What are the main properties of matter waves and how do they differ from light waves?

In one line

Matter waves are probability waves associated with moving particles; their wavelength is \(h/mv\), they are not electromagnetic, and their group velocity equals the particle's velocity.

Explained simply

A matter wave is not a ripple of water or an electromagnetic vibration. It is more like a "map of where the particle is likely to be". Where the wave is strong, you are likely to find the particle; where it is weak, you are unlikely to.

How it works

1. Wavelength depends on momentum: \(\lambda = h/mv\). Lighter or slower particles have longer wavelengths.
2. Stationary particle: if \(v = 0\), \(\lambda\) becomes infinite, so the wave has no meaning; matter waves are associated with moving particles.
3. Not electromagnetic: they are produced by any moving particle, charged or neutral (neutrons also diffract).
4. Phase velocity exceeds c: \(v_p = \omega/k = E/p = mc^2/mv = c^2/v > c\). This does not violate relativity because a single phase carries no energy or information.
5. Wave packet: a particle is described by a group of waves of slightly different wavelengths. The group velocity equals the particle velocity \(v\).
6. Probability interpretation: the amplitude is the wave function \(\psi\); \(|\psi|^2\) gives probability density, not a physical displacement.
7. They show interference and diffraction (e.g. electron diffraction), confirming their wave nature.

Key formula

\(\lambda = h/mv\); phase velocity \(v_p = c^2/v\); group velocity \(v_g = v\); and \(v_p v_g = c^2\). \(c = 3\times10^8\) m/s.

Quick example

An electron at \(v = 3\times10^6\) m/s has phase velocity \(v_p = (3\times10^8)^2/(3\times10^6) = 3\times10^{10}\) m/s, 100 times faster than light, while the "packet" (the electron) moves at only \(3\times10^6\) m/s. This shows why phase velocity has no direct physical meaning for the particle.

Common mistakes

• Claiming matter waves travel at the speed of light.
• Saying the phase velocity exceeding c breaks relativity.
• Treating \(\psi\) itself as a measurable quantity; only \(|\psi|^2\) has direct meaning.

How MU asks it

Asked as "State the properties of matter waves" (3 to 5 marks). List five or six properties with one line each, include \(v_p = c^2/v\) and \(v_g = v\), and the probability interpretation.