Quick questions on De Broglie matter waves: HSC Physics Module 8

By 1924 the photon picture had established a particle aspect of light (carrying energy $hf$ and momentum $h/\lambda$), even though wave properties were equally well established. Louis de Broglie's PhD thesis proposed the symmetric idea: any particle of momentum $p$ has an associated wavelength:

For everyday objects, the wavelength is absurdly small.

Clinton Davisson and Lester Germer at Bell Labs were studying low-energy electron scattering from a nickel target. An accident (a vacuum leak followed by a heat treatment that crystallised the nickel) left the surface as a single crystal. Subsequent scattering of electrons at 54 V from the now-crystalline surface showed a sharp angular peak at 50 degrees, exactly where Bragg-like diffraction predicted for the nickel lattice spacing and the de Broglie wavelength of 54 eV electrons (0.167 nm).

De Broglie immediately applied his hypothesis to the hydrogen atom. Picture the electron as a wave travelling around the nuclear Coulomb potential. For a stable, self-consistent orbit the wave must close on itself (a standing wave around the loop). The circumference must therefore be an integer number of wavelengths:

The wave nature of matter is the basis of much of modern science:

An electron microscope accelerates electrons through 50 kV. Find the de Broglie wavelength and compare with visible light (550 nm). (Relativistic correction is small here but indicates a real correction at higher voltages.)

The wavelength is $\lambda = h/p$, not $h/v$. Always include the mass factor.

For accelerating voltages above about 10 kV, the relativistic correction starts to matter. Use $p = \gamma m v$ if precision is required.

It directly measured a wavelength matching de Broglie to within experimental uncertainty, the kind of agreement that decides physical theories.

The matter wave is the probability amplitude for the particle. Detection always reveals a localised particle; the wave description governs interference and diffraction patterns built up over many such detections.