Planck had assumed that the idea of quanta was probably a convenient way of representing unknown physics governing the interaction of light and matter, rather than a property of light itself. Einstein had showed that, in fact, quanta are real.
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So more than 100 years after Young had conclusively demonstrated the waviness of light, Einstein showed that light of frequency f must carry energy in indivisible bits of size hf. Here, h is "Planck’s constant": 6.626 x 10⁻³⁴ J·s. This is a very particle-y behavior for a wave!
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But this is just how nature works. Light exhibits some wavy behaviors and some particle-y behaviors. There's nothing to reconcile, it just seems odd to us because our inutition, based on living in the macroscopic world, tells us that “particle" and “wave" are orthogonal concepts.
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Our intuition is great for baseballs and waves at the beach, but there’s no reason that it should extend to the subatomic world. There’s nothing weird about “wave-particle duality.” Nature is what it is. Anyway, back to de Broglie.
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Since light, our canonical example of a wave-y thing, can have particle-y properties, de Broglie suggested it might also work in the other direction. That is, perhaps something like an electron might occasional exhibit behaviors we associate with waves?
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In special relativity, a massless particle like the photon has momentum p = E/c. And its energy at frequency f comes in these E=hf chunks. Since the speed of light is c = λ*f, we can rearrange all that to get λ = h/p.
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So de Broglie proposed that a particle with momentum p should have an associated wavelength λ = h/p. A few years later, in 1927, this was confirmed in electron diffraction experiments by Thomson, and independently by Davisson and Germer.
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This led to a Nobel Prize for de Broglie in 1929, and for Thomson and Davisson in 1937. A lovely little twist is that Thomson’s father J.J. Thomson is credited with discovering the electron. Essentially, father showed that it was a particle and son showed that it was a wave.
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Demonstrating the wave properties of matter is easiest for electrons; their small mass means a small momentum and hence a large wavelength. But it has since been shown for neutrons, atoms, and entire molecules.
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Here, a team from Vienna observes the diffraction of C₆₀ molecules! https://www.nature.com/articles/44348
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