Wave–Particle Duality

De Broglie proposed that any particle with momentum $p$ has an associated wavelength $\lambda =h/p=h/mv$.

For everyday objects the wavelength is absurdly small — a moving tennis ball has $\lambda \sim 10^{-34}$ m — so wave effects are undetectable. For electrons, with tiny mass, the wavelength can be comparable to the spacing of atoms in a crystal.

For an electron accelerated from rest through a potential difference $V$, its kinetic energy is $eV$, so its momentum is $\sqrt{2meV}$, which gives the wavelength directly.

Diffraction is a wave phenomenon: particles travelling in straight lines could not spread into a pattern of rings. When a beam of electrons passes through a thin layer of polycrystalline graphite, it produces concentric rings on a screen — a diffraction pattern.

The atomic spacing in graphite is about $10^{-10}$ m, comparable to the electrons’ de Broglie wavelength, which is exactly the condition for significant diffraction.

Increasing the accelerating voltage increases the electrons’ speed and momentum, shortening the wavelength, so the rings get $\text{smaller}$ — just as the de Broglie equation predicts.

Electrons bound in an atom can occupy only certain discrete $\text{energy levels}$. These are negative, measured relative to zero for an electron that has just escaped; the lowest is the $\text{ground state}$.

An electron moves up a level by absorbing exactly the energy difference, from a photon or a collision. When it drops back, it emits a photon whose energy equals the difference between the levels.

Because only specific differences are possible, only specific photon energies — and therefore specific wavelengths — can be emitted or absorbed.

A hot gas produces an $\text{emission line spectrum}$: bright lines at specific wavelengths on a dark background, one line for each possible downward transition.

Passing white light through a cooler gas gives an $\text{absorption line spectrum}$: a continuous spectrum with dark lines where the gas has absorbed photons of exactly the right energies to excite its electrons.

The dark absorption lines of an element sit at exactly the same wavelengths as its bright emission lines, because the same energy differences are involved. Each element’s set of lines is unique, which is how the composition of distant stars is identified.