de Broglie Wavelength Calculator
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Formulas for Calculating the de Broglie Wavelength
The de Broglie wavelength is the wavelength associated with a particle, inversely proportional to its momentum, reflecting wave-particle duality.
Here are the formulas for calculating the de Broglie wavelength in terms of particle velocity and kinetic energy.
Non-Relativistic Formulas
When the velocity of a particle is much smaller than the speed of light \((v \ll c)\), classical mechanics provides a good approximation of the de Broglie wavelength.
(a) Based on Velocity:
\[ \lambda_{dB} = \frac{h}{mv}, \]
where \(\lambda_{dB}\) is the de Broglie wavelength in meters, \(h = 6.62607015 \times 10^{-34} \) J·s is Planck’s constant, \(m\) is the mass of the particle in kilograms, and \(v\) is the velocity of the particle in meters per second.
(b) Based on Kinetic Energy
\[\lambda_{dB} = \frac{h}{\sqrt{2mE_k}},\]
where \(E_k\) is the kinetic energy in joules (J).
For inputs in electronvolts (eV), energy is converted as:
\[ E_k \, (\text{J}) = E_k \, (\text{eV}) \times 1.602176634 \times 10^{-19} \]
Relativistic Formula
For particles moving at speeds comparable to the speed of light \((v \approx c)\), relativistic mechanics must be used for accurate results.
(a) Based on Velocity:
\[\lambda_{dB} = \frac{h}{\gamma mv} \quad \text{and} \quad \gamma = \frac{1}{\sqrt{1 – \left( \frac{v}{c} \right)^2}}, \]
where \(\gamma\) is the Lorentz factor and \(c = 299792458\) m/s is the speed of light in a vacuum.
(b) Based on Kinetic Energy:
For a particle whose kinetic energy is large enough that relativistic effects can not be ignored, it is convenient to begin with the special-relativistic energy–momentum relation
\[E_{\text{tot}}^{2}= (pc)^{2} + (mc^{2})^{2},\]
where \(E_{\text{tot}} = E_{k}+mc^{2}\) is the sum of kinetic and rest-mass energy, \(p\) is the relativistic momentum, \(m\) is the rest mass, and \(c = 299\,792\,458\ \text{m s}^{-1}\) is the speed of light in vacuum. Rearranging this expression for \(p\) gives
\[p = \frac{\sqrt{E_{\text{tot}}^{2} – (mc^{2})^{2}}}{c}.\]
The de Broglie wavelength is defined as \(\lambda_{dB}=h/p\). Substituting the momentum just obtained yields a compact working formula that is valid at any energy:
\[\lambda_{dB} = \frac{h}{p} = \frac{h\,c}{\sqrt{E_{\text{tot}}^{2} – (mc^{2})^{2}}}\]
Because the expression is written entirely in terms of total energy, it can be evaluated directly once the kinetic energy \(E_{k}\) and mass of the particle are known. This fully relativistic version of the de Broglie relation is indispensable when dealing with high-energy beams—such as synchrotron electrons or GeV-scale protons—whose speeds approach a significant fraction of the speed of light.
References
- P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2014,” Rev. Mod. Phys. 88, 035009 (2016). DOI: 10.1103/RevModPhys.88.035009
- E. R. Cohen and B. N. Taylor, “The 1986 CODATA Recommended Values of the Fundamental Physical Constants,” J. Phys. Chem. Ref. Data 17, 1795 (1988). DOI: 10.1063/1.555817
- C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. 1 (Wiley-Interscience, New York, 1977).
- L. de Broglie, “Waves and quanta,” Nature 112, 540 (1923). DOI: 10.1038/112540a0
- R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, Boston, MA, 1994).