De Broglie and the uncertainty principle
If a photon can behave as a particle, an electron can behave as a wave. Half of quantum mechanics came out of that symmetry, and so did the reason an atom does not collapse on itself.
01The de Broglie hypothesis
The photon has momentum without having mass: \( p = h/\lambda \), the relation that closed the previous unit. In 1924 Louis de Broglie turned the question around: if a wave —light— behaves like a particle, why could a particle —an electron, an atom, anything with mass— not behave like a wave? He proposed that every particle with momentum \( p = mv \) has an associated wavelength, with the same formula that already held for the photon.
The idea sounded like speculation —de Broglie did not yet have any data to back it— and yet Einstein endorsed it at once, and a year later it served Schrödinger as the starting point for writing the equation of quantum mechanics.
02The confirmation: electron diffraction
In 1927, Clinton Davisson and Lester Germer fired 54 eV electrons at a nickel crystal and found an intensity maximum exactly at the angle predicted by Bragg diffraction with the de Broglie wavelength: 1.67 Å. They were not looking for that —they were investigating the surface of the metal— and ended up with the first direct proof that an electron diffracts like a wave. Davisson shared the 1937 Nobel Prize with George Thomson, who had reached the same result by another route.
You choose the case and the point is placed on a logarithmic ruler of lengths, next to well-known reference scales. The distance between the point and those references decides whether wave behavior can be observed or not.
The resolution of a microscope is limited by the wavelength of what it uses to see. Visible light cannot resolve anything smaller than about 200 nm, no matter how good the lenses are. An electron accelerated through 200 kV has a wavelength of barely 2.5 pm —two hundred thousand times smaller—, and that is why an electron microscope resolves individual atoms where an optical one sees only a blur.
03Heisenberg's uncertainty principle
If a particle is also a wave, it stops making sense to ask it for an exact position: a wave, by definition, is extended. Heisenberg formalized this in 1927: there is no physical state in which the position and the momentum of a particle are both perfectly defined. It is not a limitation of the measuring instrument, it is a property of nature.
The effect is negligible at the human scale —confining a car to one meter imposes no detectable velocity on it— and decisive at the atomic scale. Confining an electron to the size of an atom, \( \Delta x \approx 10^{-10} \) m, imposes a velocity uncertainty of more than \( 5\times10^5 \) m/s: the electron cannot be at rest inside the atom. That is the real reason it does not spiral into the nucleus, as classical electromagnetism would predict: quantum mechanics demands a minimum kinetic energy of it, the zero-point energy, which grows without limit as the confinement increases.
04Where it appears in electronics
| Case | What it explains |
|---|---|
| Electron microscope (TEM, SEM) | Uses the electron wavelength to resolve what light cannot |
| Scanning tunneling microscope (STM) | The tip “feels” the wave function of the surface electrons, atom by atom |
| Tunnel diode and flash memory | An electron crosses a barrier that classical physics would forbid: tunneling, next unit |
| Quantum noise in very small devices | The more a transistor shrinks, the closer its channel gets to the regime where \( \Delta x \) matters |
| Laser linewidth | \( \Delta E\,\Delta t \) limits how narrow the emission frequency of a finite-lifetime state can be |
The miniaturization of electronics runs, sooner or later, into these two ideas. A transistor channel a few nanometers long starts to be of the order of the wavelength of its own electrons, and confinement starts to cost zero-point energy. It is one of the reasons —not the only one— why shrinking the size of a chip is not just a manufacturing problem.
05In the lab
Calculate the de Broglie wavelength of a 54 eV electron and compare it with the historical value of 1.67 Å. Using the lattice spacing of nickel, verify the diffraction angle that Davisson and Germer observed.
Calculate \( \lambda \) for an electron accelerated through 100 V, 10 kV and 200 kV. Identify from what energy the relativistic correction from the previous unit becomes necessary.
With \( \Delta x = 1 \) Å, calculate the minimum velocity uncertainty of a confined electron. Compare that velocity with the orbital velocity predicted by the Bohr model for the hydrogen atom.
06Common mistakes
- Thinking that uncertainty is a problem with the instrument. It is a property of the particle, not a limitation of whoever is measuring.
- Applying de Broglie to a macroscopic object and expecting to see something. The formula always holds, but the wavelength of a ball is inconceivably smaller than any real slit.
- Confusing \( \Delta x \) with the size of the particle. It is the uncertainty in where it is, not a physical diameter.
- Believing the electron “spins” around the nucleus like a planet. It has no definite orbit; what it has is a zero-point energy that keeps it from falling.
- Using \( h \) instead of \( \hbar = h/2\pi \) in the Heisenberg inequalities, or the other way around.
- Forgetting the relativistic correction when calculating \( \lambda \) for fast electrons, such as those in a modern electron microscope.
07Self-assessment
What did de Broglie propose in 1924?
That every particle with momentum \( p \) has an associated wavelength \( \lambda = h/p \), extending to matter the relation that already held for the photon.
What did the Davisson and Germer experiment show?
That electrons diffract just like a wave when they hit a crystal, with the exact wavelength that de Broglie predicts.
Why does an electron microscope resolve more detail than an optical one?
Because the wavelength of an accelerated electron is several orders of magnitude smaller than that of visible light.
What does Heisenberg's uncertainty principle say?
That the position and the momentum of a particle cannot both be defined with arbitrary precision: \( \Delta x\,\Delta p \geq \hbar/2 \).
Why does the electron not spiral into the nucleus?
Because confining it to a region as small as the atom requires, by Heisenberg, a minimum kinetic energy —the zero-point energy— that opposes the fall.
Why do we never see a baseball diffract?
Because its mass is so large that its de Broglie wavelength is vastly smaller than any possible physical opening: the effect exists but is unobservable.
08Further reading
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica: átomos, moléculas, sólidos, núcleos y partículas”, Limusa). Chapters 3 and 4 develop de Broglie, Davisson-Germer and uncertainty with the same rigor as the whole unit.
- C. Davisson and L. H. Germer. Diffraction of Electrons by a Crystal of Nickel. Physical Review, vol. 30, 1927. The original paper, short and accessible, of the experiment that confirmed de Broglie's hypothesis.
- Markus Arndt et al. Wave-particle duality of C₆₀ molecules. Nature, vol. 401, 1999. The interference of a 60-atom molecule (C₆₀): how far duality can be stretched before classical physics wins again.
- David J. Griffiths. Introduction to Quantum Mechanics. 3rd ed., Cambridge University Press, 2018. The formal treatment of the uncertainty principle, derived from the properties of operators, for anyone who wants to go beyond the formula.