Illustrated Physics
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Matter waves — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. An electron's wavelength against the voltage that accelerated it. The de Broglie wavelength of an electron after falling through a potential difference, in picometres. At 100 volts it is 122.6 picometres, at 400 volts it is 61.3 picometres, at 900 volts it is 40.9 picometres. The wavelength goes as the inverse square root of the voltage, so quadrupling the voltage halves it. The calculation is non-relativistic; at a kilovolt that costs a tenth of a per cent. The dashed line is the 215 picometre spacing between atomic planes in nickel, which is what makes an electron beam diffract off a crystal at all.

    Everything has a wavelength, and almost nothing shows it

    If light with a momentum can behave like a particle, a particle with a momentum can behave like a wave. The wavelength is Planck's constant over the momentum, which for anything larger than a molecule is a number too small to have consequences.

    part 1 · quantum
  2. The interference pattern arriving one particle at a time. The same double slit — 100 micrometres apart, slits 40 micrometres wide, lit at 633 nm, screen 1 metre away — recorded after 20, 200, 1000 arrivals, with the intensity that governs them plotted underneath. Each arrival is a single dot in one place, drawn at a position sampled from that intensity. After 20 there is no pattern to see; after 1000 the fringes are unmistakable, with the dark ones exactly 6.33 millimetres apart — the wavelength times the screen distance over the separation. The bright ones are not evenly spaced, because the single-slit envelope pulls each maximum toward the centre; its first zero is at 15.8 millimetres and is set by the width of one slit alone.

    One arrival at a time, and the pattern still appears

    Send particles through a double slit slowly enough that only one is ever in the apparatus, and each arrives as a single dot in one place. Wait, and the dots assemble into fringes that no dot knew about.

    part 2 · quantum
  3. Outside the winding, where nothing is supposed to be. The mid-plane of a solenoid 20 radii long, wound at 8 turns per radius, with the field summed turn by turn rather than assumed. The upper curve is the axial field against distance from the axis: it holds up across the winding and collapses outside, reaching 0.47 per cent of its central value at 2 radii. The lower curve is the flux enclosed by a circle of that radius, which is what the vector potential integrates to. The two behave completely differently, and that difference is the whole subject: the field an electron outside can feel has effectively gone, and the flux it encircles has not. The enclosed flux does fall, by 3.29 per cent out to four radii, because the lines that leave the ends come back through the plane outside the coil — that return flux is the leak a real experiment has to defeat, and the reason the definitive versions used a closed toroidal magnet with no ends at all.

    The phase a magnet leaves on a path it never touched

    An electron beam split around a solenoid comes back with its fringes displaced, although neither path ever entered a magnetic field. What the electron responds to is the flux it went round, and the only local quantity that knows about that flux is the potential.

    part 3 · quantum
  4. Two paths through a neutron interferometer at two heights. A neutron interferometer cut from one silicon crystal, tilted by 30 degrees about its incoming beam so that one path runs higher than the other. Left: the two paths, split at the first slab, turned at the second and recombined at the third, enclosing 10.1 square centimetres; the heights are drawn exaggerated. Taken as a rectangle of the same area, the upper path runs 15.8 mm higher for 3.2 cm. There a neutron of wavelength 1.445 Å, moving at 2738 m/s, is slower by 56.5 micrometres per second, so its wavelength is longer by 20.6 parts per thousand million. Over 3.2 cm that accumulates 28.7 radians less phase than the lower leg — 4.6 whole fringes — computed by integrating the local wavenumber along both legs and checked against 2πm²gλA sin α / h².

    The fall that leaves the mass in the phase

    Every body falls the same way whatever its mass, and a neutron is no exception. But a neutron is also a wave, and the phase that wave accumulates while falling depends on the mass — as its square, at a fixed wavelength. Tilt a neutron interferometer so that one path runs a centimetre higher than the other and the neutrons swing between its two detectors, which in 1975 was the first measurement in which gravity and quantum mechanics both had to be right at once.

    part 4 · quantum
  5. Only a slow grating sends everything one way. The largest fraction of particles, arriving at Bragg's angle, that a standing light wave can put into the first diffracted order, choosing the wave's depth freely, against how long it acts, in units of ħ divided by the two-kick recoil energy, on a logarithmic axis. A brief pulse acts as a thin phase grating and can put at most 33.9 per cent into that order (dashed), the Raman–Nath limit, spreading the rest over the others. A pulse lasting longer than a few units has time to notice that only one exchange conserves energy, and can put nearly all of it there: 99.9 per cent at the longest time drawn. The same distinction between thin and thick gratings decides whether sound in a crystal diffracts light into many orders or one.

    The grating made of light

    A diffraction grating is ordinarily made of matter and diffracts light. In 1933 Pyotr Kapitza and Paul Dirac proposed the opposite: a standing wave of light, whose bright and dark bands repeat every half wavelength, should diffract a beam of electrons. It took fifty-three years to see the effect with atoms and sixty-eight to see it with electrons, because a free electron barely notices light. The grating works by momentum: it can only change a particle's motion in whole pairs of photon kicks. A brief flash scatters particles into a spread of orders in proportions fixed by Bessel functions; a slow, gentle one sends every particle one way, which is how laser light became the mirrors and beam splitters of atom interferometers.

    part 5 · quantum
  6. The heights a neutron is allowed to bounce to. A neutron above a horizontal mirror in the Earth's gravity: the potential energy mgz (straight line), the first four allowed energies and, drawn on each, the probability density of the neutron's height. The natural length is z₀ = (ħ²/2m²g)^(1/3) = 5.87 μm and the natural energy mgz₀ = 0.602 peV. The allowed energies are 1.41, 2.46, 3.32, 4.08 peV — pico-electronvolts — and a classical neutron with each energy would rise to 13.7, 24.0, 32.4, 39.8 μm. Each density is an Airy function, with n − 1 nodes, leaking a little above the classical turning height and vanishing at the mirror. The lowest state never touches the mirror and never rises above about 25 μm.

    The bounce that comes in fixed heights

    A ball dropped on a floor can bounce to any height. A neutron cannot. Slow enough, and resting on a flat mirror in the Earth's gravity, it has a lowest state about fifteen micrometres high, with an energy of 1.41 pico-electronvolts, and a staircase of higher states above it — the first quantum states ever seen in a gravitational field, measured in 2002 with a slit narrower than a hair. Shaking the mirror at 255 hertz drives a neutron from the first state to the second. The staircase is now a laboratory for gravity at distances where nothing else can test it.

    part 6 · quantum
  7. The same fringes from cooler and hotter molecules. The rate at which molecules arrive across a few periods of the interference pattern behind the last grating, relative to its mean, for molecules at 1000, 2500, 3000 K, each with the visibility the emission model gives: 46 per cent, 14 per cent, 4 per cent. The cooler molecules draw fringes nearly as strong as cold ones; at 3000 K the pattern is a faint ripple on a flat background. The average number arriving is the same in every case: the heated molecules are not lost, and none of them has been deflected out of the beam. They arrive in the same places on average and no longer interfere.

    The molecule that interferes until it glows

    A molecule of seventy carbon atoms, sent through a matter-wave interferometer, travels both arms at once and lands in fringes, as an electron or a neutron does. Heat it, and the fringes fade. Nothing has touched it from outside: a hot molecule glows, and each photon it gives off is a messenger that may carry away the information of which arm it was in. Whether it does depends on one comparison — the photon's wavelength against the separation of the arms. Long photons leave the fringes alone; short ones erase them. The loss of interference that is usually described as the onset of the classical world is here a measurement made by the molecule's own light, and its threshold can be computed.

    part 7 · quantum

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