Plain-English Meaning
Einstein showed light (a wave) has particle properties (photons). de Broglie reversed this: all particles (electrons, protons, even you) have a wave. The wavelength shrinks as momentum increases. For everyday objects, λ is unimaginably small — which is why we don't notice quantum effects at macroscopic scales.
When a quantum question feels ambiguous, translating it into state, observable, probability, and evolution language usually clarifies the answer.
Deeper Explanation
λ = h/p = h/(mv) for non-relativistic particles. Electrons with energy 1 eV have λ ≈ 1.23 nm — comparable to atom sizes, explaining electron diffraction. The wave nature shows up in Young's double-slit, Davisson–Germer diffraction, and electron microscopes.
Worked Example
Problem: Find the de Broglie wavelength of an electron accelerated through 100 V.
- KE = eV = 1.6×10⁻¹⁹ × 100 = 1.6×10⁻¹⁷ J
- p = √(2mKE) = √(2 × 9.11×10⁻³¹ × 1.6×10⁻¹⁷)
- p = √(2.917×10⁻⁴⁷) = 1.71×10⁻²⁴ kg·m/s
- λ = h/p = 6.626×10⁻³⁴/1.71×10⁻²⁴ = 3.87×10⁻¹⁰ m
Result: λ ≈ 0.387 nm (X-ray wavelength range — explains electron diffraction)
At A Glance
Category: Quantum
Levels covered: High School, College, Masters, PhD
Best use: Start with the formula meaning, then move to the worked example and quiz so the equation turns into a tool instead of a memorised line.