The de Broglie wave — matter duality

What just happened?

You doubled the particle momentum, and its matter wave shrank by half. Wavelength and momentum are inversely proportional — that is the heart of wave–particle duality.

The de Broglie hypothesis:

— every moving particle (an electron, a proton, even a ball) has a matter wave of wavelength $\lambda$. Here $h$ is the Planck constant, and $p = m\upsilon$ — the particle momentum.

What does it mean in practice?

The faster and heavier the particle (greater momentum), the shorter its matter wave. That is why wave properties show up for light, slow particles — for macroscopic bodies $\lambda$ is vanishingly small and immeasurable. Notice that the product stays constant:

How do we know?

Electron matter waves are confirmed by diffraction on a crystal (the Davisson–Germer experiment) — electrons, like waves, are diffracted and form an interference pattern. The electron microscope is built on this.

Why you need it for the exam:

The formula $\lambda = h/p$ solves most matter-wave problems. The trick: compute the momentum $p = m\upsilon$ (or $p=\sqrt{2mE_k}$), then divide the Planck constant by it. Remember: momentum rises → $\lambda$ decreases.

Remember: every particle has a matter wave $\lambda = h/p$ — the wavelength decreases as momentum rises. This is a cornerstone of quantum physics.