Experiment: Matter Waves Observed

Contents

Davisson and Germer: electron diffraction by a crystal (1927)
Thomson and Reid: transmission rings from thin films (1927)
Estermann and Stern: helium and hydrogen diffracted (1930)
The electron double slit, one electron at a time (1961–1989)
Colella, Overhauser and Werner: gravity shifts a phase (1975)
Molecule interferometry and the edge of coherence (1999–2019)
What the measurements settle

Matter Waves states a hypothesis that was made on grounds of symmetry and not of data: a carrier of momentum $p$ is accompanied by a wave of wavelength $h/p$. That chapter draws out what follows — Bohr's quantization condition ceases to be a postulate, the optical–mechanical analogy of geometrical optics and classical mechanics acquires its missing entry, and a localized particle becomes a packet with a group velocity. It names the experiments in passing. It does not show them being done, and the omission is worth repairing at length, because the hypothesis was so nearly baseless when it was made: de Broglie had one relation transposed from light to matter and no measurement whatever to support it.

Six measurements are reported, in the order in which each removes an alternative explanation rather than in strict order of date. Davisson and Germer scatter slow electrons from a single crystal and find the wavelength written into the diffraction geometry [Davisson:1927]; Thomson and Reid send fast electrons through a thin film and photograph rings, so the effect is not peculiar to a surface [Thomson:1927]; Estermann and Stern diffract helium atoms and hydrogen molecules, so it is not peculiar to charge [Estermann:1930]; Jönsson opens two slits and Merli, Missiroli and Pozzi and later Tonomura and collaborators build the pattern one electron at a time, so it is not a collective effect [Joensson:1961] [Merli:1976] [Tonomura:1989]; Colella, Overhauser and Werner tilt a neutron interferometer and read gravity off the fringe count, so the phase of a matter wave is an instrument reading [Colella:1975]; and molecular interferometry carries the relation to objects of thousands of atoms and shows what does end it [Arndt:1999] [Fein:2019]. The prediction was published before any of them [deBroglie:1923] [deBroglie:1925], and the suggestion that a crystal would serve as the grating was published two years before the first two [Elsasser:1925].

The arithmetic below uses the CODATA values of $h$, $e$, $m_{e}$ and the neutron mass [Mohr:2025].

Derivation pending.

Experiment: Matter Waves Observed: what is pending is the apparatus and procedure of each measurement and the tabulation of its data — scattered intensity against angle and accelerating voltage, ring radii against voltage, diffraction angle against source temperature, fringe visibility against count, neutron intensity against tilt angle, and molecular fringe contrast against internal temperature and residual gas — each in SI with its uncertainty budget. The phenomena stated below each carry their derivation inline.

Davisson and Germer: electron diffraction by a crystal (1927)

The experiment was an accident twice over. Davisson and Kunsman had been measuring the angular distribution of electrons scattered from a nickel target and finding it smooth; a vacuum failure oxidized the target, and the prolonged heating used to reduce it recrystallized the nickel into a few large crystals. When the measurements resumed, the smooth distribution had been replaced by sharp maxima at particular angles and particular beam energies. What makes the result a measurement rather than a curiosity is that both quantities entering the comparison — the wavelength predicted from the accelerating voltage and the wavelength read off the diffraction angle — are fixed independently, one by the electrical circuit and one by X-ray crystallography, and neither is adjusted.

Apparatus

[Reserved: the sealed vacuum vessel; the electron gun delivering a collimated beam of a few tens of electronvolts with the accelerating potential adjustable and measurable; the nickel target, a single crystal presenting its (111) face, mounted so that it can be rotated about the beam axis and about the axis perpendicular to it; the movable Faraday-box collector, with a retarding potential on its aperture so that only elastically scattered electrons are counted, and the galvanometer that reads its current. To be given: the beam current and its stability, the collector acceptance angle — which sets the angular resolution and therefore, by Equation (75.3), the accuracy of the whole measurement — and the working pressure, since the recrystallization that made the experiment possible was itself a vacuum accident.]

Procedure

[Reserved: two scans, and the logic of taking both. At fixed accelerating voltage the collector is swept in angle and the elastic current recorded, giving the angular distribution; at fixed collector angle the voltage is swept, giving the excitation curve. A maximum must appear in both scans at the place where the grating condition and Equation (75.1) agree, and the joint appearance is the test: a feature of the target alone would not move with voltage, and a feature of the beam alone would not move with the crystal azimuth. The azimuthal scan, in which the crystal is rotated about the beam and the maxima reappear with the three-fold symmetry of the (111) face, is what identifies the scatterer as the lattice.]

Observations and data

[Reserved: the elastic scattered current against collector angle at each accelerating voltage, and against voltage at each collector angle, with the resulting map of maxima in the angle–voltage plane. To be tabulated when written: for each maximum, the accelerating voltage and its uncertainty, the collector angle and its uncertainty, the order assigned, the wavelength deduced from the geometry, the wavelength predicted by Equation (75.1), and their ratio — the whole set, not the single case worked below, since the strength of the result lies in the maxima moving together as the voltage is changed.]

Phenomenon 75.1 (The wavelength of an electron beam can be read off a crystal).

A beam of electrons of well-defined kinetic energy scattered from a single crystal produces sharp maxima at particular angles, as X-rays of the same wavelength do on the same crystal, and the wavelength deduced from the diffraction geometry agrees with $h/p$ computed from the accelerating voltage. 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]. The lattice spacing entering the first number was measured with X-rays [Bragg:1913b] and the voltage entering the second with a voltmeter; no quantity was fitted.

Derivation. Two numbers are compared. The first is the wavelength the hypothesis assigns to the beam. An electron accelerated from rest through a potential difference $V$ has kinetic energy $T=eV$ and momentum given by $\left(pc\right)^{2}=T^{2}+2Tm_{e}c^{2}$, so

\begin{equation}\tag{75.1} \lambda=\frac{h}{p} =\frac{h}{\sqrt{2m_{e}eV}} \left(1+\frac{eV}{2m_{e}c^{2}}\right)^{-1/2}\ep \end{equation}

The bracket is the relativistic correction. At $V=54\,\mathrm{V}$ it departs from unity by $2.6\times 10^{-5}$ and may be dropped, leaving $\lambda=0.167\,\mathrm{nm}$; it may not be dropped in Section 75.2, where the same electrons are accelerated a thousand times harder.

The second number is the wavelength the diffraction geometry assigns to the beam. A row of scatterers of spacing $d$ in the surface reinforces at angles satisfying $d\sin\theta=n\lambda$, and the row spacing of nickel, $d=0.215\,\mathrm{nm}$, is known from X-ray crystallography [Bragg:1913b] — that is, from a measurement in which no electron appears. The observed first-order maximum at $\theta=50^\circ$ therefore reads

\begin{equation}\tag{75.2} \lambda=d\sin\theta =0.215\,\mathrm{nm}\times0.766 =0.165\,\mathrm{nm}\ec \end{equation}

which agrees with Equation (75.1) to $1.2\,\mathrm{\%}$.

It is worth differentiating the comparison, because that says which measurement limits it. From Equation (75.1), $\delta\lambda/\lambda=-\tfrac{1}{2}\,\delta V/V$, so a beam energy known to $1\,\mathrm{\%}$ fixes the predicted wavelength to $0.5\,\mathrm{\%}$. From Equation (75.2),

\begin{equation}\tag{75.3} \frac{\delta\lambda}{\lambda} =\cot\theta\,\delta\theta+\frac{\delta d}{d}\ec \end{equation}

and $\cot50^\circ=0.84$, so an error of one degree in locating the maximum is worth $1.5\,\mathrm{\%}$ of the wavelength. The angular measurement, not the beam energy, sets the accuracy of the test, and the residual $1.2\,\mathrm{\%}$ is smaller than the spread that a degree of angular uncertainty alone permits. It is also of the size of a known systematic effect that any comparison of this kind must confront: the crystal presents a mean inner potential to the electron, which is accelerated on entering the solid and therefore has a shorter wavelength inside it than outside. The reduction of that effect is the business of the interpretation, not of the arithmetic here; what the arithmetic establishes is that a wavelength computed from a voltage and a wavelength read off a lattice agree to about one percent, which no theory of the electron as a corpuscle predicts at all.

∎

Interpretation

[Reserved: what the maxima are and what they are not. The azimuthal symmetry identifies the lattice as the scatterer; the joint motion of the maxima in angle and voltage identifies the wavelength as the quantity being diffracted. The section must then treat the two systematic effects honestly — refraction by the mean inner potential of the crystal, which shifts the deduced wavelength, and the distinction between the two-dimensional surface-grating reading and the three-dimensional Bragg reading of the same maxima, which the experiment does not by itself settle. The forward link is that Elsasser's prediction [Elsasser:1925] preceded the observation by two years and was unknown to one of the two groups that confirmed it, so the priority customarily assigned to the experimenters alone is incomplete.]

Primary references

[Davisson:1927]. The prediction is [Elsasser:1925]; the independently measured lattice spacing is X-ray crystallography [Bragg:1913b]; the hypothesis under test is [deBroglie:1923] [deBroglie:1925].

Thomson and Reid: transmission rings from thin films (1927)

A reflection experiment on a single crystal leaves one evasion open: the maxima might be produced by something happening in the outermost atomic layer of that particular face of that particular metal. The transmission geometry closes it. Fast electrons sent through a thin polycrystalline film meet crystallites at every orientation, emerge on the far side, and land on a plate as a set of concentric rings; the target has no preferred direction, so there is no surface for the explanation to hide in, and the pattern is the electron counterpart of the Debye–Scherrer powder rings that X-rays make in the same apparatus.

Apparatus

[Reserved: the discharge tube and accelerating column delivering electrons at some tens of kilovolts, with the potential measured independently of the beam; the film target — a self-supporting celluloid film in the first communication, metal films evaporated or beaten thin in the later work — of thickness small enough to transmit and large enough to diffract; the drift space of known length between target and plate; the photographic plate; and the magnetic shielding, since a stray field bends the beam and the whole measurement is of a radius. To be given: film thickness and its uniformity, the drift length and its uncertainty, and the plate calibration used to convert blackening into a ring radius.]

Procedure

[Reserved: record the ring pattern at each of a series of accelerating voltages spanning the available range, then repeat with a target of a different material. Two comparisons are then available and neither needs an absolute calibration: at fixed voltage, the ratios of the ring radii within one pattern must equal the ratios of the inverse lattice spacings of the target, which are known from X-ray work; and at fixed target, every radius must scale with voltage as Equation (75.4) requires. Rotating or replacing the film must change nothing, which is the control that distinguishes a lattice effect from a surface or instrumental one.]

Observations and data

[Reserved: ring radii against accelerating voltage for each target material, with the plate scale and the drift length that convert them to angles. To be tabulated when written: voltage, ring order, measured radius, deduced scattering angle, deduced lattice spacing, and the spacing known from X-ray diffraction on the same material; and separately the test of the scaling law, as the measured radius times the square root of the voltage, which must be constant along a column to the precision of the plate measurement.]

Phenomenon 75.2 (Ring radii scale as the inverse square root of the accelerating voltage).

Electrons of some tens of kilovolts transmitted through a thin polycrystalline film produce concentric rings on a plate beyond it. The radii are set by the lattice spacings of the target material and scale with the accelerating voltage as $V^{-1/2}$, independently of the orientation of the film; the pattern is that produced by X-rays of the same wavelength on the same material [Thomson:1927]. Because a polycrystalline film has no preferred orientation and the beam passes through it, no account resting on a surface effect peculiar to one crystal face survives.

Derivation. For each set of lattice planes of spacing $d$, the crystallites oriented so as to satisfy $2d\sin\theta=n\lambda$ lie on a cone about the beam, and the plate at a distance $L$ downstream records a ring of radius $r=L\tan2\theta$. The angles are small, because the wavelength is far below the spacing: Equation (75.1) at $V=40\,\mathrm{kV}$ gives $\lambda=6.0\,\mathrm{pm}$, so for $d=0.2\,\mathrm{nm}$ the first order sits at $\theta=0.86^\circ$ and the ring at $2\theta=1.7^\circ$, which on a drift of $0.3\,\mathrm{m}$ is a radius of $9\,\mathrm{mm}$. To that accuracy $\tan2\theta\simeq2\theta\simeq\lambda/d$, so

\begin{equation}\tag{75.4} r\simeq\frac{L\lambda}{d}\propto\frac{1}{\sqrt{V}}\ec \end{equation}

the proportionality following from Equation (75.1) with the relativistic bracket set to unity.

Equation (75.4) is a parameter-free scaling law, and that is what makes it a test. The drift length $L$ and the plate scale enter every radius alike and cancel from any ratio, so the prediction that $r\sqrt{V}$ is constant can be checked without calibrating the instrument at all. The lattice spacing $d$ is likewise known from X-rays, so the ratios of radii within a single pattern are predicted too, and they are predicted differently for celluloid than for a metal.

One correction has become visible in moving from Section 75.1 to this geometry, and it is worth recording because it is a second, independent piece of physics appearing in the same data. The relativistic bracket of Equation (75.1), negligible at $54\,\mathrm{V}$, shortens the wavelength by $0.5\,\mathrm{\%}$ at $10\,\mathrm{kV}$ and by $2.8\,\mathrm{\%}$ at $60\,\mathrm{kV}$. An exact $V^{-1/2}$ law therefore cannot hold across the range: the rings at the top of it must come in about two percent tighter than the non-relativistic scaling predicts. At this precision the departure is at the edge of what plate measurements resolve, and the honest statement is that these experiments establish the scaling and are consistent with the correction rather than measuring it.

∎

Interpretation

[Reserved: the loophole closed — transmission through a target with no preferred orientation removes any surface explanation of Section 75.1 — and the loophole left open, which is that the electron might be diffracted as a corpuscle by a periodic force field rather than as a wave. That one is closed not here but by Section 75.4, where the interference is formed by two apertures in free space with no lattice at all. The historical remark belongs in a remark and not in the body: J. J. Thomson showed that the electron is a particle and G. P. Thomson that it is a wave, and both were right.]

Primary references

[Thomson:1927]. That first communication reports the effect for a celluloid film; the metal films that made the result independent of the chemistry of the target came in the fuller papers of the following year [Thomson:1928].

Estermann and Stern: helium and hydrogen diffracted (1930)

Everything so far has been done with electrons, and an electron is a charged point that a lattice of ions can grip in several ways. The decisive extension is to a projectile that offers nothing to grip. A ground-state helium atom is electrically neutral, chemically inert, and carries neither an electric nor a magnetic dipole moment; if such a thing is diffracted by a crystal surface, the wave behaviour belongs to matter as such and not to charge. The instrument is Stern's molecular-beam apparatus, the same lineage as Experiment: Stern–Gerlach, and the grating is the cleavage face of a lithium fluoride crystal.

Apparatus

[Reserved: the beam source — an oven or gas reservoir with a fine aperture, effusing into high vacuum, whose temperature is the control variable of the whole experiment and must therefore be measured, not merely set; the collimating slits defining the beam; the lithium fluoride crystal, cleaved to present a fresh (100) face, mounted on a rotatable stage; and the detector, a hot-wire or manometric gauge sensitive to a beam flux of neutral atoms, movable in angle. To be given: the source temperatures used, the beam divergence — which sets the angular resolution and hence what fraction of the predicted separation between maxima can be resolved — and the provision for cooling the source, since the temperature scan is the test.]

Procedure

[Reserved: at fixed source temperature, sweep the detector in angle and record the beam flux, locating the specular peak and the diffracted maxima either side of it. Then change the species, from helium to molecular hydrogen, and repeat; then change the source temperature and repeat again. The measurement is of the positions of the maxima and not of their heights, and the quantities compared are ratios — helium against hydrogen at one temperature, and one temperature against another for one species — because the ratios are free of the beam-averaging factor that the absolute positions are not.]

Observations and data

[Reserved: beam flux against detection angle for each species and each source temperature, with the specular peak and the first-order maxima marked. To be tabulated when written: species, source temperature in kelvin, angle of the observed maximum with its uncertainty, the wavelength deduced from the grating condition, and the wavelength predicted by Equation (75.5); and separately the two ratios that the experiment really measures, helium to hydrogen at fixed temperature and cold source to warm source at fixed species.]

Phenomenon 75.3 (Neutral atoms and molecules are diffracted, with the wavelength their temperature assigns).

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 carries no permanent electric or magnetic moment, so no account of the diffraction of Sections 75.1 and 75.2 in terms of charge, of the Coulomb field, or of anything peculiar to the electron survives this measurement.

Derivation. Write $\lambda=h/mv$ and give the particle the mean thermal kinetic energy $\tfrac{3}{2}k_{\text{B}}T$ of its source. Then

\begin{equation}\tag{75.5} \lambda_{T}=\frac{h}{\sqrt{3mk_{\text{B}}T}}\ec \end{equation}

which for helium, of mass $4.003$ atomic mass units, at $T=295\,\mathrm{K}$ is $0.073\,\mathrm{nm}$, and for molecular hydrogen, of mass $2.016$, at the same temperature $0.104\,\mathrm{nm}$. Both are of the order of the row spacing of a lithium fluoride cleavage face, which is the reason the crystal serves as a grating for them at all — exactly the reason it served for the $54\,\mathrm{eV}$ electrons of Section 75.1.

Equation (75.5) makes two predictions that no adjustable quantity can absorb, and both are ratios. The first is the mass scaling at fixed temperature: the mass ratio of helium to molecular hydrogen is $1.99$, so their wavelengths must stand in the ratio $\sqrt{1.99}=1.41$, and the diffraction angles of the two beams from the same face must differ by that factor and by nothing else. The second is the temperature scaling at fixed species: $\lambda_{T}\propto T^{-1/2}$, so cooling the source from $295\,\mathrm{K}$ to about $100\,\mathrm{K}$ must lengthen the wavelength by $1.7$ and carry every maximum out to the angle the grating condition assigns to the longer wave.

Neither ratio depends on the calibration of anything, and that is fortunate, because the absolute positions do depend on a factor this derivation cannot supply. An effusive source does not emit a monochromatic beam: it emits a flux-weighted Maxwell distribution, so the beam carries a spread of wavelengths whose most probable value differs from Equation (75.5) by a factor of order unity fixed by the weighting and by the aperture geometry. That factor is the same for helium as for hydrogen and the same at $295\,\mathrm{K}$ as at $100\,\mathrm{K}$, so it cancels from both ratios above. The spread itself does not cancel; it broadens each maximum, and the broadening is predicted too, since each velocity class contributes at its own angle.

The physical conclusion needs no arithmetic. Helium presents a lattice with nothing to couple to but its momentum, and it is diffracted. Whatever produced the maxima of Section 75.1 is therefore not a property of the charge of the electron.

∎

Interpretation

[Reserved: the generalization achieved. The de Broglie relation is confirmed for a composite, neutral, chemically inert object, which is a far stronger statement than its confirmation for an elementary charged one, and it is confirmed through a temperature scan rather than at a single point. The section should also be candid about what the surface adds: helium scatters from the outermost layer alone, so this is a two-dimensional grating and the maxima carry information about the surface corrugation as well as about the wavelength — which is why the technique survives today as a probe of surfaces rather than as a test of the relation.]

Primary references

[Estermann:1930]. The molecular-beam method is Stern's, developed for the velocity measurement of [Stern:1920] and for the experiment of Experiment: Stern–Gerlach.

The electron double slit, one electron at a time (1961–1989)

Two explanations still stand after the diffraction experiments. The first is that the pattern is somehow a property of the lattice rather than of the projectile; the second is that it is a collective effect of many electrons in flight together. Both are removed here, and by two distinct measurements taken twenty-eight years apart. Jönsson cut slits in a foil and formed interference with no crystal anywhere in the apparatus [Joensson:1961]; Merli, Missiroli and Pozzi and, later and more famously, Tonomura and collaborators reduced the source until the apparatus was almost always empty, and watched the fringes assemble from individual point-like arrivals [Merli:1976] [Tonomura:1989]. This is Young's experiment of Experiment: Wave Optics performed with matter, and then performed one particle at a time.

Apparatus

[Reserved: two instruments, described separately. Jönsson's is an electron-optical bench — a $50\,\mathrm{kV}$ gun, a copper foil carrying slits of about $0.5\,\mu\mathrm{m}$ width made by electron-beam lithography and electroplating, in sets of two, three, four and five, and a magnifying electron lens system between the slits and the recording plate, without which the fringes are too fine to record. Tonomura's is a field-emission electron microscope with an electron biprism in place of the slits — a fine charged filament between two earthed plates, which superposes two virtual sources — and a position-sensitive detector reading out single arrivals. To be given for each: the accelerating voltage, the effective separation of the two paths, the drift length, and for the second the emission rate, which is the quantity the whole argument turns on.]

Procedure

[Reserved: for the multi-slit measurement, record the pattern for two, three, four and five slits at fixed voltage, and verify that the principal maxima do not move while the subsidiary structure between them changes with the slit count, as it does for light. For the single-particle measurement, reduce the beam current until the mean number of electrons in the instrument is far below one, record the individual arrivals with their times, and store successive frames of the accumulating histogram. The essential control is the occupancy estimate of Equation (75.6), which must be made from the measured emission rate and not assumed.]

Observations and data

[Reserved: the recorded interference patterns for each slit count, with fringe spacing against accelerating voltage; and the accumulation sequence, as frames at increasing total counts, with the fringe contrast extracted from each frame. To be tabulated when written: total counts per frame, measured contrast, and the contrast expected from Poisson noise alone at that count, so that the emergence of the pattern can be compared with the statistical prediction rather than merely displayed.]

Phenomenon 75.4 (Each particle interferes with itself).

Electrons passing two or more apertures in free space form an interference pattern with no crystal present [Joensson:1961]. When the source is weakened until the mean number of electrons in the apparatus is far below one, the detector still registers individual, point-like arrivals at positions that show no pattern one by one; the fringes appear only in the accumulated histogram of many thousands of 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.

Derivation. Three estimates, of which the first explains why the experiment took thirty-four years after Section 75.1.

The fringe spacing of a two-aperture pattern of separation $a$ observed at a distance $L$ is $\Delta x=\lambda L/a$. At $V=50\,\mathrm{kV}$, Equation (75.1) gives $\lambda=5.4\,\mathrm{pm}$, five orders of magnitude below optical wavelengths and smaller than any atom. Slits ruled a millimetre apart, generous for light, would put the fringes $1.6\,\mathrm{nm}$ apart and nothing could record them. Only when the separation is pushed down to a few micrometres — which is what lithography in a foil achieves and what ruling does not — does the spacing reach about $1\,\mu\mathrm{m}$ at a drift of a few tenths of a metre, and even then it must be magnified electron-optically by some four orders of magnitude before a photographic plate can resolve it. The difficulty of the experiment is entirely in these numbers.

The second estimate is the occupancy, and it is what turns the single-particle version into an argument. A $50\,\mathrm{keV}$ electron has $\gamma=1.098$ and hence $v=0.41c$, so it crosses an instrument of length $\ell\approx1.5\,\mathrm{m}$ in

\begin{equation}\tag{75.6} t_{\text{transit}}=\frac{\ell}{v}\approx1.2\times 10^{-8}\,\mathrm{s}\ec \end{equation}

while a source emitting at a rate $R$ of order $10^{3}$ electrons per second sends them in at mean intervals of $10^{-3}\,\mathrm{s}$. The mean occupancy is $Rt_{\text{transit}}\approx1.2\times 10^{-5}$: for each electron in flight, some hundred thousand transit times pass with the apparatus empty. Two electrons are present together with a probability of order the square of that, so an explanation of the fringes as an interaction between electrons fails by ten orders of magnitude and not by one.

The third estimate is what the accumulation itself establishes, and it is less than it is often said to be. If the arrivals are independent draws from a fixed density $p(x)$ — which is what “one at a time” means operationally — then the count in a detector bin is Poisson, with standard deviation $\sqrt{\bar{N}}$ about its mean $\bar{N}$, so the relative noise on the histogram falls as $\bar{N}^{-1/2}$ and a pattern of fringe contrast $C$ emerges from the noise once $\bar{N}\gtrsim C^{-2}$. For fringes of high contrast a few tens of counts per bin suffice, which is why the published frame sequences show nothing at a hundred arrivals and a clean pattern at a few tens of thousands. The derivation stops there, and the stopping point must be stated plainly: that the density $p(x)$ is the squared modulus of a single-particle amplitude is Born's postulate [Born:1926b], an addition to the de Broglie relation and not a consequence of it. What these experiments establish is that whatever the fringes are, they are a property of each particle separately; that they are probabilities is an interpretation, formalized in The Postulates of Quantum Mechanics and left unexplained by Interpretations (Evidence-Anchored).

∎

Interpretation

[Reserved: the two loopholes closed and the one that opens. The multi-slit series closes the lattice explanation — there is no lattice — and the progression from two to five slits closes any explanation appealing to a property of a particular pair of apertures, since the principal maxima stay put while the subsidiary structure changes exactly as scalar wave optics requires. The occupancy estimate closes the collective explanation. What opens is the measurement problem: a wave passes both apertures and a point is recorded at one place, which is the tension The Postulates of Quantum Mechanics formalizes. Related neutron measurements with a single slit and a double slit extend the same demonstration to a massive neutral particle [Zeilinger:1988].]

Primary references

[Joensson:1961] [Merli:1976] [Tonomura:1989].

Colella, Overhauser and Werner: gravity shifts a phase (1975)

Everything above measures a wavelength. This experiment measures a phase, which is a stronger thing to have: once the phase along each of two separated paths is an instrument reading, anything that acts differently on the two paths becomes measurable. The instrument is the perfect-crystal neutron interferometer of Rauch, Treimer and Bonse [Rauch:1974], cut from a single silicon ingot so that three parallel lamellae split, redirect and recombine a thermal-neutron beam by Bragg reflection, with the two paths separated by centimetres and the lattice coherent across the whole device. What Colella, Overhauser and Werner then do with it is to rotate it about the incident beam, so that the two paths run at different heights in the Earth's field, and count fringes.

Apparatus

[Reserved: the reactor neutron source and monochromator setting the incident wavelength; the interferometer itself, a monolithic silicon crystal with three lamellae machined from one ingot, its lattice perfect over the whole device; the goniometer that rotates it about the incident beam axis through the angle $\alpha$; the two neutron counters at the exit beams; and the thermal and vibrational isolation, since a temperature gradient across the crystal or a sub-nanometre relative displacement of the lamellae destroys the interference. To be given: the incident wavelength and its spread, the enclosed area of the two paths, and the mechanical stiffness of the device, which enters the systematic corrections directly.]

Procedure

[Reserved: rotate the interferometer through $\alpha$ in steps from one side of the horizontal to the other, counting neutrons in each exit beam for a fixed time at each step. The counted intensity oscillates in $\sin\alpha$, and the quantity extracted is the number of oscillations between the extreme settings, compared with the prediction of Equation (75.10), which contains no free parameter once the wavelength and the enclosed area are measured. Two systematic corrections must be applied and must be described rather than merely named: the elastic bending of the crystal under its own weight as it is rotated, which changes the path geometry, and the Sagnac phase from the Earth's rotation.]

Observations and data

[Reserved: neutron counts in each exit beam against tilt angle, with counting statistics; the extracted fringe count over the full rotation; and the predicted count from the measured wavelength and area. To be tabulated when written: tilt angle, counts and uncertainty in each beam, and the residual phase after the bending and Sagnac corrections, which is the number that carries the result.]

Phenomenon 75.5 (Gravity shifts the phase of a matter wave).

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 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 proportional to $\sin\alpha$, to the area they enclose, to the neutron wavelength, and to the square of the neutron mass. Colella, Overhauser and Werner observed the predicted number of fringes as the interferometer was rotated [Colella:1975]. This is a laboratory quantity in which Planck's constant and the acceleration of free fall appear together.

Derivation. Work in the regime where the wave has a local wavenumber and the phase along a path is $\int k\,\dd l$. A neutron of total energy $E$ at height $z$ in a uniform field has kinetic energy $E-m_{\text{g}}gz$, so with $m_{\text{i}}$ the inertial and $m_{\text{g}}$ the gravitational mass,

\begin{equation}\tag{75.7} k(z)=\frac{\sqrt{2m_{\text{i}} \left(E-m_{\text{g}}gz\right)}}{\hbar}\ep \end{equation}

The expansion parameter is minute. A thermal neutron of wavelength $0.1\,\mathrm{nm}$ has $E\approx0.08\,\mathrm{eV}$, while $m_{\text{g}}gz$ over a height of $1\,\mathrm{cm}$ is about $10^{-9}\,\mathrm{eV}$: a ratio of $10^{-8}$. To first order, using $E=\hbar^{2}k_{0}^{2}/2m_{\text{i}}$,

\begin{equation}\tag{75.8} k(z)\simeq k_{0}\left(1-\frac{m_{\text{g}}gz}{2E}\right) =k_{0}-\frac{m_{\text{i}}m_{\text{g}}gz}{\hbar^{2}k_{0}}\ep \end{equation}

The neutron is slower at the top, and the phase it accumulates per unit length there is correspondingly less.

The two paths of the interferometer form a parallelogram. One path takes its horizontal leg at the lower height and rises afterwards; the other rises first and takes its horizontal leg at the greater height. The two rising legs are congruent and contribute equally, so they cancel from the difference, and what remains is one leg of length $L$ traversed at height $0$ on one path and at height $H$ on the other:

\begin{equation}\tag{75.9} \Delta\varphi=\left[k(H)-k(0)\right]L =-\frac{m_{\text{i}}m_{\text{g}}g}{\hbar^{2}k_{0}}HL =-\frac{m_{\text{i}}m_{\text{g}}gA}{\hbar^{2}k_{0}}\ec \end{equation}

with $A=HL$ the area enclosed by the paths. Rotating the instrument by $\alpha$ about the incident beam tips that area out of the vertical, so that the rise becomes $H\sin\alpha$ while the two rising legs remain congruent with each other; writing $k_{0}=2\pi/\lambda$ and $m_{\text{i}}=m_{\text{g}}=m$ then gives

\begin{equation}\tag{75.10} \left|\Delta\varphi\right| =\frac{2\pi m^{2}gA\lambda}{h^{2}}\left|\sin\alpha\right|\ep \end{equation}

Three things should be read off Equation (75.10).

It is large. With an enclosed area of order $10^{-3}\,\mathrm{m}^{2}$, a wavelength of $0.14\,\mathrm{nm}$ and the neutron mass, the coefficient is about $55$ radians, so the intensity passes through some nine fringes as the instrument goes from horizontal to vertical and through twice that over the full rotation. Gravity is the feeblest interaction available in a laboratory and the phase it produces is enormous, because $\hbar^{2}$ sits in the denominator. The smallness that makes the effect delicate is not in the phase but in the fringe spacing in $\alpha$, which is what demands the mechanical stability.

It measures a product of two masses. The wavelength carries the inertial mass through $\lambda=h/m_{\text{i}}v$ and the potential carries the gravitational mass, so the observable is $m_{\text{i}}m_{\text{g}}$ and not $m$. This is a quantum interference experiment sensitive to the ratio that the torsion-balance experiments of The Equivalence Principle and Classical Tests bound classically, and it is the reason the result is quoted as a test of the equivalence principle for a quantum object rather than merely as a demonstration.

It is not a test of general relativity. Equation (75.7) treats gravity as a Newtonian potential energy in a non-relativistic wave equation, and at this accuracy that is all the experiment is sensitive to. What it establishes is that a matter wave responds to a gravitational potential exactly as a wave with that potential in its dispersion relation must, which is a statement about quantum mechanics in a weak static field and about nothing beyond it.

∎

Interpretation

[Reserved: what the fringe count establishes and the two systematics that limit it. The result confirms Equation (75.10) in form and in magnitude; the residual discrepancies reported by this and later runs, at the level of a percent or so of the phase, are attributed to crystal bending and are themselves a measurement of the elastic response of the device, which is worth stating because it is a case where the systematic became the object of study. The section should also record the companion result obtained on the same class of instrument, that the interference contrast returns only after a rotation of $720^\circ$ of the neutron spin and not after $360^\circ$ [Werner:1975] [Rauch:1975] — the double cover of the rotation group of Lie Groups, Lie Algebras, and Fibre Bundles and Angular Momentum and Spin made into a fringe count. The 1974 proposal that preceded the measurement is [Overhauser:1974].]

Primary references

[Colella:1975]. The interferometer is [Rauch:1974]; the spinor-periodicity measurements on the same class of instrument are [Werner:1975] [Rauch:1975].

Molecule interferometry and the edge of coherence (1999–2019)

The last question the relation invites is where it stops. If matter of every kind has a wavelength $h/p$, then a chair has one; the reason a chair does not interfere must be found somewhere, and the honest way to look for it is to increase the mass of the interfering object until something fails. Two decades of molecular interferometry have done exactly that, from a molecule of sixty atoms [Arndt:1999] to tailored molecules of more than two thousand [Fein:2019]. Nothing has failed in the relation. What fails, reproducibly and on demand, is the isolation of the molecule from its surroundings — and that failure is measurable, which makes this section the experimental entrance to Open Quantum Systems and Decoherence.

Apparatus

[Reserved: two generations. The far-field arrangement — an effusive oven sublimating fullerene at high temperature, a velocity selector, a nanofabricated free-standing grating of about $100\,\mathrm{nm}$ period, and a scanning ionization detector. The near-field arrangement — a three-grating Talbot–Lau interferometer, in which the first grating prepares the transverse coherence the source lacks, the second diffracts, and the third serves as a scanning mask, with a source of large molecules that must be volatilized without being destroyed. To be given for each: grating period and open fraction, grating separations, the velocity distribution admitted, and the vacuum and thermal environment, which are the experimental variables of the decoherence measurements rather than mere hygiene.]

Procedure

[Reserved: record the detected molecular flux against transverse detector position, extract the fringe visibility, and repeat as a function of the two quantities that carry the physics — the internal temperature of the molecules, set by the sublimation and by laser heating, and the residual gas pressure. The velocity selection matters and must be described: the fringe period depends on the wavelength, so an unselected thermal beam washes out its own pattern, and the observed visibility as a function of the width of the velocity window is itself a check on Equation (75.11).]

Observations and data

[Reserved: detected flux against detector position, with the fitted visibility and its uncertainty, for each molecular species. To be tabulated when written: species, mass in atomic mass units, selected speed, deduced de Broglie wavelength, grating period, predicted and observed fringe period, and observed visibility; and separately the two decoherence scans, visibility against internal temperature and against residual gas pressure, with the predictions of Equation (75.12) alongside.]

Phenomenon 75.6 (Objects of thousands of atoms interfere at the de Broglie wavelength).

Interference survives in objects far larger and far more complicated than the wavelength they carry. Fullerene molecules diffracted from a nanofabricated grating produce fringes at a wavelength of about $2.5\,\mathrm{pm}$, some hundreds of times smaller than the molecule itself, even though the molecules are hot enough to populate very large numbers 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 beyond $25000$ atomic mass units, at wavelengths of order $50\,\mathrm{fm}$ [Fein:2019]. No deviation from $\lambda=h/p$ has appeared at any mass reached, and no boundary beyond which the relation fails has been found.

Derivation. The wavelengths first, since they are what make the claim surprising. A fullerene of sixty carbon atoms has mass $m=720$ atomic mass units, and an oven beam of it selected near $220\,\mathrm{m}/\mathrm{s}$ has

\begin{equation}\tag{75.11} \lambda=\frac{h}{mv}=2.5\,\mathrm{pm}\ec \end{equation}

some hundreds of times smaller than the molecule that carries it. For a molecule of $25000$ atomic mass units at a beam speed of order $300\,\mathrm{m}/\mathrm{s}$ the same relation gives $\lambda$ of order $53\,\mathrm{fm}$: seven orders of magnitude below the wavelength of visible light, and five below the size of the object itself. Nothing in $\lambda=h/p$ objects to either, and nothing in the observed fringe periods does either.

Why, then, does a chair not interfere? Not because it is heavy. Suppose a molecule, while in a superposition of two paths separated by $d$, emits a single photon of wavenumber $k$ in a random direction. The photon state emitted from one path differs from that emitted from the other by the phase $\ee^{\ii\vect{k}\cdot\vect{d}}$, and the fringe visibility is multiplied by the overlap of the two photon states, averaged over emission direction:

\begin{equation}\tag{75.12} \left|\langle\gamma_{1}|\gamma_{2}\rangle\right| =\left|\frac{1}{4\pi}\int\dd\Omega\; \ee^{\ii\vect{k}\cdot\vect{d}}\right| =\left|\frac{\sin kd}{kd}\right|\ep \end{equation}

Coherence survives while $kd\ll1$ — while the emitted wavelength is long compared with the separation of the paths, so that the photon cannot say which path was taken — and is lost when $kd$ approaches unity.

Two consequences follow, and they are the content of the modern experiments. The criterion contains no mass at all: a heavy object is not harder to interfere for being heavy, but for being harder to keep dark and harder to keep from being touched. And because a molecule in flight emits many photons, the visibility is the product of Equation (75.12) over all of them, so it does not decline gently but collapses, the number of emitted photons rising steeply with internal temperature while each factor falls below unity. Both knobs have been turned deliberately. Heating the molecules until their thermal emission reaches short enough wavelengths destroys the fringes [Hackermueller:2004]; admitting a background gas, so that a single collision localizes the molecule to within a distance fixed by the momentum transfer, destroys them by the same argument and with the predicted pressure dependence [Hornberger:2003]. In neither case does the wavelength change, and in neither case is $\lambda=h/p$ at fault.

∎

Phenomenon 75.7 (What ends the interference is information, not mass).

The fringe visibility of a molecular interferometer is destroyed controllably by two means, both of which allow the environment to record which path the molecule took: raising the internal temperature of the molecules until the thermal photons they emit have wavelengths comparable with the path separation [Hackermueller:2004], and admitting a background gas so that molecules are scattered from [Hornberger:2003]. In both cases the visibility falls with the measured dependence on temperature or pressure, and in both cases the de Broglie wavelength is unchanged. The transition from quantum to classical behaviour observed here is a loss of isolation and not a breakdown of the wave relation.

Derivation pending.

The quantitative decoherence rates: the thermal photon emission spectrum of a hot molecule and the resulting product of which-path factors as a function of internal temperature, giving the predicted visibility curve; and the collisional case, in which the localization length follows from the momentum transfer distribution of the residual gas at a given pressure and temperature, giving the predicted decay of visibility with pressure. Both are stated qualitatively in the derivation above from the single-photon overlap; what must be supplied is the master equation whose decoherence rate they are, together with the statement of which of its assumptions the data actually test.

Interpretation

[Reserved: what the mass record does and does not bound. It does not establish that there is no limit; a null result at $25000$ atomic mass units is a lower bound on where any limit lies. It does constrain proposed objective-collapse modifications of quantum mechanics, which predict a mass-dependent breakdown of superposition, and that constraint is a live experimental frontier reported as such in What We Observe but Do Not Understand rather than as a settled matter. The section should also record the honest reading of the decoherence measurements: they show a mechanism sufficient to explain why large objects do not interfere, and they do not show that it is the only one — which is the distinction Open Quantum Systems and Decoherence and Interpretations (Evidence-Anchored) respectively make and decline to resolve.]

Primary references

[Arndt:1999] [Fein:2019]. The decoherence measurements are [Hornberger:2003] [Hackermueller:2004].

What the measurements settle

Experiment	Wavelength	What it establishes
Davisson–Germer 1927	$0.165\,\mathrm{nm}$	a wavelength computed from a voltage and one read off a lattice agree to about one percent
Thomson–Reid 1927	$5\text{–}12\,\mathrm{pm}$	ring radii scale as $V^{-1/2}$ in transmission through an unoriented target; not a surface effect
Estermann–Stern 1930	$0.07\,\mathrm{nm}$	neutral, inert helium is diffracted, and the maxima track the source temperature; not an effect of charge
Jönsson 1961; Tonomura 1989	$5.4\,\mathrm{pm}$	interference with no lattice, and with the apparatus almost always empty; not a collective effect
Colella et al. 1975	$0.14\,\mathrm{nm}$	the phase is an instrument reading and gravity shifts it by the predicted amount, in $m_{\text{i}}m_{\text{g}}$
Arndt 1999; Fein 2019	$0.05\text{–}2.5\,\mathrm{pm}$	no deviation up to $25000$ atomic mass units; what ends the fringes is which-path information

The measurements of this chapter, and the alternative explanation each removes. The third column states what the experiment is entitled to claim, which in the last row is markedly weaker than what it is often credited with.

The rows of Table 75.1 are not repetitions of one another. Each was performed because the previous one left an explanation standing, and the sequence is the removal of those explanations one at a time until nothing is left but the relation itself. Taken together they confirm $\lambda=h/p$ over more than seven orders of magnitude in mass, from the electron to a molecule of two thousand atoms, and over three in wavelength, with no adjustable parameter anywhere and no observed departure. That is a stronger evidential position than the relation enjoyed when Matter Waves adopted it, and stronger than most relations in this treatise enjoy.

Three things are not settled here, and the chapter is worth closing on them. The first is the meaning of the wave: the accumulation experiments establish that each particle interferes with itself and say nothing about what the amplitude is, and the identification of its squared modulus with a probability density is a postulate added by Born [Born:1926b] and examined in The Postulates of Quantum Mechanics. The second is the equation of motion: a wavelength for a free particle is not a wave equation, and everything about a particle in a potential — including the bound states that Atomic Models and Spectra needed — is outside what any measurement in this chapter reaches. The third is the upper limit. No mass at which the relation fails has been found, but no experiment can establish that none exists; what the molecular interferometers deliver is a bound that moves upward with each generation, and the proposals it constrains are named honestly, and without endorsement, in What We Observe but Do Not Understand.

A closing observation of a different kind. The relation has stopped being a hypothesis under test and become an instrument: electron and neutron diffraction are the standard probes of crystal and magnetic structure, and atom interferometers built on the same principle [Keith:1991] serve as gravimeters and gyroscopes and deliver $h/m$, and thence the fine-structure constant, to a precision that feeds back into the tests of the constants themselves. A hypothesis proposed from a symmetry, with no data behind it, now calibrates the measurements by which other hypotheses are judged; and the neutron bound states in the Earth's field [Nesvizhevsky:2002], in which the same wave is quantized by gravity alone, are the most recent reminder that the instrument is still finding new things to weigh.

Experiment	Wavelength	What it establishes
Davisson–Germer 1927	\(0.165\,\mathrm{nm}\)	a wavelength computed from a voltage and one read off a lattice agree to about one percent
Thomson–Reid 1927	\(5\text{–}12\,\mathrm{pm}\)	ring radii scale as $V^{-1/2}$ in transmission through an unoriented target; not a surface effect
Estermann–Stern 1930	\(0.07\,\mathrm{nm}\)	neutral, inert helium is diffracted, and the maxima track the source temperature; not an effect of charge
Jönsson 1961; Tonomura 1989	\(5.4\,\mathrm{pm}\)	interference with no lattice, and with the apparatus almost always empty; not a collective effect
Colella et al. 1975	\(0.14\,\mathrm{nm}\)	the phase is an instrument reading and gravity shifts it by the predicted amount, in $m_{\text{i}}m_{\text{g}}$
Arndt 1999; Fein 2019	\(0.05\text{–}2.5\,\mathrm{pm}\)	no deviation up to \(25000\) atomic mass units; what ends the fringes is which-path information