De Broglie Wavelength

The argument is one line long and it is the boldest line in twentieth-century physics. Light had been treated as a wave for a century, and by 1923 it was clear it also carried momentum in packets of $p = h/\lambda$. De Broglie's move was to read that equation backwards and apply it to everything: if a wave can carry momentum, then anything carrying momentum has a wavelength, $\lambda = h/p = h/mv$. No new constant, no new mechanism, just the refusal to believe that light was a special case. The reason the world does not obviously look like this is the size of $h$. It is so small that the wavelength of anything you can hold is far below any length that could reveal it, and that smallness is the entire explanation for why classical mechanics works.

Work an electron through a 100 V accelerating gap. It gains 100 eV, or $1.602 \times 10^{-17}$ J, so $v = \sqrt{2E/m} = \sqrt{2 \times 1.602 \times 10^{-17} / 9.109 \times 10^{-31}} = 5.93 \times 10^{6}$ m/s. Then $\lambda = 6.626 \times 10^{-34} / (9.109 \times 10^{-31} \times 5.93 \times 10^{6}) = 1.23 \times 10^{-10}$ m, or 0.123 nm. That is the spacing between atoms in a crystal, which is precisely why electrons diffract off crystals and why an electron microscope resolves structures light can never reach. Now do the same for a person: 70 kg walking at 1.4 m/s gives $6.8 \times 10^{-36}$ m — twenty orders of magnitude smaller than a proton. Nothing in the universe has a slit that narrow.

De Broglie put this in his 1924 doctoral thesis at Paris, and his examiners did not know what to make of it. Paul Langevin sent a copy to Einstein, who wrote back that de Broglie had lifted a corner of the great veil — an endorsement that both passed the thesis and made physicists take it seriously. Confirmation came in 1927 from two directions at once: Clinton Davisson and Lester Germer at Bell Labs, scattering electrons off a nickel crystal that had accidentally recrystallized after a laboratory vacuum accident, and George Paget Thomson in Aberdeen, firing electrons through thin foils. Both measured wavelengths matching $h/p$. The tidy historical coincidence is that J. J. Thomson won a Nobel Prize for showing the electron is a particle and his son G. P. Thomson won one for showing it is a wave, and neither was wrong. Two years after the thesis, Schrödinger built his wave equation on it.

Where this goes wrong in practice. The formula as written is non-relativistic, and $m$ is the rest mass. At 100 V that is harmless — the electron is at 2% of $c$ — but a real transmission electron microscope runs at 100 to 300 kV, where the electron is travelling at 55% to 78% of $c$, and the correct momentum is $\gamma mv$. Ignore that and the wavelength comes out about 5% too large at 100 kV and over 20% too large at 300 kV. Second, the wave is not the particle wiggling along a wavy path, and it is not a vibration of anything material; it is the wavelength of the quantum amplitude whose squared magnitude gives the probability of finding the particle. Picture it as a physical corrugation and you will make predictions the theory does not support. Third, the $v$ is the particle's own speed, so for anything accelerated through a voltage you have to find $v$ first rather than substituting the voltage. And for a photon, use $p = h/\lambda$ directly — the mass in this denominator is zero and the page has nothing to tell you.