De Broglie's hypothesis

If light, which we thought was a wave, can behave like a particle, should not the reverse also hold? In 1924 Louis de Broglie made this leap of symmetry in his doctoral thesis: a particle endowed with mass must also be able to show wave-like behaviour. To every particle, he proposed, is associated a wave whose length is fixed by its momentum.

The idea arises by inverting the photon relation $p = h/\lambda$. For the photon, momentum determines the wavelength; de Broglie postulates that the same relation holds for any particle, read the other way round: given the momentum $p = mv$ (in the non-relativistic limit $v \ll c$), the associated wave has length

$\lambda = \frac{h}{p}.$

Key formula

To every particle of momentum $p = mv$ is associated a matter wave of length $\ev{\lambda = \frac{h}{p} = \frac{h}{m\,v}}$

Experimental confirmation came in 1927 with Davisson and Germer: a beam of electrons made to strike a nickel crystal produces a diffraction pattern exactly like light through a grating, with maxima and minima at the positions predicted using de Broglie’s wavelength. Electrons — particles par excellence — diffracted like waves. On this principle rests today’s electron microscope: since the wavelength of fast electrons is thousands of times smaller than that of visible light, it achieves a resolving power enormously greater than that of an optical microscope.

Why then do we never see a ball or a person behave like a wave? Because Planck’s constant is minuscule: for macroscopic objects $\lambda$ turns out to be so tiny as to make any wave-like effect completely unobservable. The wave nature manifests itself only when $\lambda$ is comparable to the size of the system involved.

Example — When wave-like nature makes itself felt

The rule of thumb: a particle behaves like a wave when it is confined to a space small compared to its de Broglie $\lambda$.

• Baseball ($m = 0{,}15$ kg, $v = 40$ m/s): $\lambda \approx 10^{-34}$ m, much smaller than an atomic nucleus. Wave-like behaviour is completely unobservable: the ball is always a classical particle.
• 50 eV electron ($v \approx 4\cdot 10^6$ m/s): $\lambda \approx 1{,}7\cdot 10^{-10}$ m, of the order of atomic distances. Wave-like behaviour manifests clearly — this is the basis of the electron microscope and of the Davisson-Germer diffraction.
• Electron in an atom: $\lambda$ comparable to the size of the atom itself. It cannot be treated as a classical particle: this is where the quantisation of energy levels comes into play.

The comparison is illuminating: going from the ball to the atomic electron, the wavelength grows by more than twenty orders of magnitude relative to the size of the system, and with it the relevance of quantum effects. The world appears “classical” to us not because quantum mechanics stops holding, but because at our scales $\lambda$ is negligible.

Links

Topics: Quantum physics Concepts: De Broglie wavelength · Wave-particle duality · Momentum

Related exercises: Problem — Why the atom doesn’t collapse (standing wave) · Problem — De Broglie wavelength of an electron at 100 V · Connecting Malus’s law and a single photon