De Broglie Wavelength Formula
The de Broglie formula λ = h/mv calculates the wavelength of matter waves.
Understand wave-particle duality with worked examples.
The Formula
In 1924, French physicist Louis de Broglie proposed that all matter has wave-like properties. The wavelength of these "matter waves" depends on the particle's momentum.
For everyday objects, the de Broglie wavelength is incredibly tiny — far too small to observe. But for tiny particles like electrons, the wavelength becomes significant and produces measurable effects such as electron diffraction.
This idea was revolutionary because it extended wave-particle duality (previously known only for light) to all matter. De Broglie received the Nobel Prize in Physics in 1929 for this discovery.
Variables
| Symbol | Meaning |
|---|---|
| λ | De Broglie wavelength (measured in meters, m) |
| h | Planck's constant = 6.626 × 10⁻³⁴ J·s |
| m | Mass of the particle (measured in kilograms, kg) |
| v | Velocity of the particle (measured in meters per second, m/s) |
| p | Momentum of the particle (p = mv, measured in kg·m/s) |
Example 1
An electron (mass = 9.109 × 10⁻³¹ kg) is accelerated to a speed of 1.0 × 10⁶ m/s. What is its de Broglie wavelength?
Identify the values: h = 6.626 × 10⁻³⁴ J·s, m = 9.109 × 10⁻³¹ kg, v = 1.0 × 10⁶ m/s
Calculate momentum: p = mv = 9.109 × 10⁻³¹ × 1.0 × 10⁶ = 9.109 × 10⁻²⁵ kg·m/s
Apply the formula: λ = h / p = 6.626 × 10⁻³⁴ / 9.109 × 10⁻²⁵
λ ≈ 7.27 × 10⁻¹⁰ m = 0.727 nm (comparable to atomic spacing — this is why electron diffraction works)
Example 2
A baseball (mass = 0.145 kg) is thrown at 40 m/s. What is its de Broglie wavelength?
Calculate momentum: p = mv = 0.145 × 40 = 5.8 kg·m/s
Apply the formula: λ = h / p = 6.626 × 10⁻³⁴ / 5.8
λ ≈ 1.14 × 10⁻³⁴ m (incredibly small — far below any measurable scale, which is why baseballs don't show wave behavior)
When to Use It
Use the de Broglie wavelength formula to find the wave properties of particles.
- Predicting electron diffraction patterns in crystallography
- Designing electron microscopes (shorter wavelength means higher resolution)
- Understanding quantum tunneling in semiconductors
- Calculating the wavelength of neutrons used in neutron scattering experiments
- Demonstrating why macroscopic objects do not exhibit wave behavior