Matter Waves
The Photon: Photoelectric and Compton Effects established that light, which every nineteenth-century experiment had shown to be a wave, delivers energy and momentum in indivisible amounts. De Broglie's proposal [deBroglie:1923] [deBroglie:1925] was the converse, and it was made on grounds of symmetry rather than of data: if a wave carries momentum \(p=h/\lambda\), then a particle of momentum \(p\) should be accompanied by a wave of wavelength \(h/p\). The hypothesis was not idle. It supplies at once the one thing Bohr's model of Atomic Models and Spectra had to assume — the quantization of angular momentum becomes the condition that an integer number of wavelengths fit the orbit — and it completes the analogy of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy, in which classical mechanics stands to an unknown wave theory as geometrical optics stands to wave optics. The unknown wavelength is \(h/p\), and the chapter's business is the evidence that it is real.
That evidence is unusually good and it keeps improving, which is why this chapter is long on experiment. Electrons were diffracted by crystals within three years [Davisson:1927] [Thomson:1927], neutral atoms within five [Estermann:1930]; the interference pattern was later built up one electron at a time [Merli:1976] [Tonomura:1989], so that no account in terms of a collective effect survives; neutron interferometers made the phase itself a laboratory observable and measured shifts due to gravity [Colella:1975] and to a \(360^\circ\) rotation of a spin-half particle [Werner:1975]; and molecules of thousands of atoms interfere today [Arndt:1999] [Fein:2019] with no sign of a boundary beyond which the relation fails. The chapter closes on wave packets and group velocity — the machinery that reconciles a wave with a localized particle — and hands the problem of finding the wave's equation of motion to The Postulates of Quantum Mechanics.
Matter Waves: all derivations of this chapter are pending.
The de Broglie hypothesis
The relation
[Reserved: de Broglie's short note of 1923 [deBroglie:1923] and the thesis that expanded it [deBroglie:1925]; the pair of relations \(E=h\nu\) and \(p=h/\lambda\) asserted for every carrier of energy and momentum, massive or not. The relativistic argument that motivated them: a particle at rest has an internal frequency \(m c^{2}/h\), and requiring the phase of that oscillation to stay in step with a wave when the particle is seen from a moving frame (Lorentz Transformations) forces a phase velocity \(c^{2}/v\) and a group velocity \(v\). Numerical scale: an electron accelerated through \(54\,\mathrm{V}\) has \(\lambda=0.167\,\mathrm{nm}\), comparable to a lattice spacing, which is what makes the effect observable at all.]
Phase waves and the Bohr condition
[Reserved: the standing-wave condition \(n\lambda=2\pi r\) on a circular orbit is \(L=n\hbar\), so Bohr's postulate [Bohr:1913a] ceases to be an arbitrary selection rule and becomes the requirement that the accompanying wave be single-valued. The same reading applies to Sommerfeld's action integrals [Sommerfeld:1916], which become the statement that the phase accumulated round a closed path is a multiple of \(2\pi\). This is the first argument in physics that explains an old quantum condition rather than positing it, and it is why the hypothesis was taken seriously before any experiment.]
The optical–mechanical analogy completed
[Reserved: the dictionary of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy had one incomplete entry — classical mechanics is the short-wavelength limit of some wave theory, with the part of the wavelength played by an unidentified quantity. De Broglie identifies it as \(h/p\), and the Hamilton–Jacobi equation becomes the eikonal equation of the matter wave, valid where the potential varies slowly on the scale of \(\lambda\). The criterion for the classical limit follows, and it is quantitative: it fails for an electron in an atom and holds for a marble. Schrödinger constructed the wave equation whose eikonal limit this is, and said so in his first communication [Schroedinger:1926a].]
Electron diffraction
Elsasser's prediction
[Reserved: Elsasser's one-page note [Elsasser:1925] pointing out that de Broglie wavelengths of slow electrons match crystal lattice spacings, so a crystal should diffract an electron beam exactly as it diffracts X-rays [Bragg:1913b], and that the anomalous scattering already reported by Davisson and Kunsman might be this. The prediction was published two years before the confirming experiments and was unknown to one of the two groups that performed them; the section records it because the priority is routinely assigned to the experimenters alone.]
Davisson and Germer
[Reserved: the Bell Telephone Laboratories experiment [Davisson:1927]. A vacuum accident recrystallized a nickel target into a few large crystals; the scattered intensity then showed sharp maxima at particular angles and energies where before it had been smooth. At \(54\,\mathrm{eV}\) the beam scattered from the (111) face peaks near \(50^\circ\), giving \(\lambda=0.165\,\mathrm{nm}\) against the de Broglie value \(0.167\,\mathrm{nm}\). Apparatus, the Faraday-box collector, the angle and voltage scans, and the systematic correction for the inner potential of the crystal are all to be given in full, since this is the primary evidence of the chapter.]
A beam of electrons of well-defined kinetic energy scattered from a crystal produces sharp intensity maxima at particular angles, exactly as X-rays of the same wavelength do on the same crystal, and the wavelength read off the diffraction geometry is \(h/p\). Davisson and Germer, scattering electrons of \(54\,\mathrm{eV}\) from the (111) face of a nickel single crystal, found a pronounced maximum near \(50^\circ\) corresponding to a wavelength of \(0.165\,\mathrm{nm}\), against the de Broglie value \(0.167\,\mathrm{nm}\) [Davisson:1927]. Thomson and Reid passed electrons of \(10\text{–}60\,\mathrm{keV}\) through thin polycrystalline films of celluloid, gold and aluminium and photographed concentric rings whose radii scale as the inverse square root of the accelerating voltage, independently of the orientation of the target [Thomson:1927].
Derivation. Assume de Broglie's relation \(\lambda=h/p\) [deBroglie:1923] [deBroglie:1925] and compute the wavelength of the beam. An electron accelerated from rest through a potential difference \(V\) small enough that \(eV\ll m_{\text{e}}c^{2}\) acquires \(p=\left(2m_{\text{e}}eV\right)^{1/2}\), so
which at \(54\,\mathrm{V}\) is \(0.167\,\mathrm{nm}\) — comparable with a lattice spacing, which is the whole reason the effect is observable at all. The rest is the geometry of a grating. A row of scatterers of spacing \(d\) on the crystal surface reinforces at angles satisfying \(d\sin\theta=n\lambda\); the nickel row spacing is \(0.215\,\mathrm{nm}\), measured independently by X-ray crystallography [Bragg:1913b], and this puts the first order at \(50^\circ\), where the observed maximum lies. Reading \(\lambda=d\sin\theta\) back off that geometry gives \(0.165\,\mathrm{nm}\). Nothing was adjusted: \(d\) comes from X-rays and \(\lambda\) from the accelerating voltage, and the two independently determined numbers agree.
Equation (71.1) also accounts for the transmission geometry. In a polycrystalline film the crystallites take every orientation, so the reinforcement condition is met on cones about the beam and the pattern is a set of rings whose radii are proportional to \(\lambda\) and therefore to \(V^{-1/2}\), which is what is photographed [Thomson:1927]. Because a polycrystal has no preferred orientation, that observation also removes any explanation resting on a surface effect peculiar to one crystal face.
∎Thomson's transmission experiment
[Reserved: G. P. Thomson and Reid passed electrons of \(10\text{–}60\,\mathrm{keV}\) through thin films of celluloid, gold and aluminium and photographed concentric Debye–Scherrer rings whose radii scale as the inverse square root of the accelerating voltage [Thomson:1927] — a polycrystalline pattern, independent of target orientation and therefore of any surface effect, complementing the single-crystal geometry of Section 71.2.2. The historical remark is irresistible and belongs in a remark rather than the body: J. J. Thomson showed the electron is a particle [Thomson:1897] and G. P. Thomson that it is a wave, and both were right.]
Interference of atoms and molecules
Helium beams: Estermann and Stern
[Reserved: diffraction of neutral helium atoms and of hydrogen molecules from the cleavage face of lithium fluoride [Estermann:1930], with the wavelength varied by changing the source temperature and checked against \(h/p\) over the thermal distribution. This is the decisive extension: helium is electrically neutral and structureless, so no explanation in terms of charge, of the Coulomb field, or of anything peculiar to the electron survives. The molecular-beam technique is Stern's, the same apparatus lineage as Experiment: Stern–Gerlach.]
Beams of helium atoms and of hydrogen molecules reflected from the cleavage face of a lithium fluoride crystal show diffraction maxima at the angles the grating condition assigns to the wavelength \(h/p\), and those maxima move as predicted when the wavelength is changed by changing the temperature, and hence the mean momentum, of the source [Estermann:1930]. Helium is electrically neutral, chemically inert and has no permanent electric or magnetic moment, so no account of the phenomenon in terms of charge, of the Coulomb field, or of anything peculiar to the electron survives this measurement: the wave behaviour belongs to matter as such.
Diffraction of a thermal beam: the grating condition for a two-dimensional surface lattice, and the shape of the observed maxima when the incident beam is not monochromatic but carries the Maxwellian velocity distribution of the oven, each velocity class contributing its own de Broglie wavelength. What must be shown is that the position of each maximum tracks the most probable momentum in the predicted way as the source temperature is varied, which is the quantitative content of the experiment.
The electron double slit
[Reserved: Jönsson's fabrication of slits of about \(0.5\,\mu\mathrm{m}\) width in copper foil by electron lithography, and the resulting two-, three-, four- and five-slit interference patterns formed with \(50\,\mathrm{keV}\) electrons and magnified electron- optically [Joensson:1961] — Young's experiment (Experiment: Wave Optics) performed with matter. Why it took thirty-four years after Section 71.2.2: the wavelength is four orders of magnitude below optical, so the slit separation and the detection geometry are the whole difficulty.]
Building the pattern one particle at a time
[Reserved: the experiment that removes the last collective explanation. Merli, Missiroli and Pozzi [Merli:1976] and, with a field-emission source, a biprism and a position-sensitive detector, Tonomura and collaborators [Tonomura:1989] recorded interference with at most one electron in the apparatus at a time; individual detections arrive as points at apparently random places and the fringes emerge only in the accumulated histogram. The wave is therefore a wave of probability amplitude, in Born's sense [Born:1926b], and not a wave of anything that is spread out — the statement that The Postulates of Quantum Mechanics formalizes and Interpretations (Evidence-Anchored) declines to explain further.]
When the source is weakened until at most one electron is inside the apparatus at any moment, the detector still registers individual, point-like arrivals, at positions that show no pattern one by one. The interference fringes appear only in the accumulated histogram of many thousands of such arrivals, and they are the same fringes, in the same places and with the same spacing, as those formed at high intensity [Merli:1976] [Tonomura:1989]. The pattern is therefore not an effect of electrons acting on one another, and the multiple-slit fringes formed with \(50\,\mathrm{keV}\) electrons [Joensson:1961] are a property of each electron separately.
The accumulation statistics: that the arrival positions are independent samples drawn from the squared modulus of the single-particle amplitude, so that the histogram of many arrivals converges on the intensity pattern of the wave while no individual arrival carries any fringe at all. What must be derived is the convergence itself and its rate, and what must be stated is that the probabilistic reading of the amplitude is an additional postulate and not a consequence of the de Broglie relation.
Neutron interferometry
The perfect-crystal interferometer
[Reserved: Rauch, Treimer and Bonse's interferometer cut from a single silicon crystal, in which three parallel lamellae split, redirect and recombine a thermal-neutron beam by Bragg reflection [Rauch:1974], with an arm separation of centimetres and a wavelength near \(0.18\,\mathrm{nm}\). It converts the phase of a matter wave into a countable intensity and thereby makes phase an experimental quantity for a massive particle. Requirements: lattice perfection over the whole device, thermal and vibrational stability; this is the instrument behind the next three subsections.]
Gravitationally induced phase
[Reserved: rotating the interferometer about the incident beam tilts the plane containing the two paths out of the horizontal, so they run at different heights in the Earth's field and accumulate a relative phase proportional to \(m^{2}gA\lambda/\hbar^{2}\); Colella, Overhauser and Werner observed the predicted fringe count [Colella:1975]. Two things make it remarkable and both belong here: the inertial and gravitational masses appear as a product \(m^{2}\), so this is a quantum experiment in which the equivalence principle of The Equivalence Principle and Classical Tests is probed; and \(\hbar\) and \(g\) appear in the same formula, which no other laboratory measurement of the period achieved.]
Rotating a neutron interferometer about the direction of the incident beam tilts the plane containing its two paths out of the horizontal, so that the two paths run at different heights in the Earth's gravitational field. The counted intensity in each exit beam then oscillates with the tilt angle \(\alpha\), with a phase difference
between the paths, where \(A\) is the area they enclose and \(\lambda\) the neutron wavelength. Colella, Overhauser and Werner observed the predicted number of fringes as the interferometer was rotated [Colella:1975]. Two features are worth naming, and both are unusual. The mass enters squared, once inertially through the wavelength and once gravitationally through the potential, so this is a quantum interference experiment that probes the equivalence principle of The Equivalence Principle and Classical Tests; and \(\hbar\) and \(g\) appear together in a single measured quantity, which no other laboratory experiment of its period achieved.
The gravitationally induced phase: the local wavenumber of a neutron in a linear potential, obtained from the free-particle dispersion relation with the gravitational potential energy subtracted from the total energy; the phase accumulated along each arm as the line integral of that wavenumber; and the reduction of the difference to the stated product of the squared mass, the acceleration of free fall, the enclosed area and the wavelength. The systematic corrections that must accompany it — bending of the crystal under its own weight as the device is rotated, and the Sagnac phase from the Earth's rotation — belong with it.
Rotation by a full turn changes the sign
[Reserved: a magnetic field on one arm precesses the neutron spin; the interference contrast returns to its original value only after a rotation of \(720^\circ\), the amplitude changing sign at \(360^\circ\). Measured independently by Werner and collaborators [Werner:1975] and by Rauch's group [Rauch:1975]. This is a direct observation of the double cover \(\Spin(3)\to\SO(3)\) of Lie Groups, Lie Algebras, and Fibre Bundles, that is of the fact that a spin-half state transforms under a representation of the covering group and not of the rotation group itself (Angular Momentum and Spin); the sign is unobservable for a whole system and observable in an interferometer, which is the distinction the experiment makes concrete.]
A magnetic field applied on one arm of a neutron interferometer precesses the neutron spin through a controlled angle, and the interference contrast is recorded as a function of that angle. The contrast does not return to its initial value after a precession of \(360^\circ\): the amplitude has changed sign there, and the original state is recovered only after \(720^\circ\). The double-turn periodicity was measured independently by two groups, on different instruments [Werner:1975] [Rauch:1975]. The sign is unobservable for an isolated system and becomes observable here only because the interferometer compares it against an amplitude that was not rotated.
The rotation operator for a spin-half state: that the generator of rotations has half-integer eigenvalues, so that a rotation by a full turn acts as minus the identity rather than the identity, and that a spin-half state therefore carries a representation of the covering group and not of the rotation group. The interferometric argument that turns this global sign into a measurable relative phase must accompany it, together with the relation between the applied magnetic field, the transit time and the precession angle.
Bound states of a neutron in the Earth's field
[Reserved: ultracold neutrons above a horizontal mirror form quantized vertical bound states in the linear gravitational potential; the transmission through a slit of variable height rises in steps at the predicted heights, with the lowest state around \(13\,\mu\mathrm{m}\) and an energy of a few \(\mathrm{peV}\) [Nesvizhevsky:2002]. Airy functions as the eigenfunctions, worked in Elementary Quantum Systems; the experiment is the cleanest existing observation of quantized motion in a gravitational field and is cited again in Quantum Gravity: The Honest Status for what it does not test.]
Ultracold neutrons travelling horizontally above a flat, polished mirror are bound vertically between the mirror below and the linear gravitational potential above, and the states of that vertical motion are discrete. The transmitted flux, measured as an absorber is lowered towards the mirror, does not fall smoothly with the gap but in steps: no neutron passes at all below a gap of about \(13\,\mu\mathrm{m}\), which is the extent of the lowest state, and further steps follow at the heights of the states above it [Nesvizhevsky:2002]. The energy of the lowest state is about \(1.4\,\mathrm{peV}\). This is the cleanest existing observation of quantized motion in a gravitational field.
The eigenvalue problem for a particle in a linear potential above a hard wall: the equation reduces to Airy's equation, the admissible solutions are the decaying Airy function shifted so as to vanish at the mirror, and the eigenvalues are therefore fixed by its zeros. The classical turning-point heights follow from the eigenvalues, and it is those heights that the absorber measures; the step positions predicted this way are to be compared with the observed ones.
How large can an interfering object be?
Fullerenes
[Reserved: far-field diffraction of C\(_{60}\) from a nanofabricated grating, with a de Broglie wavelength near \(2.5\,\mathrm{pm}\), some four hundred times smaller than the molecule itself [Arndt:1999]; the molecules are hot, populating millions of internal vibrational states and radiating thermal photons, and the fringes survive — so internal complexity does not by itself destroy coherence. What does destroy it is the emission or scattering of a photon carrying which-path information, which the same apparatus demonstrates by admitting a background gas, and which is the experimental basis of Open Quantum Systems and Decoherence.]
Interference survives in objects far larger than their own de Broglie wavelength and far more complicated than a point. C\(_{60}\) molecules diffracted from a nanofabricated grating produce fringes at a wavelength near \(2.5\,\mathrm{pm}\), some four hundred times smaller than the molecule itself, even though the molecules are hot enough to populate millions of internal vibrational states and to radiate thermal photons while in flight [Arndt:1999]. Near-field interferometry has since shown fringes for tailored molecules built from more than two thousand atoms, with masses above \(25000\) atomic mass units, at wavelengths of order \(50\,\mathrm{fm}\) [Fein:2019]. No deviation from \(\lambda=h/p\) has been found at any mass yet reached, and no boundary beyond which the relation fails has appeared.
Two things, of which only the first is optics. The near-field Talbot–Lau fringe condition and its scaling with mass and wavelength, which is what fixes the grating periods and the source coherence a given molecular mass demands. And the decoherence criterion: how much which-path information the emission or scattering of a single photon of given wavelength carries, and hence the residual gas pressure and internal temperature at which the fringes must disappear — the quantitative statement of what these experiments do and do not bound.
The present limits
[Reserved: Talbot–Lau near-field interferometry has since shown interference for tailored molecules of more than two thousand atoms and masses above \(25000\) atomic mass units [Fein:2019], at wavelengths of order \(50\,\mathrm{fm}\). No deviation from the de Broglie relation has been seen at any mass. The section should state plainly what this does and does not bound: it constrains proposed objective-collapse modifications of quantum mechanics, which is a genuine experimental frontier, and it is reported honestly in What We Observe but Do Not Understand rather than as a settled matter.]
Wave packets
Superposition and localization
[Reserved: a single plane wave \(\ee^{\ii(kx-\omega t)}\) describes a particle of definite momentum and no position at all, so a localized particle must be a superposition — a Fourier integral over \(k\) with an envelope, in the sense of Fourier Analysis and Integral Transforms. The Gaussian packet as the worked case: its width in \(x\), its width in \(k\), and the product of the two. Nothing in this subsection is specific to quantum mechanics; it is the classical mathematics of Oscillations and Mechanical Waves applied to the de Broglie wave, which is exactly why it settles the apparent paradox.]
Phase velocity, group velocity and spreading
[Reserved: the free-particle dispersion relation \(\omega=\hbar k^{2}/2m\) gives a phase velocity \(\hbar k/2m\) — half the particle's speed, and for the relativistic relation greater than \(c\), which alarms nobody once it is seen that no signal travels at it — and a group velocity \(\dd\omega/\dd k=\hbar k/m=v\), the particle's own velocity. The dispersion is quadratic and therefore non-vanishing, so every free matter wave packet spreads, at a rate computed here and observed in the expansion of released cold-atom clouds (Experiment: Bose–Einstein Condensation). Compare the non-dispersive string of Oscillations and Mechanical Waves.]
Bandwidth and the uncertainty relation
[Reserved: for any wave, the product of the spatial width of a packet and the width of its spectrum is bounded below by a number of order unity — a theorem about Fourier transforms [Griffiths:2018] that predates quantum mechanics. Insert \(p=\hbar k\) and it reads \(\Delta x\,\Delta p\gtrsim\hbar/2\): the uncertainty relation is what the de Broglie relation makes of a bandwidth theorem. Heisenberg's own argument was operational, through the disturbance caused by a measurement [Heisenberg:1927], and the two readings are not the same statement; which is which, and which one the modern inequality proves, is settled in The Postulates of Quantum Mechanics.]
From a wavelength to a wave equation
What the hypothesis does not supply
[Reserved: \(\lambda=h/p\) fixes the wavelength of a free particle and says nothing about a particle in a potential, about the equation the wave obeys, about what the wave is a wave of, or about spin. Each gap is named here with the chapter that closes it: the equation of motion [Schroedinger:1926a] in The Postulates of Quantum Mechanics, the probabilistic reading [Born:1926b] there too, the spin degree of freedom in Angular Momentum and Spin, and the relativistic treatment — for which the naive substitution into \(E^{2}=p^{2}c^{2} +m^{2}c^{4}\) is the wrong first guess — in The Klein–Gordon Equation and The Dirac Equation.]
Matter waves as instruments
[Reserved: the hypothesis is now hardware. Electron and neutron diffraction are the standard probes of crystal structure and magnetic order (Magnetism in Matter); atom interferometers built from optical gratings [Keith:1991] serve as gravimeters, gradiometers and gyroscopes; and the recoil frequency measured in an atom interferometer gives \(h/m\) and hence the fine-structure constant to parts in \(10^{10}\) [Parker:2018] [Morel:2020], an input to the tests of Experiment: The Electron Anomalous Magnetic Moment. A hypothesis that began as a symmetry argument now calibrates the constants of physics, which is the strongest form of confirmation available.]