State the properties of matter waves. (asked in MU Dec 2024; also Dec 2025)

Answer: Matter waves have \(\lambda=h/mv\); they are not electromagnetic, they exist only for moving particles, their phase velocity can exceed \(c\), and their amplitude gives the probability of finding the particle.

A moving particle of mass \(m\) and velocity \(v\) has an associated de Broglie wave of wavelength \(\lambda=\dfrac{h}{p}=\dfrac{h}{mv}\). Its main properties are:

1. Wavelength depends on momentum: \(\lambda\propto\frac1{mv}\). A lighter or slower particle has a longer wavelength, so wave effects are noticeable for electrons and neutrons but not for a cricket ball.
2. Only moving particles have them: if \(v=0\), \(\lambda\to\infty\), so no wave is associated with a particle at rest.
3. Not electromagnetic: they are associated with any moving particle, charged or neutral (e.g. neutrons), so they are not produced by accelerating charges and are not EM waves.
4. Phase velocity can exceed \(c\): \(v_p=\dfrac{E}{p}=\dfrac{c^2}{v}>c\), since \(v<c\). This phase velocity has no physical meaning; the particle travels with the group velocity of the wave packet, \(v_g=v\).
5. Probability waves: the amplitude of the wave function \(\psi\) is not a physical displacement; \(|\psi|^2\) gives the probability density of finding the particle at a point.
6. They show diffraction and interference, as proved by the Davisson–Germer electron diffraction experiment.