De Broglie Wavelength: Definition, Equation & How To Calculate

French physicist Louis de Broglie won the Nobel Prize in 1929 for groundbreaking work in quantum mechanics. His work to show mathematically how subatomic particles share some wave properties was later proven correct through experiment.

Particles that exhibit both wave and particle properties are said to have ​wave-particle duality​. This natural phenomenon was first observed in electromagnetic radiation, or light, which can be described as either an electromagnetic wave or a particle known as the photon.

When acting as a wave, light follows the same rules as other waves in nature. For example, in a double-slit experiment, the resulting patterns of wave interference show light's wave nature.

In other situations, light exhibits particle-like behavior, such as when observing the photoelectric effect or Compton scattering. In these cases, photons appear to move in discrete packets of kinetic energy following the same rules of motion as any other particle (although photons are massless).

The de Broglie hypothesis is the idea that matter (anything with mass) can also exhibit wavelike properties. Moreover, these resulting matter waves are central to a quantum mechanical understanding of the world – without them, scientists would not be able to describe nature on its smallest scale.

Thus, the wave nature of matter is most noticeable in quantum theory, for example when studying the behavior of electrons. De Broglie was able to mathematically determine what the wavelength of an electron should be by connecting Albert Einstein's mass-energy equivalency equation (E = mc²) with Planck's equation (E = hf), the wave speed equation (v = λf ) and momentum in a series of substitutions.

Setting the first two equations equal to one another under the assumption that particles and their wave forms would have equal energies:

\(E = mc^2 = hf\)

(where ​E​ is energy, ​m​ is mass and ​c​ is the speed of light in a vacuum, ​h​ is the Planck constant and ​f​ is frequency).

Then, because massive particles do not travel at the speed of light, replacing ​c​ with the velocity of the particle ​v​:

\(mv^2 = hf\)

Next replacing ​f​ with ​v/λ​ (from the wave speed equation, where ​λ​ [lambda] is wavelength), and simplifying:

\(\lambda = \frac {h}{mv}\)

Finally, because momentum ​p​ is equal to mass ​m​ times velocity ​v:​

\(\lambda = \frac {h}{p}\)

This is known as the de Broglie equation. As with any wavelength, standard unit of measure for the de Broglie wavelength is meters (m).

where ​_λ_​ is wavelength in meters (m), ​h​ is Planck's constant in joule-seconds (6.63 × 10^-34 Js) and ​p​ is momentum in kilogram-meters per second (kgm/s).

​Example:​ What is the de Broglie wavelength of 9.1 × 10^-31 × 10⁶ m/s?

Since:

\(\begin{aligned} & p = m v \
& p = (9.1 × 10^{-31} \text{ kg}) × (4.5 × 10^6 \text{ m/s})\
& p = 4.095 × 10^{-24} \text{ kgm/s} \
& λ = \frac {h}{p} \
& λ = (6.63 × 10^{-34} \text{ Js}) / (4.095 × 10^{-24} \text{ kgm/s} ) \newline λ = 1.62 × 10^{-10} \text{ m} \end{aligned}\)

Note that for very large masses – meaning something on the scale of everyday objects, like a baseball or a car – this wavelength becomes vanishingly small. In other words, the de Broglie wavelength doesn't have much impact on the behavior of objects we can observe unaided; it isn't needed to determine where a baseball pitch will land or how much force it takes to push a car down the road. The de Broglie wavelength of an electron, however, is a significant value in describing what electrons do, since the rest mass of an electron is small enough to put it on the quantum scale.