de Broglie Wavelength Formula
The de Broglie wavelength formula calculates the quantum wavelength of a particle from its momentum.
Learn with examples.
The Formula
The de Broglie wavelength describes the wave-like nature of matter. Every moving particle has an associated wavelength that is inversely proportional to its momentum.
French physicist Louis de Broglie proposed this idea in his 1924 doctoral thesis. He suggested that just as light can behave as particles (photons), particles of matter can behave as waves. This insight earned him the Nobel Prize in Physics in 1929 and became a cornerstone of quantum mechanics.
For everyday objects, the wavelength is incredibly tiny and undetectable. But for small particles like electrons, the wavelength becomes significant enough to produce observable wave effects such as diffraction and interference. Electron microscopes exploit this principle to achieve much higher resolution than light microscopes.
Variables
| Symbol | Meaning |
|---|---|
| λ | de Broglie wavelength (meters, m) |
| h | Planck's constant (6.626 × 10⁻³⁴ J·s) |
| p | Momentum of the particle (kg·m/s) |
| m | Mass of the particle (kg) |
| v | Velocity of the particle (m/s) |
Example 1
An electron (mass 9.11 × 10⁻³¹ kg) is accelerated to 1 × 10⁶ m/s. What is its de Broglie wavelength?
Calculate momentum: p = mv = 9.11 × 10⁻³¹ × 1 × 10⁶ = 9.11 × 10⁻²⁵ kg·m/s
Apply the formula: λ = h/p = 6.626 × 10⁻³⁴ / 9.11 × 10⁻²⁵
λ ≈ 7.27 × 10⁻¹⁰ m = 0.727 nm (comparable to atomic spacing, hence electron diffraction is observable)
Example 2
A 0.145 kg baseball 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 (inconceivably small — quantum effects are undetectable at this scale)
When to Use It
Use the de Broglie wavelength formula to find the quantum wavelength associated with a moving particle.
- Predicting electron diffraction patterns in crystallography
- Understanding the resolution limits of electron microscopes
- Determining when quantum effects become significant for a particle
- Calculating wavelengths for neutron diffraction experiments