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# Quantum Physics Explains Newton's Laws of Motion

FEATURES www.iop.org/journals/physed explains Newton’s laws of

Jon Ogborn1 and Edwin F Taylor2

1 Institute of Education, University of London, UK 2 Massachusetts Institute of , Cambridge, MA 02139, USA

Abstract Newton was obliged to give his laws of motion as fundamental . But today we know that the quantum is fundamental, and Newton’s laws can be seen as consequences of fundamental quantum laws. This article traces this transition from fundamental quantum to derived .

Explaining Fermat’s : source of the key It is common to quantum physics and the quantum idea behaviour of quantum objects such as or Geometrical predicts the formation of as mysterious and peculiar. And indeed, images by rays. Examples are images due in ’s words, electrons do ‘behave to reﬂection and images formed by eyeglasses and in their own inimitable way . . . in a way that is like camera lenses. All of —every nothing that you have ever seen before.’ (Feynman path of every light ray—can be predicted from a 1965, p 128). single principle: Between source and reception But it was also Richard Feynman who point light travels along a path that takes the devised a way to describe quantum behaviour with shortest possible . This is called Fermat’s astounding simplicity and clarity (Feynman 1985). principle after the Frenchman Pierre de Fermat There is no longer any need for the mystery that (1601–1665). comes from trying to describe quantum behaviour A simple example of Fermat’s principle is the as some strange approximation to the classical law of reﬂection: angle of incidence equals angle behaviour of and . Instead we of reﬂection. Fermat’s principle also accounts turn the job of explaining around. We start from for the of a lens: a lens places different quantum behaviour and show how this explains classical behaviour. thicknesses of glass along different paths so that This may make you uncomfortable at ﬁrst. every ray takes the same time to travel from a Why explain familiar things in terms of something point on the source to the corresponding point on unfamiliar? But this is the way explanations have the image. Fermat’s principle accounts for how to . Explanations that don’t start somewhere a curved mirror in a telescope works: the mirror else than what they explain don’t explain! is bent so that each path takes the same time to In this article we show that quantum reach the focus. These and other examples are mechanics actually explains why Newton’s laws discussed in the Advancing Physics AS Student’s of motion are good enough to predict how footballs Book (Ogborn and Whitehouse 2000). and satellites move. For Newton, fundamental Fermat’s contemporaries had a fundamental laws had to be axioms—starting points. For us, objection to his principle, asking him a profound Newton’s laws are seen to be consequences of the question, ‘How could the light possibly know in fundamental way the quantum world works. advance which path is the quickest?’ The answer

January 2005 P HYSICS E DUCATION 27 J Ogborn andEFTaylor in many directions, so total contributions from SPthese end-segments never amount to much. Even if we extend the mirror AM on each side to make it longer, the contributions to the resulting arrow made by reﬂections from the sections of these ABCDEFGHI J KLM right-and-left extensions add almost nothing to the total resulting arrow. Why? Because they will all curl even more tightly than arrows ABC and KLM at the two ends of the resulting arrow shown in the bottom panel of the ﬁgure. Most of the

28 P HYSICS E DUCATION January 2005 Quantum physics explains Newton’s laws of motion Electrons do it too! In our analyses of photon reﬂection and straight- line transmission we assumed that the hand of the photon stopclock rotates at the frequency f of the b b classical wave. If we are to use a similar analysis d a a for an electron or other ‘ordinary’ submicroscopic source d eye b b particle, we need to know the corresponding frequency of rotation of its quantum stopclock. With this question we have reached bottom: there is nothing more fundamental with which Figure 2. Extreme paths through a slit (not to scale). to answer this question than simply to give the answer that underlies nonrelativistic quantum mechanics. This answer forms Feynman’s the inﬁnite number of possible paths to those new basis for quantum physics, then propagates consisting of two straight segments of equal length between source and eye. How wide (width 2d in upward, forming the bridge by which quantum the ﬁgure) does the slit have to be in order to pass mechanics explains Newtonian mechanics. most of the light from the source that we would Here then is that fundamental answer: For observe by eye? an ‘ordinary’ particle, a particle with mass, the We can get numbers quickly. Apply quantum stopclock rotates at a frequency Pythagoras’ theorem to one of the right triangles L K − U abd in the ﬁgure: f = = (nonrelativistic particles). h h a2 + d2 = b2 (2) In this h is the famous quantum of or action known as the . L is called d2 = b2 − a2 = (b + a)(b − a) ≈ 2a(b − a). the Lagrangian. For single particles moving at nonrelativistic the Lagrangian is given by In the last step we have made the assumption that a the difference between the kinetic K and and b are nearly the same length; that is, we make the U. a small percentage error by equating (b + a) to 2a. Is this weird? Of course it is weird. We will check this assumption after substituting Remember: ‘Explanations that don’t start some- numerical values. where else than what they explain don’t explain!’ Our criterion is that the difference 2(b − a) If equation (2) for a particle were not weird, the between the paths be equal to half a , ancients would have discovered it. The ancients the distance over which the stopclock hand were just as smart as we are, but they had reverses direction. Take the distance a between slit not experienced the long, slow development of and either source or receptor to be a = 1 metre and physical theory needed to arrive at this equation. −8 use green light for which λ = 600 nm = 60×10 Equation (2) is the kernel of how the microworld metres. Then works: accept it; celebrate it! λ − For a free electron equation (2) reduces to d2 ≈ a = 1 × 30 × 10 8 m2 (1) 2 K × –4 f = (nonrelativistic ). (3) so that d is about 5 10 m. Therefore 2d, the h width of the slit, is about one millimetre. (Check: b2 = a2 + d2 = (1+3× 10−7) m2, so our assump- This looks a lot like the corresponding equation tion that b + a ≈ 2a is justiﬁed.)2 for photons: 2 Try looking at a nearby bright object through the slit formed E by two ﬁngers held parallel and close together at arm’s length. f = (photon) At a ﬁnger separation of a millimetre or so, the object looks just h as bright. When the gap between ﬁngers closes up, the image spreads; the result of diffraction. For a very narrow range of (though mere likeness proves nothing, of course). alternative paths, geometrical optics and Fermat’s principle no With the fundamental assumption of equa- longer rule. But arrow-adding still works. tion (2), all the analysis above concerning the

30 P HYSICS E DUCATION January 2005 Quantum physics explains Newton’s laws of motion

varying the worldline by a small amount over a short segment

AB

y raise midpoint of by segment by by

bt bt vertical distance

time t

the action must not vary as the path varies

b(action) = (bLA + bLB)bt where by varies and bt is fixed

bLA = b(KA – UA)

bLB = b(KB – UB) action will not vary if (L + L ) = (K + K ) – (U + U ) b A B b A B b A B changes in are equal to changes in potential energy

change in kinetic energy change in potential energy ( b(KA + KB) b UA + UB) A B A B

by/2 p by momentum by A p B by/2 bt bt over both segments y is increased on average by by/2 slope increases slope decreases by by/bt by by/bt bUA = mg by/2 and bUB = mg by/2 by by bv = + bv = – b(UA + UB) = mg by A bt B bt

1 since K = mv2 2 bK = mv bv = p bv to get no change in action set changes in kinetic and potential energy equal:

b(UA + UB) = mg by b(K + K ) = p bv + p bv bp A B A A B B b(K + K ) = – by A B bt by b(K + K ) = (p – p ) A B A B bt bp by bp = – mg b(K + K ) = – bp = – by bt A B bt bt Newton’s Law is the condition for the action not to vary Figure 3. From least action to Newton.

January 2005 P HYSICS E DUCATION 31 J Ogborn andEFTaylor energy function U(y) is given by the expression action is unvarying with respect to such a worldline mgy and the Lagrangian L becomes shift. Take ﬁrst the change in potential energy. As = − = 1 2 − L K U 2 mv mgy can be seen in ﬁgure 3, the average change in y along both parts A and B due to the shift δy in with the v in the y-direction. the centre point is δy/2. The football is higher up Now look at ﬁgure 3, which plots the vertical by δy/2 in both parts of the segment of worldline. position of the centre of a rising and falling football Thus the total change in potential energy is as a function of time. This position–time curve is called the worldline. The worldline stretches from δ(UA + UB) = mg δy. (6) the initial position and time—the initial event of launch—to the ﬁnal event of impact. Suppose that Thus we have the change in the sum of potential the worldline shown is the one actually followed , i.e. the second term in equation (5). An by the football. This means that the value of the easy ﬁrst wrestling move! action along this worldline is a minimum. Getting the ﬁrst term, the change in the sum Now use scientiﬁc martial arts to throw the of kinetic energies, needs a triﬂe more agility. problem onto the mat in one overhand ﬂip: If the The effect of the shift δy in the centre point of action is a minimum along the entire worldline the worldline is to increase the steepness of the with respect to adjacent worldlines, then it is a worldline in part A, and to decrease it in part minimum along every segment of that worldline B (remember the end points are ﬁxed). But the with respect to adjacent worldlines along that steepness of the worldline (also the graph of y segment. ‘Otherwise you could just ﬁddle with against t) is just the velocity v in the y-direction. just that piece of the path and make the whole And these changes in v are equal and opposite, as integral a little lower.’ (Feynman 1964, p 19-8). can be seen from ﬁgure 3. In parts A and B the So all we have to do is to ensure that the action is velocity changes by a minimum along any arbitrary small segment. δy δy We take a short segment of the worldline and δvA = δvB =− . δt δt vary the y-position of its centre point, shifting it up by a small amount δy. Then we demand the But what we want to know is the change in the condition that such a shift does not change the kinetic energy. Since action along the segment (leaving the end-points K = 1 mv2 ﬁxed). 2 We consider the change in the action over the then for small changes two parts A and B of the segment, which occupy equal times δt. The change in the action is δK = mv δv = p δv.

δSAB = δSA + δSB = (δLA + δLB)δt. Thus we get for the sum of changes in kinetic energy Since δt is ﬁxed, the only changes that are = those in δLA and δLB. These are δ(KA + KB) pAδvA + pBδvB or, remembering that the changes in velocity are δLA = δ(KA − UA)δLB = δ(KB − UB). equal and opposite, Rearrange the terms to write their sum as δy δ(KA + KB) = (pA − pB) . δ(LA + LB) = δ(KA + KB) − δ(UA + UB). (5) δt The change in momentum from part A to part B It remains to see how the two sums KA + KB and of the segment is the change δp = pB − pA. UA + UB change when the centre of the worldline Thus the total change in kinetic energy needed for is shifted by δy. We shall require the changes to equation (5) is simply be equal, so that the change in the action, which is their difference multiplied by the ﬁxed time δp δ(K + K ) =− δy. (7) interval δt, then vanishes, and we know that the A B δt

32 P HYSICS E DUCATION January 2005 Quantum physics explains Newton’s laws of motion Now for the ﬁnal throw to the mat! conservative —where all can be Equation (5) says that if the changes of kinetic gotten from a potential function." (Feynman 1964, energy (from equation (7)) are equal to the changes p 19-7). dissipates organized mechanical in the potential energy (from equation (6)) then energy into disorganized internal energy; we are their difference is zero, and the action does not not trying to explain in this change (for ﬁxed δt). Thus we must equate the two article! to ﬁnd the condition for no change in the action: δp Classical and quantum? − δy = mg δy. δt Let’s get away from the and try to describe how it all works at the fundamental That is, the change in the action is zero if level. Newton’s law ﬁxes the path so that changes δp in phase from changes in kinetic energy exactly =−mg. match those from changes in potential energy. δt This is the modern quantum ﬁeld theory view In the special case chosen, mg is the gravitational of forces: that forces change phases of quantum in the negative (downward) direction. In the amplitudes. We see it here in elemental form. limit of small segments this result becomes What Newtonian physics treats as cause and effect (force producing ) the quantum ‘many dp F = Newton’s Second Law. paths’ view treats as a balance of changes in dt phase produced by changes in kinetic and potential energy. It’s all over: the problem lies at our feet! The force So ﬁnally we have come all the way from must be equal to the rate of change of momentum. the deepest principle of nonrelativistic quantum Newton’s law is a consequence of the principle mechanics—Explore all paths!—to the deepest of least action, which is itself a consequence of principle of classical mechanics in a conservative quantum physics. potential—Follow the path of least action! And What about a more general potential energy from there to the classical mechanics taught in function? To begin with, every actual potential every high school. The old of the classical energy function is effectively linear for a small increment of . So the above analysis world come straight out of the new truths of the still works for a small enough increment along quantum world. Better still, we can now estimate every small segment of every actual potential the limits of accuracy of the old classical truths. energy function. By slightly modifying the derivation above, you can show that the general Half-truths we have told case leads to In this article we have deliberately stressed an dU dp − = y important half-, that every quantum object dy dt (a photon, an electron etc) is signiﬁcantly like where −dU/dy is the more general expression every other quantum object: namely, that all for force, and we have added the subscript y to obey the same elemental quantum command Try the momentum, since the motion took place up ! But if electrons as a behaved and down along the y-axis. For three-dimensional exactly like photons as a group, no atom would motion there are two more of similar exist and neither would our current , our form, one for the x-direction and one for the , our Earth, nor we who write and read this z-direction (and, to be technically correct, the article. of U becomes a partial derivative with The Try everything! half-truth does a good job respect to that coordinate). of describing the motion of a single photon or a “I have been saying that we get Newton’s single electron. In that sense it is fundamental. law. That is not quite true, because Newton’s But the behaviour of and the structure of law includes nonconservative forces like friction. depend respectively on collaboration among Newton said that ma is equal to any F . But photons and collaboration among electrons. And the principle of least action only works for collaboration is very different between electrons