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==de Broglie==
In 1923, ten years after Bohr had derived the spectrum of atomic hydrogen by postulating the quantization of angular momentum, [[wikipedia:Louis de Broglie|Louis de Broglie]] hit on an explanation of why the atom's angular momentum comes in multiples of <math>\hbar.</math> Since 1905, [[wikipedia:Albert Einstein|Einstein]] had argued that electromagnetic radiation itself was quantized (and not merely its emission and absorption, as Planck held). If electromagnetic waves can behave like particles (now known as [[wikipedia:Photon|photons]]), de Broglie reasoned, why cannot electrons behave like waves?

Suppose that the electron in a hydrogen atom is a [[wikipedia:Standing wave|standing wave]] on what has so far been thought of as the electron's circular orbit. (The [[wikipedia:Crest (physics)|crests, troughs]], and [[wikipedia:Node (physics)|nodes]] of a standing wave are stationary.) For such a wave to exist on a circle, the circumference of the latter must be an integral multiple of the [[wikipedia:Wavelength|wavelength]] <math>\lambda</math> of the former: <math>2\pi r = n\lambda.</math>
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<br>Einstein had established not only that electromagnetic radiation of frequency <math>\nu</math> comes in quanta of energy <math>E=h\nu</math> but also that these quanta carry a momentum <math>p=h/\lambda.</math> Using this formula to eliminate <math>\lambda</math> from the condition <math>2\pi r = n\lambda,</math> one obtains <math>pr=n\hbar.</math> But <math>pr=mvr</math> is just the angular momentum <math>L</math> of a classical electron with an orbit of radius <math>r.</math> In this way de Broglie derived the condition <math>L=n\hbar</math> that Bohr had simply postulated.
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