The electron as a standing wave

In 1913 Niels Bohr proposed to save the atom by introducing an ad hoc quantisation rule: electrons can only occupy certain “allowed” orbits, in which — by decree — they do not radiate. The rule worked very well at reproducing the hydrogen spectrum, but it was a postulate handed down from above, without any real physical reason.

The explanation came ten years later from Louis de Broglie, with a surprising idea: if light, which we believed was a wave, also behaves like a particle (the photon), then the electron, which we believed was a particle, must also behave like a wave. To every electron of momentum $p = mv$ is associated a wavelength $\lambda = h/(mv)$. And a wave circling the nucleus, in order not to cancel itself out, must “close up” after one complete turn: it must be a standing wave.

Principle — Standing wave in the hydrogen atom

For the electron’s wave to return to its starting value after travelling around a circular orbit of radius $r$, the circumference must contain an integer number of wavelengths: $\ev{2\pi r = n\,\lambda} \qquad (n = 1, 2, 3, \dots)$

The intuition is the same as for a guitar string or a wave on a ring: only certain wavelengths “fit” an integer number of times along the closed path. If the circumference did not contain an integer number of $\lambda$, after every turn the wave would find itself out of phase with itself and, overlapping turn after turn, would cancel out through destructive interference. The only orbits that “survive” are those in which the wave latches perfectly onto its own tail.

The electron’s wave as a standing wave on a circular orbit: here $n=4$ wavelengths close exactly along the circumference. Only for an integer number of wavelengths does the wave “find itself again” after one turn.

From the wave condition to the quantisation of angular momentum. The strength of this idea is that, by combining the closure condition with de Broglie’s relation $\lambda = h/(mv)$, Bohr’s ad hoc rule turns into a consequence:

$2\pi r = n\cdot\frac{h}{m\,v} \quad\Longleftrightarrow\quad \ev{m\,v\,r = n\,\hbar}$

where $\hbar = h/(2\pi)$. The left-hand side, $mvr$, is precisely the electron’s angular momentum in the orbit. The result therefore says that angular momentum cannot take just any value: it is always an integer multiple of $\hbar$. Here, derived rather than postulated, is Bohr’s quantisation rule — born from the simple requirement that the electron-wave closes back on itself.

Links

Topics: Quantum physics Concepts: Bohr model · De Broglie wavelength · Standing waves · Angular momentum · Wave-particle duality Objects: Hydrogen atom

Related exercises: Problem — Why the atom doesn’t collapse (standing wave) · True or false on quantum physics · Problem — De Broglie wavelength of an electron at 100 V