Thermal de Broglie Wavelength Calculator
The thermal de Broglie wavelength sets the boundary between classical and quantum behaviour in a gas. It is the typical quantum wavelength of a particle at a given temperature, and when it grows to the size of the spacing between particles, quantum statistics take over and exotic states like Bose-Einstein condensates can form. This calculator takes the mass of a single particle and the absolute temperature, then returns the thermal de Broglie wavelength in metres and nanometres. It is essential in statistical mechanics and ultracold atomic physics.
Thermal de Broglie wavelength formula
Lambda = h / sqrt(2 * pi * m * k * T)
h = 6.62607015e-34 J*s (Planck constant)
k = 1.380649e-23 J/K (Boltzmann constant)
m = particle mass (kg), T = temperature (K)
The thermal wavelength shrinks with increasing mass and temperature. The result is given in metres and nanometres. Compare it to the interparticle spacing to judge quantum versus classical behaviour.
Quantum gas context
- Planck's constant is exactly 6.62607015 times ten to the minus thirty-fourth joule-seconds.
- The Boltzmann constant is exactly 1.380649 times ten to the minus twenty-third joules per kelvin.
- A proton has a mass of about 1.673 times ten to the minus twenty-seventh kilograms.
- Quantum effects dominate when the thermal wavelength rivals the particle spacing.
- Ultracold experiments cool gases to nanokelvin to stretch the thermal wavelength.
Thermal de Broglie wavelength: frequently asked questions
What is the thermal de Broglie wavelength?
The thermal de Broglie wavelength is the average quantum wavelength of a particle in a gas at a given temperature. It is Lambda = h / sqrt(2 * pi * m * k * T), where h is Planck's constant, m the particle mass, k the Boltzmann constant and T the absolute temperature. It measures the spatial extent of a particle's quantum wave packet.
Why does the thermal de Broglie wavelength matter?
It tells you whether a gas behaves classically or quantum mechanically. When the thermal wavelength is much smaller than the spacing between particles, classical statistics apply. When it becomes comparable to or larger than the spacing, quantum effects dominate, leading to Bose-Einstein condensation or Fermi degeneracy.
What mass should I enter?
Enter the mass of a single particle in kilograms. A hydrogen atom is about 1.67 times ten to the minus twenty-seventh kilograms, and an electron is about 9.11 times ten to the minus thirty-first. Lighter particles have longer thermal wavelengths and reach quantum behaviour at higher temperatures.
Why does the wavelength grow as temperature falls?
The thermal de Broglie wavelength is inversely proportional to the square root of temperature, so cooling a gas lengthens it. That is why ultracold atom experiments chill gases to nanokelvin temperatures: it stretches the thermal wavelength until the atomic wave packets overlap and quantum condensation appears.
What units does this calculator return?
It returns the thermal de Broglie wavelength in metres and in nanometres. For everyday gases at room temperature the value is a tiny fraction of a nanometre, far smaller than the spacing between molecules, which is why ordinary air behaves classically.
Official sources
- U.S. National Institute of Standards and Technology: Planck constant.
- U.S. National Institute of Standards and Technology: Boltzmann constant.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.