Appearance
7.2 — Wave–Particle Duality
In 1924 Louis de Broglie, a French aristocrat who had switched from history to physics, submitted a doctoral thesis containing one idea and almost no supporting evidence.
His reasoning was symmetry. Light had been a wave for a century and was now also a particle. If waves can be particles, why can particles not be waves?
His examiners had no idea whether it was profound or nonsense, so they sent it to Einstein. Einstein replied that de Broglie "has lifted a corner of the great veil". The thesis passed. Three years later it was confirmed by accident, and de Broglie received the Nobel Prize in 1929 — the only person ever to win it for a doctoral thesis.
The de Broglie relation
For a photon, Chapter 7.1 established E = hf and p = h/\lambda. De Broglie proposed the second of these applies to everything:
\boxed{\lambda = \frac{h}{p} = \frac{h}{mv}}
Read aloud: lambda equals h over p — the wavelength of any object is Planck's constant divided by its momentum.
Worked numbers, to see why nobody noticed.
A cricket ball, 0.16 kg at 40 m/s:
\lambda = \frac{6.626\times10^{-34}}{0.16\times40} = \frac{6.626\times10^{-34}}{6.4} = 1.04\times10^{-34}\ \text{m}
A hundred million billion times smaller than a proton. No aperture in the universe could diffract it, so its wave nature is unobservable in principle, not merely in practice.
A person walking, 70 kg at 1.5 m/s: \lambda = 6.3\times10^{-36} m. Same conclusion.
An electron accelerated through 100 V. Its kinetic energy is 100 eV = 1.602\times10^{-17} J, so:
p = \sqrt{2m_eE} = \sqrt{2(9.109\times10^{-31})(1.602\times10^{-17})} = \sqrt{2.919\times10^{-47}} = 5.40\times10^{-24}
\lambda = \frac{6.626\times10^{-34}}{5.40\times10^{-24}} = 1.23\times10^{-10}\ \text{m} = 1.23\ \text{Å}
That is the spacing between atoms in a crystal. Which means a crystal is a ready-made diffraction grating for 100 eV electrons — and it means somebody was going to see the effect as soon as they fired electrons at one.
A useful shortcut, for non-relativistic electrons:
\lambda\ (\text{in Å}) = \sqrt{\frac{150.4}{V\ (\text{in volts})}}
Davisson and Germer
Clinton Davisson and Lester Germer at Bell Labs were not testing de Broglie. They were studying how electrons bounce off nickel, for entirely practical reasons to do with vacuum tubes.
In 1925 a liquid-air bottle exploded and their vacuum chamber filled with air, oxidising the nickel target. To clean it they heated it strongly, which had a side effect they did not intend: the many small nickel crystals fused into a few large ones.
When they restarted, the scattering pattern had completely changed. Instead of a smooth spread, there were sharp peaks at particular angles.
The peaks fit the Bragg condition for X-ray diffraction from a crystal:
n\lambda = d\sin\theta
At 54 V they found a strong peak at 50°. With nickel's atomic spacing d = 2.15 Å:
\lambda = (2.15)\sin(50°) = (2.15)(0.766) = 1.65\ \text{Å}
De Broglie's prediction for 54 V:
\lambda = \sqrt{\frac{150.4}{54}} = \sqrt{2.785} = 1.67\ \text{Å}
Agreement to 1 %. George Paget Thomson independently demonstrated the same thing by firing electrons through thin metal foils, and he and Davisson shared the 1937 Nobel Prize.
There is a piece of history worth noting. J. J. Thomson won the Nobel Prize in 1906 for showing the electron is a particle. His son George Paget Thomson won it in 1937 for showing the electron is a wave. Both were right.
It works for everything
The experiment has been repeated with steadily larger objects, and it has never failed.
| Object | Year | Note |
|---|---|---|
| Electrons | 1927 | Davisson–Germer |
| Helium atoms | 1930 | Estermann and Stern |
| Neutrons | 1936 | Now a standard materials technique |
| Buckyballs (C₆₀, 720 amu) | 1999 | Vienna |
| Molecules of 810 atoms | 2011 | Vienna |
| Molecules of 25,000 amu | 2019 | Vienna |
Neutron diffraction is a working industrial tool. Neutrons at room temperature have \lambda \approx 1.8 Å, ideal for crystals, and unlike X-rays they scatter strongly off hydrogen and can distinguish neighbouring elements. Reactor and spallation sources worldwide exist for this.
The C₆₀ result is the striking one. A buckyball is a soccer-ball molecule of sixty carbon atoms, 1 nm across, with a mass of 720 atomic units, internal vibrations, and a temperature of several hundred kelvin. It is unambiguously an object. At 200 m/s its de Broglie wavelength is:
\lambda = \frac{6.626\times10^{-34}}{(1.2\times10^{-24})(200)} = 2.8\times10^{-12}\ \text{m}
400 times smaller than the molecule itself, and interference fringes were still observed. The 2019 experiments reached molecules of 2000 atoms.
Is there a limit? No known one in principle. In practice, larger objects have shorter wavelengths and interact more with their environment, and any interaction that could reveal which path was taken destroys the interference. That process is decoherence, and Chapter 7.8 covers it. It is the reason a cat is not observed in two places, and it is a matter of degree rather than a hard boundary.
The double slit, one particle at a time
Chapter 5.3 ran the double slit with light and got interference. Now run it with electrons, one at a time, so that only one is ever in the apparatus.
What happens:
Each electron arrives as a single localised dot. Not a smeared wave — a point, at one place, detected by one pixel.
The dots appear at unpredictable positions.
After enough of them, the interference pattern appears, with bright and dark fringes exactly where the wave calculation says.
Akira Tonomura's team at Hitachi did this definitively in 1989, and the panels in the figure show 10, 200, 6,000, 40,000 and 140,000 electrons.
The single electron interfered with itself. There was nothing else in the apparatus.
Which path?
Put a detector at one slit to see which the electron took.
The interference pattern vanishes. You get two plain bands — the classical answer.
Turn the detector off and the pattern returns. Turn it on and it disappears. This has been done with the detector switched randomly after the electron has already passed the slits (a "delayed choice" experiment, proposed by Wheeler and performed many times), and the result is the same: whether interference appears depends on whether the which-path information exists, not on when the decision was made.
Is it just that the detector disturbs the electron? That was Bohr's original argument and it is not the deepest answer. Experiments using quantum erasure mark the paths without any momentum kick — by tagging photons with polarisation, say — and the interference still vanishes. Then, if the marking is erased before the data are analysed, the interference comes back in the correlated subsets.
The rule that survives every version:
Interference occurs exactly when the which-path information does not exist anywhere in the universe, even in principle.
Not "when nobody looked". Not "when a conscious observer was present". When the information does not exist.
What is actually waving?
Not the electron's substance. Not a charge distribution — a detected electron is always a whole electron at one place, never a fraction.
Max Born supplied the answer in 1926, in a footnote:
The wave is a probability amplitude. Its squared magnitude gives the probability of finding the particle at that place.
P(x)\,dx = |\psi(x)|^2dx
where \psi (psi) is the wavefunction. Chapter 7.3 develops it properly. Born received the Nobel Prize in 1954, twenty-eight years later, largely because the interpretation took that long to be accepted.
Why amplitudes and not probabilities? Because amplitudes are complex numbers that can cancel, and probabilities cannot. Two contributions to a probability can only add; two amplitudes can add to zero. That cancellation is interference, and it is the whole reason the quantum world differs from a classical world with randomness in it.
The rule for combining:
- If the paths are indistinguishable, add the amplitudes and then square: P = |\psi_1+\psi_2|^2. Cross terms appear, and those cross terms are the fringes.
- If the paths are distinguishable, add the probabilities: P = |\psi_1|^2+|\psi_2|^2. No cross terms, no fringes.
Work it out. With \psi_1 = Ae^{i\phi_1} and \psi_2 = Ae^{i\phi_2}:
|\psi_1+\psi_2|^2 = A^2\left|e^{i\phi_1}+e^{i\phi_2}\right|^2 = 2A^2\left[1+\cos(\phi_1-\phi_2)\right]
The \cos(\Delta\phi) term oscillates between +1 and -1, giving 4A^2 at the maxima and zero at the minima. Adding probabilities instead would give a flat 2A^2 everywhere.
So interference is a statement about the arithmetic of complex numbers, and the reason the classical world looks classical is that the phases of large objects are randomised by their environment faster than any measurement can catch them.
Standing waves and why energy levels exist
De Broglie's idea does one more thing immediately: it explains Bohr's arbitrary quantisation rule.
Chapter 2.4 showed that a wave confined between boundaries can only take certain wavelengths — those that fit. Everything else interferes with itself destructively and dies.
Apply that to an electron in an orbit. For the wave to survive going round, it must join up with itself in phase after one circuit, so the circumference must be a whole number of wavelengths:
2\pi r = n\lambda = \frac{nh}{m_ev}
Rearrange:
m_evr = \frac{nh}{2\pi} = n\hbar
That is Bohr's third assumption, which he had to postulate with no justification. It is the condition for a standing wave.
This is the single most important idea in the rest of this Part:
Confine a wave and its energy becomes discrete. Quantisation is not mysterious; it is what boundaries do to waves.
A guitar string has discrete notes for the same reason an atom has discrete energy levels. Free electrons can have any energy; bound ones cannot.
Matter waves in practice
The electron microscope. Chapter 5.4 gave the resolution limit as $\approx\lambda/2$NA. Light at 500 nm stops at 200 nm. Electrons at 100 keV have:
\lambda = \frac{h}{\sqrt{2m_eE}}\times(\text{relativistic correction}) = 3.7\times10^{-12}\ \text{m}
Under four picometres, a hundred thousand times shorter than visible light. Practical resolution is limited by lens aberrations to about 0.05 nm, which is atomic. Every image of an individual atom ever taken exists because of de Broglie's thesis.
Neutron scattering maps hydrogen positions in proteins and magnetic structures in materials, because neutrons carry a magnetic moment and scatter off nuclei rather than electron clouds.
Atom interferometry uses the wave nature of whole atoms to build the most sensitive accelerometers and gravimeters in existence, measuring g to nine decimal places. They are used to find underground voids, monitor volcanoes, and test the equivalence principle of Chapter 6.6 by dropping two different isotopes and comparing.
Scanning tunnelling microscopy relies on the related effect of tunnelling, which Chapter 7.4 derives.
What duality actually means
It is worth being careful, because the popular phrasing invites confusion.
An electron is not "sometimes a wave and sometimes a particle". It is one kind of thing, described by a wavefunction, and that thing is neither a classical wave nor a classical particle. Those are two familiar concepts borrowed from everyday experience, and neither fits.
Which aspect you see depends on what you measure. Position measurements yield localised dots. Interference measurements yield fringes. You cannot arrange to see both at once, and Chapter 7.5 shows this is not a technological limit but a mathematical consequence of the uncertainty principle.
Bohr called this complementarity: the wave and particle descriptions are both needed, both incomplete, and mutually exclusive in any single experiment.
Feynman's summary, from his lectures, is worth quoting because it is honest:
"We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery."
He meant the double slit. Everything else in quantum mechanics is machinery for calculating what it does.
Where this shows up in your life
Every electron micrograph — of a virus, a chip, a fracture surface — depends on matter waves.
Semiconductor design. Electrons in a crystal are waves, and their allowed energies form bands with gaps between, exactly like the allowed modes of a confined wave. Volume III, Chapter 2 builds transistors from that.
Quantum dots in modern displays are nanocrystals small enough to confine an electron's wavefunction, so the confinement sets the energy levels and therefore the colour. Make the dot smaller and the colour shifts blue, purely from the standing-wave condition, and that is how QLED televisions produce their very pure red, green and blue.
MRI uses the wave nature of nuclear spins.
And the periodic table. Chapter 9.3 shows that the shape of the table — two elements in the first row, eight in the next — comes from the standing-wave patterns available to an electron around a nucleus. Chemistry is a consequence of what shapes a confined wave can take.
What the next chapter fixes
The wave exists, it carries probability, and confinement makes energies discrete. What is missing is the equation that says how the wave behaves — what determines its shape, how it changes in time, and how to compute it for a given situation. Chapter 7.3 introduces the wavefunction properly, states the rules it must obey, and derives the Schrödinger equation both in the way it is usually motivated and in something closer to the way Schrödinger actually found it over a Christmas holiday in 1925.