Wave–Particle Duality

You’ll be able to

Explain that light and matter each show both wave and particle behavior
Use the de Broglie relation to connect a particle’s wavelength and momentum
Calculate the momentum of a photon from its wavelength

Everything is both

Modern physics ends with a startling unification: light and matter both have a wave side and a particle side. Light shows its wave nature in interference and diffraction, and its particle nature in the photoelectric effect. Astonishingly, matter does too: electrons, normally thought of as particles, produce interference patterns when fired through a double slit. Which face nature shows depends on the experiment. This wave–particle duality is one of the deepest ideas in physics.

The de Broglie wavelength

Louis de Broglie proposed that any particle with momentum p has an associated wavelength λ = h/p. The larger the momentum, the shorter the wavelength. For everyday objects — a thrown baseball — the momentum is so large that the wavelength is unimaginably tiny, far too small to ever detect, which is why we never see a baseball diffract. Only for very light, fast particles like electrons does the de Broglie wavelength grow big enough to produce observable wave effects. A photon, likewise, carries momentum p = h/λ.

de Broglie wavelength & photon momentum

λ = h / p · p_photon = h / λ

h = 6.63 × 10⁻³⁴ J·s. Wavelength and momentum are inversely related: heavy, fast particles have vanishingly small wavelengths; light, slow particles have larger ones.

Worked example

A photon has a wavelength of 500 nm. Find its momentum. (h = 6.63 × 10⁻³⁴ J·s)

1.Use the photon momentum relation: p = h/λ.
2.Convert the wavelength: 500 nm = 5.0 × 10⁻⁷ m.
3.Substitute: p = (6.63 × 10⁻³⁴) / (5.0 × 10⁻⁷).
4.Divide: 6.63 / 5.0 = 1.33, and 10⁻³⁴ / 10⁻⁷ = 10⁻²⁷, giving 1.33 × 10⁻²⁷.

Answer: p ≈ 1.3 × 10⁻²⁷ kg·m·s⁻¹ — an extremely small momentum, as expected for a single photon.

Checkpoint

According to de Broglie, if a particle’s momentum increases, its wavelength:

Tip

The reason we never notice the wave nature of everyday objects is the tiny size of h. A walking person has a de Broglie wavelength around 10⁻³⁵ m — many orders of magnitude smaller than an atom — so no experiment could ever reveal it.

Checkpoint

A photon has a wavelength of 500 nm. What is its momentum? (h = 6.63 × 10⁻³⁴ J·s)

On the exam

Two duality relations to keep straight: for a photon E = hf and p = h/λ; for a matter particle λ = h/p. Both hinge on Planck’s constant — the bridge between the wave quantities (f, λ) and the particle quantities (E, p).

Answer the 2 checkpoints as you read.