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λ = h/p looks like the simplest formula in modern physics. It came from a PhD thesis by someone who had first taken a degree in history, and it was confirmed partly by accident in a New York laboratory. This post covers how Louis de Broglie reached the idea, how it was tested, and how to use it in a calculation.
Who was Louis de Broglie?
His Nobel Prize biography says he was born in Dieppe in 1892 and first studied literature, taking a degree in history in 1910. His interest in science won out, and he finished a science degree in 1913. During the First World War he served in the army's wireless section, stationed at the Eiffel Tower, where he spent his spare time on technical problems. After the war he turned to theoretical physics and quantum problems, while following the experimental work of his elder brother, Maurice.
What was the idea in the 1924 thesis?
In 1924 he presented his thesis, Recherches sur la théorie des quanta, at the University of Paris. It was published in Annales de Physique in 1925.
The starting point was a pattern he noticed. In his Nobel lecture he explained: "the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations." Light was already known to act as both wave and particle. If whole numbers in atoms looked like standing waves, perhaps electrons had a wave side too.
From this he reached the relation λ = h/p, with p the momentum. Fitting a whole number of these waves around an orbit reproduces the stability condition of the old quantum theory. The biography notes that the ideas "first gave rise to astonishment owing to their novelty".
How was the idea tested?
Clinton Davisson's Nobel lecture is honest about how it happened. His team at the Bell Telephone Laboratories in New York was studying electrons scattered from metals, not testing de Broglie. They found "purely by accident" that the scattering depended on the orientation of the crystal. Only in the summer of 1926, after discussing the work in England with Max Born and others, did the experiment become a test of electron waves.
In early 1927 they found the first diffraction beam, then many more. Davisson reported that for the beams they could use, the de Broglie formula "was verified to within the limit of accuracy of the measurements". The results appeared in Nature and Physical Review in 1927. In the same year George Paget Thomson and Alexander Reid reported electrons diffracting through a thin film.
De Broglie received the 1929 Nobel Prize in Physics "for his discovery of the wave nature of electrons". Davisson and Thomson shared the 1937 prize "for their experimental discovery of the diffraction of electrons by crystals".
Worked example: the wavelength of a 54 V electron
Davisson's lecture shows data for a 54-volt beam. Find the wavelength of an electron accelerated from rest through 54 V.
Kinetic energy: K = eV = 1.602 × 10⁻¹⁹ × 54 = 8.65 × 10⁻¹⁸ J.
Momentum: p = √(2mK) = √(2 × 9.109 × 10⁻³¹ × 8.65 × 10⁻¹⁸) ≈ 3.97 × 10⁻²⁴ kg m/s.
Wavelength: λ = h/p = 6.626 × 10⁻³⁴ / 3.97 × 10⁻²⁴ ≈ 1.67 × 10⁻¹⁰ m, or 0.167 nm.
That is about the spacing between atoms in a crystal, which is why a crystal acts as a diffraction grating for electrons. A useful shortcut for electrons is λ ≈ 1.227/√V nm: 1.227/√54 = 0.167 nm.
Compare a cricket ball of mass 0.16 kg at 40 m/s: λ = 6.626 × 10⁻³⁴ / (0.16 × 40) ≈ 1.0 × 10⁻³⁴ m. Far too small to show any wave effect.
Common mistakes:
Using the electron shortcut for protons or alpha particles. The 1.227/√V nm formula has the electron's mass built in.
Forgetting the charge. An alpha particle through V volts gains 2eV of energy, not eV.
Getting the scaling wrong. λ ∝ 1/√V, so four times the voltage halves the wavelength.
Using it at very high voltages. At tens of kilovolts and above, relativity starts to matter.
The JEE (Main) 2026 syllabus lists "Matter waves: wave nature of particle, de Broglie relation" in physics and de Broglie's relationship in chemistry. Check the current bulletin. For many more exam-style questions, work through our guide to de Broglie wavelength problems.
What can a student take from de Broglie's way of working?
Look for the same pattern in different places. He linked whole numbers in atoms to standing waves on strings. Connecting chapters is how many hard problems open up.
A broad education is not wasted. He came to physics after history. His biography also notes that he won the first UNESCO Kalinga Prize in 1952 for explaining modern physics to the public.
Ideas need tests. His was taken seriously because it made a prediction that experiments could check.
If interference and diffraction still feel abstract, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read. Our post on Young's double-slit problems covers the wave optics that electron diffraction borrows.
Try this now: find the de Broglie wavelength of an electron accelerated through 150 V, first with the full method and then with the shortcut.
