FEATURES www.iop.org/journals/physed Quantum physics explains Newton’s laws of motion Jon Ogborn1 and Edwin F Taylor2 1 Institute of Education, University of London, UK 2 Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract Newton was obliged to give his laws of motion as fundamental axioms. But today we know that the quantum world is fundamental, and Newton’s laws can be seen as consequences of fundamental quantum laws. This article traces this transition from fundamental quantum mechanics to derived classical mechanics. Explaining Fermat’s principle: source of the key It is common to present quantum physics and the quantum idea behaviour of quantum objects such as electrons or Geometrical optics predicts the formation of photons as mysterious and peculiar. And indeed, images by light rays. Examples are images due in Richard Feynman’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 geometrical optics—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 time. 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 waves and particles. Instead we of reflection. Fermat’s principle also accounts turn the job of explaining around. We start from for the action 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 work. 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 26 P HYSICS E DUCATION 40 (1) 0031-9120/05/010026+09$30.00 © 2005 IOP Publishing Ltd Quantum physics explains Newton’s laws of motion goes very deep and was delivered fully only in the emission of a photon at one place and time and its twentieth century. Here is the key idea: The light detection at a different place and time. (There are explores all possible paths between emission and also rules for how the length of the arrow changes reception. Later we will find a similar rule for with distance, which yield an inverse square law motion of atomic particles such as the electron: a of intensity with distance. For simplicity we particle explores all possible paths between source consider only cases where the distances vary little and detector. This is the basic idea behind the and changes in arrow length can be ignored.) formulation of quantum mechanics developed by The resultant arrow determines the probabil- Richard Feynman (1985). ity of the event. The probability is equal to the The idea of exploring all possible paths raises (suitably normalized) square of the length of the two deep questions: (1) What does it mean to arrow. In this way the classical result that the in- explore all paths? (2) How can ‘explore all tensity is proportional to the square of the wave paths’ be reconciled with the fact that everyday amplitude is recovered. objects (such as footballs) and light rays follow We have outlined Feynman’s simple and vivid unique single paths? To answer these questions description of ‘quantum behaviour’ for a photon. is to understand the bridge that connects quantum In effect he steals the mathematics of Huygens’ mechanics to Newton’s mechanics. wavelets without assuming that there are waves. For Huygens, wavelets go everywhere because What does Explore all paths! mean? that is what waves do. For Feynman, photons ‘go everywhere’ because that is what photons do. The idea of exploring all paths descends from Christiaan Huygens’ idea of wavelets (1690). Huygens explained the propagation of a wavefront The bridge from quantum to classical by imagining that each point on the wavefront physics sends out a spherical wavelet. He could then The next question is this: How can Nature’s show that the wavelets reconstitute the wavefront command Explore all paths! be made to fit with at a later time; the parts of the wavelets going our everyday observation that an object such as a everywhere else just cancel each other out. In 1819 football or a light ray follows a single path? the French road and bridge engineer Augustin The short answer is It does not follow a Fresnel put the idea on a sound mathematical single path! There is no clean limit between basis and used it to explain optical diffraction and particles that can be shown to explore many paths interference effects in precise detail. and everyday objects. What do you mean by In the 1940s Richard Feynman (following a everyday objects anyway? Things with a wide hint from Dirac) adapted Huygens’ idea to give range of masses and structural complexity are quantum physics a new foundation, starting with ‘everyday’ objects of study by many scientists. the quantum of light: the photon. Nature’s simple A recent example is interference observed for three-word command to the photon is Explore all the large molecules of the fullerene carbon-70, paths!; try every possible route from source to which has the approximate mass of 840 protons detector. Each possible path is associated with a (www.quantum.at). change of phase. One can imagine a photon having In fact quantum behaviour tapers off gradually a ‘stopclock’ whose hand rotates at the classical into classical behaviour. This and the following frequency of the light. The rotation starts when sections show you how to predict the range of the photon is emitted; the rotation stops when the this taper for various kinds of observations. In the photon arrives at the detector. The final position meantime, as you look around you, think about the of the hand gives an ‘arrow’ for that path. deep sense in which the football goes from foot to The photon explores all paths between goal by way of Japan. So do the photons by which emission event and a possible detection event. The you see your nearby friend! arrows for all paths are to be added head-to-tail The key idea is illustrated in figure 1 for the (that is, taking account of their phases, just as case of photons reflected from a mirror. The mirror wavelets are to be superposed) to find the total is conceptually divided into little segments, sub- resultant quantum amplitude (resultant arrow) for mirrors labelled A to M. The little arrows shown an event. This resultant arrow describes the under each section in the middle panel correspond 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 time resulting arrow comes from the small proportion of arrows that ‘line up’, and almost nothing comes from those that ‘curl up’. ABCDEFGHI J KLM Now you see how Explore all paths! leads to a narrow spread of paths that contribute significantly to the resulting arrow at P. And that narrow spread I J must lie near the path of minimum time of travel, H M B because that is where the time, and so the phase, C K L varies only slightly from path to nearby path. A G Here then is the answer to Fermat’s critics. D F E They asked ‘How could the light possibly know in Figure 1. Many paths account of reflection at a mirrror advance which path is the quickest?’ Answer: the (adapted from Feynman 1985, p 43). Each path the light photon does not know in advance: it explores all could go (in this simplified situation) is shown at the top, with a point on the graph below showing the time paths.