a dfferet guess 0 P 0, ad solve aga for { x, }. By tral ad error, oe ca evetually coverge o a tal choce for 0 such that xn Xf. For that choce of tal mometum, the corresodg set of ots { x }, coected by straght-le segmets, gves the aroxmate trajectory of the baseball. Ths rocess s llustrated fgure.. Now let us retur to the secod-order form of Newto s laws, wrtte equato (.). Remarably, F ma ca be exressed as the codto that a certa fucto s statoary. Aga usg (.3) to relace dervatves by fte dffereces, the equato F ma at each tme t becomes -3

7 A Itroducto to Quatum Theory ma { } m x+ x x x d V( x) ϵ ϵ ϵ (.6) The equatos have to be solved for,,, N, wth x 0 X ad x N X f et fxed. Now otce that equato (.6) ca be wrtte as a total dervatve + m x x d ( ) ( + m x x ) ϵvx ( ) 0 (.7) ϵ ϵ so that F ma at some tme t ca be terreted as a codto that the fucto bracets { } should be statoary wth resect to varatos the osto x. But what about all other tmes? We ow troduce a very mortat exresso, crucal both classcal ad quatum hyscs, whch s ow as the acto of the trajectory. The acto s a fucto whch deeds o all the ots { x}, 0,,, N of the trajectory, ad ths case t s N + S x m x x ( ) [{ }] ϵvx ( ) (.8) ϵ 0 The Newto s law F ma ca be restated as the codto that the acto fuctoal S[{ x}] s statoary wth resect to varato of ay of the x (excet for the edots x 0 ad x N, whch are held fxed). I other words d x S x d [{ }] d Fgure.. The shootg method for fdg a trajectory wth fxed ed ots. d N 0 + m x x ( ) ϵvx ( ) ϵ + m x x ( ) ( + m x x ) ϵvx ( ) ϵ ϵ ϵ{ ma( t ) + F ( t )} 0 for,,, N (.9) Ths set of codtos s ow as the rcle of least acto. It s the rcle that the acto S s statoary at ay trajectory { x } satsfyg the equatos of moto F ma, equato (.6), at every tme { t }. -4

8 A Itroducto to Quatum Theory. Euler Lagrage ad Hamlto s equatos I bref, the Euler Lagrage equatos are the secod-order form of the equatos of moto (.), whle Hamlto s equatos are the frst-order form (.). I ether form, the equatos of moto ca be regarded as a cosequece of the rcle of least acto. We wll ow re-wrte those equatos a very geeral way, whch ca be aled to ay mechacal system, cludg those whch are much more comlcated tha a baseball. We beg by wrtg where ad where N S[{ x}] ϵl[ x, x ] (.0) 0 Lx [, x ] mx V( x) (.) x x + x ϵ (.) L[ x, x ] s ow as the Lagraga fucto. The the rcle of least acto requres that, for each, N, d ϵ x S x d 0 [{ }] d Lx [, x ] ϵ N 0 N [, ] + ϵ x Lx x Lx x [, ] x 0 (.3) ad, sce ϵ (.4) ϵ 0 otherwse ths becomes x Lx x [, ] ϵ x Lx [, x ] Lx [, x ] 0 (.5) x Recallg that x x( t ), ths last equato ca be wrtte Lx [, x ] Lx [, x] Lx [, x] x ϵ x x t t t t t tϵ 0 (.6) -5

9 A Itroducto to Quatum Theory Ths s the Euler Lagrage equato for the baseball. It becomes smler whe we tae the ϵ 0 lmt (the cotuum lmt). I that lmt, we have x x N x ϵ xt () S ϵl[ x, x ] S d t L[ x( t), x ( t)] + t0+δt where the Lagraga fucto for the baseball s t0 (.7) Lxt [ ( ), xt ( )] mx () t Vxt [ ()] (.8) ad the Euler Lagrage equato, the cotuum lmt, becomes L xt () d L 0 (.9) d t xt ( ) I tag artal dervatves of L wth resect to x ad ẋ, t s mortat to uderstad that we have to reted that x ad ẋ are deedet varables whch have othg to do wth oe aother, deste the fact that ẋ s really the tme dervatve of x. That s just the meag of the artal dervatve otato. For the Lagraga of the baseball, equato (.8), the relevat artal dervatves are L d Vxt [ ( )] xt () d xt ( ) (.0) L mx () t xt () whch, whe substtuted to equato (.9) gve m x dv + 0 (.) t Ths s smly Newto s law F ma, the secod-order form of equato (.). We ow wat to rewrte the Euler Lagrage equato frst-order form. Of course, we already ow the aswer, whch s equato (.), but let us forget ths aswer for a momet, order to troduce a very geeral method. The reaso the Euler Lagrage equato s secod-order the tme dervatves s that L/ x s frstorder the tme dervatve. So let us defe the mometum corresodg to the coordate x to be L x (.) Ths gves as a fucto of x ad ẋ, but, alteratvely, we ca solve for ẋ as a fucto of x ad,.e. x x ( x, ) (.3) -6

10 A Itroducto to Quatum Theory Next, we troduce the Hamltoa fucto H[, x] x ( x, ) L[ x, x ( x, )] (.4) Sce ẋ s a fucto of x ad, H s also a fucto of x ad. The reaso for troducg the Hamltoa s that ts frst dervatves wth resect to x ad have a remarable roerty; amely, o a trajectory satsfyg the Euler Lagrage equatos, the x ad dervatves of H are roortoal to the tme dervatves of ad x. To see ths, frst dfferetate the Hamltoa wth resect to, H + x x xx (, ) L x (, x) x where we have aled (.). Next, dfferetatg H wth resect to x, H xx (, ) L L x (, x) x x x x x L x (.5) (.6) Usg the Euler Lagrage equato (.9) (ad ths s where the equatos of moto eter), we fd H d L x x d (.7) Thus, wth the hel of the Hamltoa fucto, we have rewrtte the sgle secod-order Euler Lagrage equato (.9) as a ar of frst-order dfferetal equatos H (.8) d H x whch are ow as Hamlto s equatos. For a baseball, the Lagraga s gve by equato (.8), ad therefore the mometum s L mx x (.9) Ths s verted to gve x x (, x) (.30) m -7

11 A Itroducto to Quatum Theory ad the Hamltoa s H x ( x, ) L[ x, x ( x, )] m Vx ( ) m m + m Vx ( ) (.3) Note that the Hamltoa for the baseball s smly the etc eergy lus the otetal eergy;.e. the Hamltoa s a exresso for the total eergy of the baseball. Substtutg H to Hamlto s equatos, oe fds + t m Vx ( ) d m d + t x m Vx dv ( ) d whch s smly the frst-order form of Newto s law (.)..3 Classcal mechacs a utshell (.3) All the machery of the least acto rcle, the Lagraga fucto, ad Hamlto s equatos, s overll the case of a baseball. I that case, we ew the equato of moto from the begg. But for more volved dyamcal systems, volvg, say, wheels, srgs, levers, ad edulums, all couled together some comlcated way, the equatos of moto are ofte far from obvous, ad what s eeded s some systematc way to derve them. For ay mechacal system, the geeralzed coordates { q } are a set of varables eeded to descrbe the cofgurato of the system at a gve tme. These could be a set of Cartesa coordates of a umber of dfferet artcles, or the agular dslacemet of a edulum, or the dslacemet of a srg from equlbrum, or all of the above. The dyamcs of the system, terms of these coordates, s gve by a Lagraga fucto L, whch deeds o the geeralzed coordates { q } ad ther frst tme dervatves { q }. Normally, o-relatvstc mechacs, we frst secfy. The Lagraga L { q, q} Ketc Eergy Potetal Eergy (.33) Oe the defes. The acto S d t L { q, q} (.34) From the least acto rcle, followg a method smlar to the oe we used for the baseball, we derve -8

12 A Itroducto to Quatum Theory 3. The Euler Lagrage equatos L q d L q 0 (.35) These are the secod-order equatos of moto. To go to the frst-order form, frst defe 4. The geeralzed mometa L q (.36) whch ca be verted to gve the tme dervatves q of the geeralzed coordates terms of the geeralzed coordates ad mometa q q { q, } (.37) Vewg q as a fucto of ad q, oe the defes 5. The Hamltoa { } { } H q, q L q, q (.38) Usually the Hamltoa has the form H [, q] Ketc Eergy + Potetal Eergy (.39) Fally, the equatos of moto frst-order form are gve by 6. Hamlto s equatos H q H q (.40) If the otetal eergy does ot deed exlctly o tme, the Hamlto s equatos mly that H s tme-deedet, dh H q + H q t t + q q t t t t 0 (.4) as we would exect f H s the total eergy of the system. -9

13 A Itroducto to Quatum Theory Examle: the lae edulum Our edulum s a mass m at the ed of a weghtless rgd rod of legth l, whch vots a lae aroud the ot P. The geeralzed coordate, whch secfes the osto of the edulum at ay gve tme, s the agle θ betwee the rod ad the vertcal axs.. The Lagraga L θ ml ( V0 mgl cos( θ)) (.4) where V 0 s the gravtatoal otetal at the heght of ot P, whch the edulum reaches at θ π/. Sce V 0 s arbtrary, we wll just set t to V0 0.. The acto 3. The Euler Lagrage equatos We have t S θ + θ ml mgl cos( ) (.43) t0 L θ θ mgl s L θ θ ml (.44) ad therefore ml θ + mgl s θ 0 (.45) s the Euler Lagrage form of the equatos of moto. 4. The geeralzed mometum L θ θ ml (.46) 5. The Hamltoa Isert to θ ml (.47) θ θ H + θ ml mgl cos( ) (.48) -0

14 A Itroducto to Quatum Theory to obta H mgl cos( θ) (.49) ml 6. Hamlto s equatos θ H ml H mgl s θ θ whch are easly see to be equvalet to the Euler Lagrage equatos..4 The classcal state (.50) Predcto s rather mortat hyscs, sce the oly relable test of a scetfc theory s the ablty, gve the state of affars at reset, to redct the future. Stated rather abstractly, the rocess of redcto wors as follows. By a slght dsturbace ow as a measuremet, a object s assged a mathematcal reresetato whch we wll call ts hyscal state. The laws of moto are mathematcal rules by whch, gve a hyscal state at a artcular tme, oe ca deduce the hyscal state of the object at some later tme. The later hyscal state s the redcto, whch ca be checed by a subsequet measuremet of the object. From the dscusso so far, t s easy to see that what s meat classcal hyscs by the hyscal state of a system s smly ts set of geeralzed coordates ad the geeralzed mometa a { q, a }. These are suosed to be obtaed, at some tme t 0, by the measuremet rocess. Gve the hyscal state at some tme t, the state at t + ϵ s obtaed by the rule a a H q ( t + ϵ) q ( t) + ϵ a t H a( t + ϵ) a( t) ϵ a q t (.5) I ths way, the hyscal state at ay later tme ca be obtaed ( rcle) to a arbtrary degree of accuracy, by mag the tme-ste ϵ suffcetly small (or else, f ossble, by solvg the equatos of moto exactly). Note that the coordates { q a } aloe are ot eough to secfy the hyscal state, because they are ot suffcet to redct the future. Iformato about the mometa { a } s also requred. The sace of all ossble a { q, a } s ow as hase sace. For a sgle artcle movg three dmesos, there are three comoets of osto ad three comoets of mometum, so the hyscal state s secfed by sx umbers ( x, y, z, x, y, z ), whch ca be vewed as a ot sx-dmesoal hase sace. Lewse, the hyscal state of a system of N artcles cossts of three coordates -

15 A Itroducto to Quatum Theory for each artcle ( 3N coordates all), ad three comoets of mometum for each artcle ( 3N mometum comoets all), so the state s gve by a set of 6N umbers, whch ca be vewed as a sgle ot 6N-dmesoal sace. As we wll see the ext chaters, classcal mechacs fals to redct correctly the behavor of both lght ad matter at the atomc level, ad s relaced by quatum mechacs. But classcal ad quatum mechacs have a lot commo: they both assg hyscal states to objects, ad these hyscal states evolve accordg to dfferetal equatos whch are frst order the tme dervatves. The dfferece les maly the cotrast betwee a hyscal state as uderstood by classcal mechacs, the classcal state, ad ts quatum couterart, the quatum state. Ths dfferece wll be exlored the ext few chaters. Problems. Two ot-le artcles movg three dmesos have masses m ad m resectvely, ad teract va a otetal V ( x x ). Fd Hamlto s equatos of moto for the artcles.. Suose, stead of a rgd rod, the mass of the lae edulum s coected to ot P by a weghtless srg. The otetal eergy of the srg s L ( L) 0, where L s the legth of the srg, ad L 0 s ts legth whe ot dslaced by a exteral force. Choosg L ad θ as the geeralzed coordates, fd Hamlto s equatos. -