Wave-particle duality and the de Broglie wavelength

Another major development in quantum mechanics was pioneered by French physicist Louis de Broglie. Based on work by Planck and Einstein that showed how light waves could exhibit particle-like properties, de Broglie hypothesized that particles could also have wavelike properties.

[What are wavelike properties?]

Examples of observable wavelike behavior are interference and diffraction. For example, when light is shined through a barrier with two slits, as in Young's double-slit experiment, the light waves will diffract through the slits. The destructive and constructive interference between the light waves produces a pattern of dark and light areas on the detector.

De Broglie derived the following equation for the wavelength of a particle of mass $\text m$mm (in kilograms $\text{kg}$kgk, g), traveling at velocity $\text v$vv (in $\dfrac{\text m}{\text s}$​s​​m​​start fraction, m, divided by, s, end fraction), where $\lambda$λlambda is the de Broglie wavelength of the particle in meters and $h$hh is Planck's constant, $6.626 \times 10^{-34} \,\dfrac{\text{kg} \cdot \text m^2}{\text s}$6.626×10​−34​​​s​​kg⋅m​2​​​​6, point, 626, times, 10, start superscript, minus, 34, end superscript, space, start fraction, k, g, dot, m, start superscript, 2, end superscript, divided by, s, end fraction:

$\lambda=\dfrac{h}{\text {mv}}$λ=​mv​​h​​lambda, equals, start fraction, h, divided by, m, v, end fraction

Note that the de Broglie wavelength and particle mass are inversely proportional. The inverse relationship is why we don't notice any wavelike behavior for the macroscopic objects we encounter in everyday life. It turns out that the wavelike behavior of matter is most significant when a wave encounters an obstacle or slit that is a similar size to its de Broglie wavelength. However, when a particle has a mass on the order of $10^{-31}$10​−31​​10, start superscript, minus, 31, end superscript kg, as an electron does, the wavelike behavior becomes significant enough to lead to some very interesting phenomena.

Concept check: The fastest baseball pitch ever recorded was approximately 46.7 $\dfrac{\text{m}}{\text s}$​s​​m​​start fraction, m, divided by, s, end fraction. If a baseball has a mass of 0.145 kg, what is its de Broglie wavelength?

 

Example 1: Calculating the de Broglie wavelength of an electron

The velocity of an electron in the ground-state energy level of hydrogen is $2.2\times10^6\,\dfrac{\text{m}}{\text s}$2.2×10​6​​​s​​m​​2, point, 2, times, 10, start superscript, 6, end superscript, space, start fraction, m, divided by, s, end fraction. If the electron's mass is $9.1\times10^{-31}$9.1×10​−31​​9, point, 1, times, 10, start superscript, minus, 31, end superscript kg, what is the de Broglie wavelength of this electron?

We can substitute Planck's constant and the mass and velocity of the electron into de Broglie's equation:

λ=hmv=6.626×10−34kg⋅m2s(9.1×10−31kg)(2.2×106ms)=3.3×10−10 m

The wavelength of our electron, $3.3\times10^{-10}\,$3.3×10​−10​​3, point, 3, times, 10, start superscript, minus, 10, end superscript, space meters, is on the same order of magnitude as the diameter of a hydrogen atom, ~$1\times 10^{-10}\,$1×10​−10​​1, times, 10, start superscript, minus, 10, end superscript, space meters. That means the de Broglie wavelength of our electron is such that it will often be encountering things with a similar size as its wavelength—for instance, a neutron or atom. When that happens, the electron will be likely to demonstrate wavelike behavior!