From classical to quantum and back

Transcription

1 FROM CLASSICAL TO QUANTUM AND BACK Beyond The Bsics I: Chpter 5 From clssicl to quntum nd bck In the first four chpters of this book we reviewed Newtonin mechnics nd Einstein s expnsion of it into the reltivistic regime; introduced the vritionl clculus nd used it to present the elegnt formultion of mechnics known s Hmilton s principle of sttionry ction; nd derived Lgrnge s equtions of motion. However, in spite of its power, clssicl mechnics, even extending it into the domin of specil reltivity, hs its limittions; it rises s specil cse of the vstly more comprehensive theory of quntum mechnics. Where does clssicl mechnics fll short, nd why is it limited? The key turns out to be Hmilton s principle. We will show how this principle comes bout s specil cse of the lrger theory. And so, since we cn use Hmilton s principle to derive clssicl mechnics, we will rech good understnding of when clssicl mechnics is vlid nd when we hve to use the full pprtus of quntum theory. So this chpter is not prt of

2 BEYOND THE BASICS clssicl mechnics per se; reders short on time cn skip it without compromising their preprtion for future chpters. But it would be too bd to skip this chpter forever, becuse here we will understnd how clssicl mechnics comes bout, nd how it fits into the grnd scheme of physics. We do this by showing how Hmilton s principle, tht most compct, elegnt sttement of clssicl mechnics, which emerges rther mysteriously in Chpter 4, is in fct nturl consequence of quntum mechnics in certin limit. This chpter sets clssicl mechnics in context. We begin the chpter with the behvior of wves in clssicl physics, nd then show results of some criticl experiments with light nd toms, upsetting trditionl notions of light s wves nd toms s prticles. We proceed to give brief review of Feynmn s sum-over-pths formultion of quntum mechnics, which describes the ctul behvior of light nd toms, nd then show tht Hmilton s Principle nturlly emerges in certin limiting cse. Once equipped with Hmilton s Principle, we lredy know from Chpter 4 how to derive clssicl motion. 5.1 Clssicl wves A string with uniform mss per unit length µ is held in horizontl position under uniform tension T. Wht hppens if we disturb the string? In prticulr, suppose tht t time t = 0 we give the string some prticulr shpe y(x), nd some velocity distribution y(x, t)/ t t=0, where x is the horizontl coordinte long the string nd y is the trnsverse displcement (see Figure 5.1). Our gol is to find y(x, t), the shpe of the string t ny lter time. Consider very smll slice of string of length x nd mss m = µ x, s shown in Figure 5.2(). We ignore grvity, so the only forces cting on this piece re the string tensions to the right nd to the left of it. If the string is displced from equilibrium, the slice will generlly be slightly curved. The two tension forces on the right nd left therefore pull in slightly different directions, s shown in Figure 5.2(b), so the resulting unblnced force cuses m to ccelerte. For smll verticl displcements, the horizontl component of T remins essentilly constnt long the string, so m ccelertes verticlly, not horizontlly. The verticl component of tension is F y = T tn θ, where θ is the ngle of the string t some point reltive to the horizontl.

3 FROM CLASSICAL TO QUANTUM AND BACK FIG 5.1 : A trnsverse smll displcement of string. The tngent of this ngle, the slope of the string, cn lso be written s the prtil derivtive y(x, t)/ x; we hve to use the prtil derivtive here, becuse wheres y depends upon both the position x long the string nd the time t, the slope t fixed time t is the derivtive of y with respect to x lone. Let the left-hnd end of m be locted t x 0 nd the right-hnd end t x 0 + x. Using Tylor series, the slopes of the string t the left nd right re relted by y = y x x + 2 y x0 x 2 x y x0 2! x 3 ( x) (5.1) x0 x0 + x If y(x) is smooth nd x is sufficiently smll, we cn neglect ll but the first two terms on the right. The verticl forces on the right-hnd nd left-hnd sides of the slice m re F y (right) = T y nd F y (left) = T y x x, (5.2) x0 x0 + x with the force t the left upwrd nd the force t the right downwrd. The net verticl force on m is therefore [ y F net = T y ] = T 2 y x x x 2 x. (5.3) x0 x0 + x x0

4 BEYOND THE BASICS () (b) FIG 5.2 : () A smll slice of string; (b) Tension forces on the slice. The ccelertion of m is 2 y/ t 2, the second derivtive of y with respect to time, keeping now the position x fixed. The left end of the slice of string moves up nd down verticlly, with position y(x 0, t), velocity y/ t x0, nd ccelertion 2 y/ t 2 x0. Newton s second lw F net = m for m therefore becomes T 2 y x 2 x = m 2 y x0 t 2 = µ x 2 y x0 t 2, (5.4) x0 so 2 y ( µ ) 2 x y = 0, (5.5) 2 T t2 which is the wve eqution of the string for smll trnsverse displcements. It represents n infinite number of F = m equtions for the infinite number of infinitesiml slices of the string. For single prticle, n initil position nd initil velocity determine the future position for given mss nd forces by solving the ordinry differentil eqution F = m. For string, n initil shpe y(x, 0) nd velocity distribution y/ t t=0 determine the future shpe y(x, t) for given mss density µ nd tension T by solving prtil differentil eqution, the wve eqution. It is most convenient to write the wve eqution in more generl form 2 y x 2 1 v 2 2 y t 2 = 0 (5.6)

5 FROM CLASSICAL TO QUANTUM AND BACK where v is constnt whose role will become pprent soon. The wve eqution is liner, which is consequence of ssuming tht the displcements re smll. As result, it stisfies the superposition principle: two seprte solutions of the eqution cn be simply dded, nd the result still stisfies the wve eqution. A solution of the wve eqution is ny two functions f(u) nd g(u) of the combintion u = x ± v t y sol = f(x + v t) + g(x v t) (5.7) s cn be esily verified by substituting this expression into (5.6). This implies tht, once y(x, 0) nd y (x, 0) re given, f nd g re fixed. The wve eqution simply tells tht the profiles of f nd g evolve in time, without distortion, t speed v towrds negtive nd positive x respectively. An esy wy to see this is to sketch the two functions t two instnts in time. Note lso tht the wve eqution is rel in ddition to being liner. This mens tht we cn solve it with complex functions s well, with the rel nd imginry prts s seprte solutions: the eqution then splits into rel prt nd n imginry prt, ech looking identicl in form, pplied to the rel nd imginry prts of the complex solution. Clssicl wves propgte lso in fluids like ir or wter chrcterized by mcroscopic properties such s mss density nd pressure. If fluid is loclly perturbed, sound wves cn be set up, in which both the locl density nd pressure oscillte, nd the oscilltions re propgted from the initil site throughout the mteril. In the cse of smll-mplitude wves, the density of the mteril hs the form ρ = ρ 0 + ρ nd the pressure is p = p 0 + p, where ρ 0 nd p 0 re the mbient density nd pressure, nd ρ nd p re the smll perturbtions tht cn propgte from plce to plce. We won t prove it here, but the disturbnce ρ obeys wve eqution which in three dimensions hs the form ( ρ ) 2 2 ( ρ) ( ρ) = 0, (5.8) B t 2 where B is the bulk modulus of the mteril. If chnge of pressure p t point in the mteril cuses frctionl chnge in density ρ/ρ, then B p ρ/ρ, (5.9) so B is mesure of stiffness: the less the frctionl chnge in density for given pressure chnge, the greter the stiffness, nd the greter the bulk modulus.

6 BEYOND THE BASICS The differentil opertor 2 is the Lplcin, which In Crtesin coordintes tkes the form 2 = 2 x y z 2. (5.10) In the cse of n infinite plne wve propgting in the x direction tht is, ρ(x, y, z, t) = ρ(x, t), the wve eqution becomes 2 ( ρ) ( ρ ) 2 ( ρ) = 0, (5.11) x 2 B t 2 which hs the sme form s the wve eqution for string. Another well-known clssicl wve is the electromgnetic wve, one of the fmous consequences of Mxwell s equtions of electrodynmics. In vcuum the equtions cn be combined to produce wve eqution 2 E 1 c 2 2 E t 2 = 0 (5.12) for the electric field E, with similr eqution for the mgnetic field B. The electric nd mgnetic fields propgte together t the speed of light c, so Mxwell ws ble to chieve grnd synthesis of electricity, mgnetism, nd optics by showing tht light wves re in fct electromgnetic wves. Agin, the one-dimensionl form of these equtions, corresponding to n electromgnetic plne wve propgting in the x direction, hs the sme form s wves on string. A difference here is tht the electromgnetic wve eqution is vlid for ny clssicl electromgnetic wve, whtever its mplitude. No liner pproximtion hs to be mde in this cse. Among the infinite vriety of solutions of the one-dimensionl wve eqution, whether for wves on string, sound wves, or light wves, re sinusoidl trveling wves tht propgte to the right or to the left. These hve the form ( y(x, t) = A 0 sin k (x ω ) ) k t ϕ (5.13) where A 0 is the mplitude, k is the wve number, ω is the ngulr frequency, nd ϕ is the phse ngle of the wve. Notice tht this solution is indeed function of x ± v t s rgued erlier. The upper (minus) sign corresponds to wve trveling to the right, nd the lower (plus) sign corresponds to wve trveling to the left. The wve number is relted to the wvelength λ

7 FROM CLASSICAL TO QUANTUM AND BACK by k = 2π/λ, nd the ngulr frequency is relted to the frequency ν (i.e.,, cycles/s) by ω = 2πν. If the phse ngle ϕ = 0, then y(x, t) = y 0 sin(kx ωt), so tht initilly the wve is sine wve with y = 0 t x = 0, etc. The phse ngle simply displces the sinusoidl shpe to the right (if ϕ is positive) or to the left (if ϕ is negtive). We cn find the velocity of the wve by discovering how x chnges s t increses, so s to keep the overll phse θ kx ωt ϕ constnt. Tht is, how fr is prticulr wve shpe displced during some intervl of time? Setting dθ/dt = 0 gives v = dx /dt = ω/k, or in terms of wvelength nd frequency, v = λν. Substituting the wve form into the wve eqution then shows tht the trveling wve solves the wve eqution for wves on string if nd only if v = ω/k = T/µ, (5.14) nd for sound wves v = ω/k = B/ρ. (5.15) The intensity I of plne wve of sound or light, i.e.,, wve solution tht is independent of two of the Crtesin coordintes nd moves in the third direction (x), is proportionl to the squre of its mplitude A 0 ; tht is, I = CA 2 0, (5.16) where C is constnt tht depends upon the type of wve. The intensity is the energy/second pssing through squre meter perpendiculr to the wve velocity. The wve eqution lso hs complex exponentil trveling-wve solutions of the form y(x, t) = A 0 e i(kx ωt ϕ), (5.17) which is simple to verify by substitution, but is quite obvious from the fct tht the rel nd imginry prts of the complex exponentil re given by Euler s formul e iθ = cos θ + i sin θ, (5.18) so tht if we choose θ = kx ωt ϕ, it is in fct the sum of two sinusoidl trveling wves with the sme mplitude, frequency, nd wve number, but

8 BEYOND THE BASICS differing in phse by π/2. Complex exponentil solutions re often used in prt becuse they re esier to work with mthemticlly (the derivtive or n exponentil is n exponentil, for exmple). Then we cn lwys tke the rel (or imginry) prt of the finl result to get the physicl result, which in clssicl physics corresponds to n observble quntity nd must therefore be rel. In quntum mechnics, s we will introduce in this chpter, the complex exponentil form turns out to be the nturl form to use. The intensity of complex wve is proportionl to the product of the wve mplitude nd the complex conjugte of the wve mplitude; this gives the rel quntity I = Cy(x, t)y (x, t) = C [ A 0 e i(kx ωt ϕ)] [ A 0e i(kx ωt ϕ)] = C A 0 2 (5.19) s expected. EXAMPLE 5-1: Two-slit interference of wves When two or more trveling wves combine, we observe interference effects. Direct plne sinusoidl wve from left to right t double-slit system, for exmple, s shown in Figure 5.3. Only wves tht pss through one of the slits mke it through to the right-hnd side. The resulting wve is then detected on detecting plne, which is screen or bnk of detectors much frther long to the right. Wht will be observed by the detectors? (We ssume for simplicity tht the detecting plne is very fr from the slit system compred with the distnce between the two slits, so the wve disturbnces from ech slit propgte essentilly prllel to one nother.) Using the complex exponentil form t the position of the detector, y(x, t) = y 1 + y 2 = A 0 (e i(ks1 ωt ϕ) + e i(ks2 ωt ϕ)), (5.20) where s 1 is the distnce of the detector from slit 1 nd s 2 is the distnce of the detector from slit 2. Here we hve used the fct tht the wve number, frequency, nd phse ngle of ech prt of the wve re the sme (the phse ngles re the sme becuse the phse of both wves t the plne of the slits is the sme.). We hve lso ssumed tht the mplitude of ech wve s it reches the detector is the sme, which is n excellent pproximtion s long s the detecting plne is fr wy compred with the distnce between the two slits. The totl wve mplitude t the right is then the sum of the two wve mplitudes, y T = A 0 e i(ωt+φ) ( e iks1 + e iks2). (5.21) The intensity of the wve t the detecting plne is I = Cy T y T = C A 0 ) 2 ( e iks1 + e iks2) ( e iks1 + e iks2) = 2C A 0 ) 2 (1 + cos(k(s 2 s 1 )) = 4C A 0 ) 2 cos 2 (k(s 2 s 1 )/2) (5.22)

9 FROM CLASSICAL TO QUANTUM AND BACK slit system detector FIG 5.3 : Two pths for wves from slit system to detectors. using the identities cos q = (e iq + e iq )/2 nd cos 2 (q/2) = (1/2)(1 + cos q). The difference s 2 s 1 of the pth lengths from the two slits to point on the detecting plne is s 2 s 1 = d sin θ, s shown in Figure 5.4(), where θ is the ngle between the two rys nd the forwrd direction. The phse difference between the two wves is Φ k(s 2 s 1 ) = (2πd/λ) sin θ, so the intensity t n rbitrry ngle θ, in terms of the intensity I 0 in the forwrd direction θ = 0, is I(θ) = I 0 cos 2 (Φ/2) where Φ = 2πd λ sin θ, (5.23) s illustrted in Figure 5.4(b). There re lternting mxim nd minim, with the mxim occurring t ngles θ for which nλ = d sin θ, with n = 0, ±1, ±2,... If we direct plne wve of sound t double-slit system, where the wvelength λ is smller thn the slit seprtion d, then we do observe the lternting mxim nd minim predicted by eqution (5.23). 1 1 We cn detect the sound intensities by microphones plced long the detecting plne. The microphones must of course be lrge compred with the distnce between molecules in the sound-trnsmitting medium; the wve eqution for sound models the medium s continuum.

10 BEYOND THE BASICS () (b) FIG 5.4 : () The reltionship between s 2 s 1, d, nd θ; (b) The two-slit interference pttern. 5.2 Two-slit experiments with light nd toms According to Mxwell s equtions, light is n electromgnetic wve, so if we direct bem of light t double slit we should observe wve interference. But if we direct bem of toms t double slit, clssicl mechnics teches us tht we should observe bunch of toms downstrem of ech slit, much like wht would hppen if we tossed bll berings t pir of slits. Atoms re prticles, fter ll, so should exhibit no interference t ll. Now wht bout ctul experiments? LIGHT Vrious light detectors cn be used on the detecting plne, including photogrphic film, photomultipliers, CCDs, nd others, depending upon the wvelength. If the wvelength of the light bem is smller thn the slit seprtion, firly bright light source is used, nd firly long exposures re mde (the mening of firly here will soon become cler), the experimentl intensities gin show lternting mxim nd minim, with mxim occurring where nλ = d sin θ. But now crnk the brightness of the light source wy down, nd observe wht hppens over short time intervls. Insted of seeing very low intensity light spred immeditely over the detecting plne, s predicted by the interference/diffrction formul, one finds tht t first the light rrives t pprently rndom discrete loctions. If the detector is bnk of photocells,

11 FROM CLASSICAL TO QUANTUM AND BACK () (b) FIG 5.5 : () At very low intensity light, individul photons pper to lnd on the screen rndomly; (b) s the intensity is crnked up, the interference pttern emerges. for exmple, only certin cells will register the reception of light, while others (even t loctions where the intensity should be mximum) receive nothing t first. Tht is, light is seen to rrive in discrete lumps, or photons (see Figure 5.5()). The remrkble fct is tht even though the photons rrive one t time t the detectors, if we wit long enough the lrge number of photons distribute themselves mong the detectors exctly s the interference formul predicts (see Figure 5.5(b))! Tht is, in some sense light hs both prticle nture (we observe single prticles only in the detectors) nd wve nture (when huge numbers of photons hve rrived t the detecting plne, the overll distribution shows the interference pttern predicted by wve theory.) If we close off one of the two slits, the pttern of photons shows no such interference. By observing the number of photons rriving t the screen, nd knowing the intensity, the wvelength λ, nd frequency ν = c/λ of the bem, one finds tht ech photon must hve n energy E = hν nd momentum p = E/c = h/λ, where h is Plnck s constnt, h = J s. It ws Albert Einstein in 1905 who first relized tht light is not continuous, Mxwellin wve fter ll, but consists of discrete photons, nd tht ech photon is mssless nd hs energy E = hν nd momentum p = hν/c = h/λ. The centrl puzzle is: If light consists of strem of individul photons,

12 BEYOND THE BASICS so tht in the cse of two slits ech photon presumbly goes through one slit or the other slit nd not both, how cn they develop n interference pttern? How do photons know whether two slits re open or only one? ATOMS Now project bem of helium toms t pir of slits nd observe their distribution on the detecting plne. A double-slit system hs slits of width = 1 µm nd slit seprtion d = 8 µm. Ech helium tom hs mss m = kg, nd ech cn be detected by vrious counters s discrete prticle, where the detecting plne is distnce D = 1.95 m behind the slits. Our bem of helium toms trvels t speeds between 2.1 nd 2.2 km/s. A distinct interference pttern is observed! 2 The toms do rrive t the screen in discrete lumps, s expected, but the distribution shows interference effects similr to wht we observe with light! Figure 5.6 shows the ctul results of this experiment. The obvious question is: for helium toms, s with photons, wht exctly is interfering? The bem intensity cn be turned so low tht there is t most single tom in flight t ny given time, so toms re not interfering with other toms; ech tom must be interfering with itself in some wy. The interference distribution emerges only fter mny toms hve been detected. We cn crry out similr experiments with toms with different msses moving with different velocities. The results show tht the wvelength λ deduced from prticulr interference pttern on the screen is inversely proportionl to both the tomic mss m nd the velocity v of the toms. Tht is, λ = h/p (5.24) where p = mv is the momentum of the nonreltivistic tom nd h gin is Plnck s constnt. This is exctly the sme reltion between λ nd p s for photons. 3 If one of the two slits is blocked off, so toms cn only penetrte one of the slits, the two-slit interference pttern goes wy. The centrl puzzle once gin: Even though n tom is detected t specific spot much like clssicl prticle, the interference ptterns show 2 reference 3 The wvelength λ = h/p is clled the de Broglie wvelength, becuse it ws in his doctorl disserttion tht the French physicist Louis de Broglie proposed tht ll prticles hve wvelength λ = h/p. In the cse of photons, we cn increse the momentum by incresing the frequency of the light, since p = h/λ = hν/c. In the cse of toms, we cn increse the momentum by incresing their velocity.

13 FROM CLASSICAL TO QUANTUM AND BACK FIG 5.6 : Helium toms with speeds between 2.1 nd 2.2 km/s reching the rer detectors, with both slits open. The detectors observe the rrivl of individul toms, but the distribution shows cler interference pttern s we would expect for wves!. We see how the interference pttern builds up one tom t time. The first dt set is tken fter 5 minutes of counting, while the lst is tken fter 42 hours of counting. The experiments were crried out by Ch. Kurtsiefer, T. Pfu, nd J. Mlynek; see their rticle in Nture 386, 150 (1997). (The hotspot in the dt rises from n enhnced drk count due to n impurity in the microchnnel plte detector.)

14 BEYOND THE BASICS tht n individul tom somehow knows whether there re two slits open or only one. How does it know tht? If both slits re open, does it somehow probe both pths? Does it in some sense tke both pths? 5.3 Feynmn sum-over-pths Thirty-one yers go, Dick Feynmn told me bout his sum over histories version of quntum mechnics. The electron does nything it likes, he sid. It just goes in ny direction t ny speed,... however it likes, nd then you dd up the mplitudes nd it gives you the wve function. I sid to him You re crzy. But he wsn t.- Freemn Dyson, According to the Americn physicist Richrd Feynmn ( ) the nswer to the question posed t the end of the preceding section is yes! In his sum-over-pths formultion of quntum mechnics, toms (or electrons or molecules or photons or bll berings or... ) do tke ll vilble pths between two points. If both slits re open in the experiments we hve described, the prticle in some sense goes through both slits. If one of the slits is closed, tht pth is not vilble; or if we perform n experiment (perhps t the slits themselves) showing which slit ech tom psses through, then ech tom tkes only one of the pths, nd no interference pttern is observed t the screen. How do we predict wht will be observed in ech cse? According to quntum mechnics, there is no wy we cn tell where prticulr photon or helium tom will go. This is not becuse our mesuring devices do not yet hve sufficient precision; it is becuse prticle does not hve definite position or momentum t ny given time, nd it does not trvel by ny single clssicl pth. The best we cn do is find the probbility P tht prticle will be observed t ny prticulr loction. How do we find the probbility distribution? Here re the rules: (1) The probbility P tht prticle will be observed t prticulr loction is given by the bsolute squre of totl complex probbility mplitude z T to rrive there, P = z T 2 z T z T (5.25) where z T is the complex conjugte of z T.

15 FROM CLASSICAL TO QUANTUM AND BACK (2) The totl probbility mplitude for prticle to go from to b is simply the sum of the probbility mplitudes to go by every pth vilble to it, z T = z 1 + z (5.26) (3) The probbility mplitude z to go from source t to detector t b by some prticulr pth is given by z = z(t 0 )e iφ, (5.27) where z(t 0 ) is the mplitude of the source when the prticle leves, nd φ is phse tht depends upon the pth. The phse is the Lorentz-invrint quntity φ = 1 b η µν p µ dx ν = 1 b (p ds Edt) = b (k ds ωdt) (5.28) where the coordintes t re the initil position nd time (x 0, y 0, z 0, t 0 ), nd the coordintes t b re the finl position nd time (x, y, z, t), for prticulr pth. Here the mgnitude of the wve number three-vector k is k = 2π/λ = p/ nd ω = 2πν = E/, where h/2π, p is the mgnitude of the prticle s momentum, nd E is its energy. Note tht both reltionships p = h/λ nd E = hν re vlid for mssless photons s well s mssive prticles like helium toms. These re the three simple rules for clculting the probbility tht prticle will be detected t time t. Note from Rule 3 tht if prticulr pth from source t to detector t d is thought of s sequence of pth segments, for exmple, (1) ( b), (2) (b c), (3) (c d), then the phse φ, being n integrl over the entire pth from to d, is the sum of integrls for ech segment of the pth. Tht is, φ = φ 1 + φ 2 + φ , so the phse fctor cn be written e iφ = e i(φ 1+φ 2 +φ ) = e iφ 1 e iφ 2 e iφ 3... (5.29) The mplitude to go by prticulr pth ll the wy from source to detector is therefore z = z(t 0 )e iφ 1 e iφ 2 e iφ 3..., (5.30)

16 BEYOND THE BASICS the mplitude t the source multiplied by the product of phse fctors for ech segment of the pth. Suppose for now tht both the energy of the prticle nd the mgnitude of its momentum re conserved nd tht they hve the sme vlue long ech pth. (Lter on we will generlize to llow chnges in ech.) In tht cse k ds = ks, where s is the pth length, nd ωdt = ω(t t 0 ). The mplitude for prticulr pth cn therefore be written z = z(t 0 )e iφ = z(t 0 ) e iks e iω(t t 0). (5.31) The mplitude z(t 0 ) when the prticle leves the source is itself generlly complex, so z(t 0 ) = z(t 0 ) e iφ 0, (5.32) where φ 0 is rel. The mplitude for prticulr pth to rech the detector is therefore z = z(t 0 )e iφ = z(t 0 ) e i(φ+φ 0), (5.33) where φ = ks ω(t t 0 ). This probbility mplitude z cn be displyed s two-dimensionl vector clled phsor in the complex plne. The horizontl xis represents rel numbers nd the verticl xis imginry numbers. Points not on either xis represent complex numbers with both rel nd imginry prt. Plcing the til of the phsor t the origin, the length of the phsor is z(t 0 ) nd its ngle with the rel xis is the totl phse (φ + φ 0 ), where φ 0 is the ngle of z(t 0 ) reltive to the rel xis, s illustrted in Figure 5.7. Now suppose the source emits stedy bem of prticles, ll with the sme energy. The mgnitude of the mplitude t the source therefore remins constnt, but the phse t tht point chnges with time. Tht is becuse t the source itself we cn set s = 0 in the mplitude, so t the source z(t) = z(t 0 )e iω(t t 0), (5.34) which is phsor tht spins clockwise in the complex plne s time progresses. It is convenient to refer the mplitude to some stndrd time t = 0, so z(0) = z(t 0 )e iωt 0. (5.35)

17 FROM CLASSICAL TO QUANTUM AND BACK FIG 5.7 : A phsor z(t 0 )e iφ z(t 0 ) e i(φ+φ0) drwn in the complex plne. The rel xis is horizontl nd the imginry xis is verticl. The bsolute length of the phsor is z(t 0 ) nd the ngle between the phsor nd the rel xis is the phse (φ + φ 0 ), where φ 0 is the phse of z(t 0 ) lone. () (b) (c) (d) (e) FIG 5.8 : The sum of two individul phsors with the sme mgnitudes z(t 0 ) but different phses. The result is phsor tht extends from the til of the first to the tip of the second, s in vector ddition. The difference in their ngles in the complex plne is the difference in their phse ngles. Shown re exmples with phse differences equl to () zero (b) 45 (c) 90 (d) 135 (e) 180.

18 BEYOND THE BASICS In terms of z(0), the mplitude for pth of length s is z = z(t 0 )e i(ks ω(t t 0)) = z(0)e i(ks ωt), (5.36) so for monoenergetic bem of prticles the totl mplitude for prticle to rech the detector t time t is z T = z(0) ( e i(ks 1 ωt) + e i(ks 2 ωt) +... ) = z(0)e iωt ( e iks 1 + e iks ). (5.37) Therefore even though, for monoenergetic bem of mssive prticles, those tking longer pths must hve left the source erlier to rrive t the sme time t, the strting time t 0 hs been eliminted, so the totl mplitude is exctly the sme s it would be if the prticles were ll emitted from the source t t = 0 with energy E = ω, no mtter which pth they tke! Note tht in summing over pths, we men every pth llowed by the physicl circumstnces. Pths cn zig-zg, go bck nd forth, in circles, ny wy they like s long s there re no physicl brriers to prevent them. 5.4 Two slits nd two pths We cn now derive the probbility distribution for prticles pssing through double slit using the quntum rules. We will mke huge simplifiction for now, llowing prticles to move long just two pths from the source to detector, ech pth consisting of two stright-line segments joined t slit, s illustrted in Figure 5.9. We ssume lso tht both the source nd the detector re fr from the slit system, so the two pths from the source to the slits re essentilly prllel to one nother, nd the two pths from the slits to the detector re lso essentilly prllel to one nother. The totl probbility mplitude is z T = z(0)e iωt ( e iks 1 + e iks 2 ), (5.38) so the probbility of observing photon or tom t prticulr detector is P = z T z T = z(0) 2 ( e iks 1 + e iks 2 ) ( e iks 1 + e iks 2 ) = 2 z(0) 2 (1 + cos(k(s 2 s 1 )) = 4 z(0) 2 cos 2 (k(s 2 s 1 )/2). (5.39)

19 FROM CLASSICAL TO QUANTUM AND BACK detector 1 source 2 FIG 5.9 : Two pths from source to detector. The probbility tht prticle is detected t rbitrry ngle θ, in terms of the probbility P (0) of detecting it in the forwrd direction θ = 0, is therefore P (θ) = P (0) cos 2 (Φ/2) where Φ = k(s 2 s 1 ) = 2π λ (d sin θ) (5.40) using the sme trig identities we used erlier for clssicl wves, where we found n intensity distribution I(θ) = I(0) cos 2 (Φ/2). Nturlly enough, if the probbilities of single-prticle events obey the two-slit pttern, then if we collect gret mny prticles the intensity will hve the sme distribution. The formul grees with the experimentl results for photons or helium toms whose wvelengths re not extremely smll compred with the slit seprtion d, s shown in Figure 5.4. Now wht hppens if the wvelength is extremely smll, i.e.,, λ < d? From λ = h/p it follows tht if the prticles hppen to be nonreltivistic bll berings, the momentum p = mv is enormous becuse their msses re so lrge. If we toss bll berings t double-slit system, we expect bunches of blls to ccumulte downstrem of ech slit, with no interference pttern t ll. It is true tht some blls might nick the slit edges nd be deflected to one side or the other, yet we would certinly see no interference pttern. So something else must be going on to explin why in tht cse we do not see the two-slit interference pttern of eqution (5.40). If the prticles hppen to be photons with λ d, they hve very lrge moment nd therefore very high energies; or if the prticles hppen to be nonreltivistic helium

20 BEYOND THE BASICS toms, their velocities must be quite lrge to hve very smll wvelengths. If quntum mechnics pplies to everything, including bll berings, highenergy photons, nd fst helium toms, the sum-over-pths rules must still pply, even though two-slit interference is not evident. Consider n ctul experiment with fst helium toms. As before, ech slit hs width = 1 µm nd the two slits re seprted by distnce d = 8 µm. Ech tom hs mss m = kg, nd ech cn be detected by vrious counters s discrete prticle. For toms with velocities bove 30 km/s the results of ctul experiments with both slits open re shown in Figure They strike the screen with bunch downstrem of ech slit, much like wht we would find if we tossed bll berings t much lrger slit system. Tht is, ny bll bering tht penetrted two-slit system would go through either one slit or the other, nd for those going through the top slit it would mke no difference whether the bottom slit is open or not, nd for those going through the bottom slit it would mke no difference whether the top slit is open or not. The distribution with both slits open is simply the sum of the distributions with only one slit open t time. And tht is wht we observe for these fst helium toms. In fct, the two-slit interference pttern is incomplete, becuse even with just two slits, mny more thn two pths re vilble. For exmple, becuse ech individul slit hs finite width, there is n infinite number of nerby pths pssing through ech slit. These pths hve slightly different phses, especilly if the wvelength is smll, so they interfere with one nother. Consider nrrow slice of single slit of width dy. If we mesure y up from the bottom of the slit, then s = s 0 +y sin θ, where s 0 is the distnce of the bottom of the slit from the detector, s shown in Figure 5.11(). The mplitude dz of ll pths pssing through the nrrow slice will be proportionl to dy, the width of the slice, so dz = (b dy) e iks 0 e i(ky sin θ ωt) (5.41) where b is constnt. The totl mplitude to rech the detector, pssing through single slit of width, is therefore z T = b e i(ks 0 ωt) 0 ( ) dy e iky sin θ = b e i(ks 0 ωt) e ik sin θ 1. (5.42) ik sin θ

21 FROM CLASSICAL TO QUANTUM AND BACK FIG 5.10 : High-velocity helium toms, with speeds bove 30 km/s, reching the rer detectors, with both slits open. The detectors observe the rrivl of individul toms, nd the distribution is wht we would expect for clssicl prticles. Experiments crried out by Ch. Kurtsiever, T. Pfu, nd J. Mlynek, Nture 386, 150 (1997)

22 BEYOND THE BASICS () (b) FIG 5.11 : () Pth length s function of position y within the slit. (b) The single-slit diffrction pttern. The probbility is equl to the bsolute squre of z, ( ) ( ) e P = zt z T = b 2 iky sin θ 1 e iky sin θ 1 ik sin θ ik sin θ ( ) 2b 2 sin = k 2 sin 2 θ (1 cos(k sin θ)) = 2b2 2 2 α, (5.43) α 2 where α (k sin θ)/2 nd we hve used the identity sin 2 α = (1/2)(1 cos 2α). Now (sin 2 α)/(α 2 ) 1 s α 0, which is the mximum vlue this rtio cn chieve. This distribution is clled single slit diffrction. The probbility pttern in this cse is ( ) sin 2 α P = P (mx) α 2 where α k sin θ 2 = π λ sin θ. (5.44) The diffrction pttern P (α) is shown in Figure The distribution hs mximum in the middle where α = 0, nd the first minimum t ech side corresponds to α = ±π. The hlf-width of the centrl pek is therefore α = π, which occurs t n ngle θ 0 for which sin θ 0 = λ/. So fr, using Feynmn s sum-over-pths, we hve found both the twoslit interference pttern neglecting single-slit diffrction nd the single-slit diffrction pttern in the bsence of two-slit interference. Of course, the

23 FROM CLASSICAL TO QUANTUM AND BACK ctul two-slit probbility distribution t the screen must include both interference nd diffrction. Assume for now tht the wvelength λ is smll enough tht λ. Then for the centrl diffrction pek, sin θ 1, so sin θ = θ. Let D be the distnce from the slit system to the detecting screen, nd x be the distnce on the screen from the midpoint of the wve pttern on the screen, s shown in Figure Now x/d = tn θ = θ if θ 1, so in this cse the centrl pek of the single-slit diffrction pttern hs hlf-width on the screen of mgnitude x 1/2 = Dθ = Dλ. (5.45) Figure 5.13 illustrtes interference/diffrction curves for double slit system with d = 4 nd d/d = In Figure 5.13() the wvelength of the prticle bem obeys λ/ = 0.01, so the ngle of the first minimum of the diffrction pttern is θ 1/2 = λ/ = 0.01, nd the distnce on the detecting plne from the center of the pttern to the first diffrction minimum is x 1/2 = Dθ 1/2 = 0.01D. This is ten times the distnce d between the two slits, so the diffrction curves for the two slits essentilly overlp. In this cse the overll probbility distribution is simply the product of the interference oscilltion with the centrl diffrction envelope, ( ) 2 sin α P = P mx cos 2 β. (5.46) α where β πd sin θ/λ nd α π sin θ/λ. Now keep the sme pir of slits nd the sme distnce to the detecting plne, but decrese the bem wvelength by fctor of 20. The ngle to the first minimum of ech diffrction pttern, mesured from the centrl pek, then becomes θ 1/2 = λ/ = , so the distnce on the detecting plne between the pek nd first minimum is x 1/2 = d/2. The entire pttern now looks s shown in Figure 5.13(b). The two peks re quite well seprted, s one would expect for clssicl prticles. Clerly s the wvelength is further reduced, the pttern becomes closer nd closer to tht corresponding to two bunches of prticles formed downstrem of ech of the two slits. In these experiments clssicl mechnics is shown to be specil cse of quntum mechnics, corresponding to the limit of smll debroglie wvelength.

24 BEYOND THE BASICS FIG 5.12 : The double slit, with screen t distnce D. We cn view the intensity on the screen s function of the trnsverse distnce x. () (b) FIG 5.13 : Interference/diffrction ptterns for double slit with = d/4 nd D = 1000d. The diffrction curves, shown in dshed lines, serve s envelopes for the more rpidly oscillting interference pttern. () The pttern in the cse d = 0.1x 1/2, where x 1/2 is the distnce on the detecting plne between the center nd the first minimum of the diffrction envelope. The diffrction curves of the two slits strongly overlp in this cse, giving in effect single diffrction envelope. (b) The pttern in the cse d = 2x 1/2, showing tht the two diffrction ptterns hve become seprted, with the first minimum due to ech slit t the sme loction in the center. This cse corresponds to wvelength smller by fctor of 20 thn the pttern shown in ().

25 FROM CLASSICAL TO QUANTUM AND BACK Figure 5.13() grees very well with the observtions of slower helium toms, s we showed in Figure 5.4. We lredy found in Section 6.? tht helium toms with v = 2.15 km/s hve the wvelength λ = m. The rtio λ/ is therefore m/ (10 6 m) = , so the condition used to derive eqution (5.46) for the hlf-width of the single-slit diffrction pttern is vlid in this cse. The results illustrted in Figure 5.13 used distnce D = 1.95 m from the slits to the screen, so the trnsverse hlf-width t the detectors is x = Dθ = 1.95 m( ) = m. (5.47) This is more thn ten times the distnce d between the two slits, so there is strong overlp between the two single-slit diffrction ptterns in the experiment shown in Figure 5.6?. Now wht bout the experimentl result shown in Figure 5.6 for velocities bove 30 km/s? The wvelength of 30 km/s helium toms is shorter thn 2.15 km/s toms by fctor (2.15)/(30) = , so the hlf-width of the single-slit diffrction pttern on the screen is now x = Dθ = ( m)(0.0717) = m. (5.48) In this cse the distnce 2 x is only bit lrger thn d, so there is some seprtion in the peks of the two diffrction ptterns. Most of the toms in this smple hve even higher velocities nd even shorter wvelengths, so their diffrction ptterns re more distinctly seprted. This is consistent with the experimentl results shown in Figure 5.6. Tht is, the pprent clssicl behvior of fst helium toms is relly well-seprted diffrction ptterns. The two-slit interference within ech diffrction curve is msked in the experiment becuse of the wide rnge of speeds nd therefore wvelengths represented in the smple, with correspondingly different positions of the mxim nd minim within the diffrction envelopes, nd lso by fuzziness in the detectors. Our conclusion from tomic-bem experiments is tht helium toms generlly behve like neither bll berings nor sound wves: They re neither clssicl prticles nor clssicl wves, but retin some properties of ech. They re detected s loclized units like prticles, but they show interference ptterns like wves. In fct, we find tht their prticle nd wve properties re relted by p = h/λ k where h is Plnck s constnt nd h/2π.

26 BEYOND THE BASICS () (b) FIG 5.14 : The sum of lrge number of phsors () tht re bout the sme (b) tht differ by constnt mounts. 5.5 No brriers t ll Now remove the system of slits so there re no brriers t ll between the source nd detector, nd consider ll pths between them. According to the quntum rules, the totl probbility mplitude for the prticle to be detected t time t f is z = z(0)e iωt f (e iks 1 + e iks 2 + e iks ), (5.49) the sum of n infinite number of terms, where we hve ssumed tht ll of the mgnitudes z(0) (which my be complex numbers) re equl. 4 The phses ks 1, ks 2, ks 3... re obviously proportionl to the pth lengths. Suppose tht the pth lengths (nd therefore the phses) re ll bout the sme for some prticulr set of pths. In tht cse the phsors ssocited with ech term in the set dd up to give lrge totl mplitude, s shown in Figure 5.14(). If the pth lengths re ll quite different for nother set of pths, then those phsors tend to cncel one nother out, s in Figure 5.14(b). Under wht circumstnces will the phses be bout the sme for set of pths? In the nlogous cse of continuous function y(x), the Tylor series 4 In fct, the mplitude for longer pths is less thn the mplitude for shorter pths, but this turns out to mke no difference in mking the trnsition to clssicl mechnics, s we shll soon see.

27 FROM CLASSICAL TO QUANTUM AND BACK bout point x 0 is y(x) = y(x 0 ) + dy(x) dx (x x 0 ) + 1 x0 2! d 2 y(x) dx 2 (x x 0 ) (5.50) x0 so the vlue of y(x) is the sme to first order in x x x 0 if x 0 hppens to be mximum or minimum of the function. If x 0 does not correspond to mximum or minimum, then y(x) chnges more rpidly s x vries. The sme is true of the phses in the sum-over-pths. If the set of pths is nerby the pth of minimum length, for exmple, then the phses will ll be bout the sme nd the corresponding phsors will dd up to give lrge totl. If the set of pths is nerby some other rbitrry pth, their phses will differ sufficiently from one nother tht the totl phsor will be smll. In the cse of free prticle, the shortest pth is stright line, nd the phse of nerby pths will be nerly the sme, so the totl probbility mplitude will be lrge. Tht is, if free prticle trvels from to b by stright-line pth, the neighboring pths ll hve bout the sme length, so their phses dd up constructively. But if the prticle trvels by some rbitrry pth, the neighboring pths differ more mrkedly in length from it, so the phses for these surrounding pths tend to cncel one nother out. EXAMPLE 5-2: A clss of pths ner stright-line pth We cn show wht hppens for specil set of free-prticle pths. Let s 0 be the shortest distnce between the source nd detector, corresponding to stright-line pth. We will sum the probbility mplitudes for certin clss of pths ner this pth. These prticulr lternte pths consist of stright line with kink in the middle, where the kink is distnce D = n D 0 (n = ±1, ±2,...) from the stright line, s shown in Figure If the pths hve length s n, then by the Pythgoren theorem (s/2) 2 = (s 0 /2) 2 + (nd 0 ) 2. (5.51) We will ssume tht n D 0 s 0, so using the binomil pproximtion, s = s (2nD0 /s 0 ) 2 = s 0 (1 + 2n 2 D0/s 2 2 0). (5.52) Therefore the probbility mplitude to go by prticulr pth of length s is proportionl to e iks = e iks0 e iθn (5.53)

28 BEYOND THE BASICS FIG 5.15 : A clss of kinked pths between source nd detector. The stright line is the shortest pth, nd the midpoint of the others is distnce D = n D 0 from the stright line, where (n = ±1, ±2,...). where ( ) 2kD 2 θ n = 0 n 2 n = 0, 1, 2,... (5.54) s 0 Note tht θ n is the ngle of the ssocited phsor with respect to tht of the stright-line pth. As prticulr cse, suppose the phsor corresponding to the stright-line pth is horizontl, nd tht 2kD0/s 2 0 = π/200. Then the ngles θ n re given in Tble 6.1 for n = 0, n = ±1, n = ±2,...n = ±25, The sum of these phsors, ll with the sme length but in directions θ n reltive to the horizontl, will give the totl phsor for these pths. n θ n n θ n n θ n n θ n n θ n 0 0 ± ± ± ± ±1 0.9 ± ± ± ± ±2 3.6 ± ± ± ± ±3 8.1 ± ± ± ± ± ± ± ± ± ± ± ± ± ± Tble 5.1. The ngles of bent-line segments in terms of the integer n tht chrcterizes them. The phsors re drwn in Figure Those corresponding to n = 0 through n = ±5 re more or less ligned, so the pths neighboring the stright-line pth re enhncing it. As n increses the ngles between successive phsors grdully

29 FROM CLASSICAL TO QUANTUM AND BACK FIG 5.16 : Phsors up to n = ±25. The more distnt pths wind up in spirls, contributing very little to the overll phsor sum. increse, so the phsors begin to loop round, winding up in tighter nd tighter spirls so they no longer mke ny importnt contribution to the totl mplitude. The shpe of these phsors is clled Cornu spirl. The sum of ll the phsors up to n = ±25 is the long rrow shown. If we were to include dditionl phsors we would not chnge this sum very much. It is the stright-line pth nd its neighbors tht contribute the most to the overll phsor, nd therefore to the overll probbility mplitude for the tom to go from the source to the detector. The clssicl pth is the stright-line pth in this cse, but it is not the only pth. How do we know tht the clssicl pth is not the only pth tht prticle tkes? The best wy to show the importnce of the other pths is to block them off. If highly collimted bem of prticles trvels from source to detecting screen, detectors will find tht the prticles re confined to nrrow region on the screen, in ccord with the ide tht the prticles follow stright-line pth from source to screen. Therefore if we were to introduce nrrow slit t loction directly between the source nd screen, it should mke no difference, becuse ccording to clssicl ides the prticles re tking only tht pth nywy. But it does mke difference. With the nrrow slit in plce, the prticles show diffrction pttern on the screen. Therefore without the screen, prticles must be tking more thn the stright

30 BEYOND THE BASICS line pth fter ll; in fct, ccording to quntum mechnics they tke ll pths from source to screen tht re not blocked off by some brrier. EXAMPLE 5-3: How clssicl is the pth? The phse of free prticle involves the product ks = 2πs/λ. If the wvelength is very smll, so tht s/λ 1, then slight chnges in s men lrge chnges in phse. The stright-line pth between nd b is the minimum-distnce pth, so we lredy know tht is the clssicl pth. Neighboring pths hve lmost the sme length, so they tend to dd up in phse. But for given pth ner the stright line, s the wvelength becomes smller nd smller, the phse 2πs/λ chnges more nd more, so the corresponding phsors tend to spirl round nd cncel out. Tht is, for very smll wvelengths the set of mutully reinforcing pths becomes more nd more constrined, closer nd closer to the stright line; in the limit s/λ 1, non-clssicl pths become less nd less importnt in the overll sum, so clssicl mechnics becomes better nd better pproximtion to the true sitution. Tke for exmple n electron moving t speed v. Clssicl motion is vlid in the limit s λ = p s h = mvs 1. (5.55) h An electron is cthode-ry tube (such s n old-fshioned TV picture tube), with s = 0.5 m nd v = 10 8 m/s, hs the rtio mvs h = kg 10 8 m/s 0.5 m J s 10 11, (5.56) so such n electron moves on clssicl pth (in TV tube we obviously wnt the electrons to move long clssicl, deterministic pth.) But consider the electron in hydrogen tom, where it hs typicl speed of 10 6 m/s nd pth length from one side of the tom to the other of order m. The rtio in this cse is mvs h = kg 10 6 m/s m J s 0.1, (5.57) so the pth in this cse is not t ll clssicl, but quntum mechniclly fuzzy. The electron in hydrogen does not move long clssicl orbit nything like the motion of plnets round the Sun. On the other hnd, the rtio for Erth orbiting the Sun is mvs h = kg m/s m J s which corresponds to supremely clssicl motion , (5.58) There re no shrp boundries between clssicl nd quntum behvior. All motion is relly quntum mechnicl, but clssicl behvior cn tke plce s specil cse. So quntum mechnics shows the role of clssicl mechnics nd its rnge of vlidity.

31 FROM CLASSICAL TO QUANTUM AND BACK 5.6 Pth shpes for light rys nd prticles We cn now use the formlism of sum-over-pths to find the shpe of pths tken by light rys nd by non-reltivistic mssive prticles. In Chpter 8 we will be ble to use the methods derived here to find the pth shpes of reltivistic prticles moving in electromgnetic fields. LIGHT RAYS When light psses from one medium to nother, sy ir to refrctive medium like glss, its frequency (i.e.,, the energy/photon) remins constnt, but the wvelength chnges, becuse the velocity v = λν is less in glss thn in ir. If the medium hs index of refrction n, where n my depend upon position, the speed of light in the medium is v = c/n, so with constnt frequency the wvelength decreses by the fctor 1/n, nd the wvenumber k = 2π/λ increses by the fctor n. So wheres in vcuum ω/k = c (or k = ω/c), in refrctive medium k = ω/v = nω/c. The quntum mechnicl mplitude for photons to trvel by prticulr pth therefore becomes where z = z(t 0 )e iφ (5.59) φ = b kds ω(t t 0 ) = (ω/c) b nds ω(t t 0 ). (5.60) The corresponding phsors dd up for pths ner tht pth which extremizes the phse b n ds. Tht is, they dd up long the pth tht extremizes the opticl pth b n ds. This is just Fermt s principle of sttionry time, since the time for light to follow prticulr pth is t = b ds/v = 1 c b n ds. (5.61) According to Fermt, light rys tke minimum-time pths between nd b.. According to Feynmn, photons tke ll pths between nd b, but it is minly pths ner the sttionry pth tht contribute to the totl mplitude. Who is right? Tht is esy to test: lthough we my not think tht photons tke pths tht differ from the sttionry-time pth, if we try to block off these lterntive pths the signl t the detector chnges, so they re relly there. For exmple, if the source is in ir nd the detector is in glss, ccording to Fermt s principle nd Snell s lw, the light trvels only by the pth

32 BEYOND THE BASICS for which the ngles of incidence nd reflection obey n 1 sin θ 1 = n 2 sin θ 2, nd the pth tht stisfies this lw crosses the ir-glss interfce t only single point. If so, it would mke no difference if we blocked off ll other points on the interfce. But it does mke difference, becuse if we block off ll other points, leving only tiny hole for the light to pss through, it will be diffrcted, chnging the mount of light observed t the detector, nd llowing other detectors in the glss to receive light, even in directions for which Snell s lw is not obeyed. NONRELATIVISTIC MASSIVE PARTICLES Up to now we hve described the behvior of mssive prticles only when they re free, in which cse their clssicl pths re stright lines. The probbility mplitude for pth of length s in this cse is z = z(0)e (i/ )pµxµ = z(0)e (i/ )(ps Et), (5.62) where z(0) is the mplitude t the source t time t = 0, nd p nd E re the momentum mgnitude nd energy, both constnt long the pth. For nonreltivistic prticles encountering forces s they move from to b, this expression cnnot be correct, becuse even with conserved energy E = T + U, the momentum mgnitude p = 2mT = 2m(E U(r)) is not conserved becuse the potentil energy generlly depends upon position. The expression for the mplitude z = z(0)e (i/ )(ps Et) therefore no longer mkes sense, becuse p keeps chnging; it hs to be replced by z = z(0)e (i/ )[(p 1 s 1 +p 2 s 2 +p 3 s ) Et] = z(0)e (i/ )( b p ds Et), (5.63) where the vrying momentum is integrted over the pth between the source nd detector. The totl probbility mplitude of the prticle beginning t point nd ending t b t time t f is therefore z T = z(0)e iet/ e (i/ ) b p ds, (5.64) summing over ll pths between nd b. For ny prticulr pth the integrl is b p ds = b 2m(E U(r)) ds. (5.65) In the cse of free prticle, the clssicl pth is the pth for which the product ps is minimized, which mens (becuse p is fixed in this cse) tht s is