Louis de Broglie (1892 – 1987) was a French physicist who extended the ideas of radiant energy to matter. He suggested that matter has wavelike properties. To develop his theory, de Broglie looked at the inverse relationship between the wavelength and energy for photons.

\(\displaystyle E\;=\;\frac{hc}{\lambda}\;\;\;and\;\;\;\lambda\;=\;\frac{hc}{E}\)

We can substitute Einstein’s equation, E = mc² for E,

\(\displaystyle \lambda\;=\;\frac{hc}{mc^2}\;=\;\frac{h}{mc}\)

and then replace the speed of the particle, ν, for c.

\(\displaystyle \lambda\;=\;\frac{h}{m\nu}\;\;\;\;(de\; Broglie\; Equation)\)

The quantity, mν, is the momentum of an object. The mass, m, has units of kJ and the speed, ν has units of m/s. The term matter wave is used to relate wave properties to matter. An example of a matter wave is an electron beam. Matter waves are different from electromagnetic waves because they are the result of the motion of matter. There are no significant fields associated with matter waves whereas with electromagnetic waves there is an electrical and a magnetic field. Matter waves cannot propagate through a vacuum, therefore, the speeds, ν, are less than the speed of light, c. Matter is composed of particles and de Broglie stated that all matter possesses wave like characteristics.

If we have an electron with a mass of 9.11 x 10^-31 kg moving at 2.3 x 10⁶ m/s, the wavelength is:

\(\displaystyle \lambda\;=\;\frac{6.626\times\;10^{-34}\;\frac{kg⋅m^2}{s^2}⋅s}{9.11\times\;10^{-31}\;kg\;\times(2.3\times\;10^6\;m/s)}\;=\;3.2\times\;10^{-10}\;m\)

Recall, 1 J = kg⋅m²/s². The wavelength, 3.2 x 10^-10 m, is easily measured in the lab. If we have a golf ball with a mass of 0.0457 kg moving at a speed of 53.6 m/s, the wavelength is 2.7 x 10^-34 m. This is not a measurable wavelength — it is much too small. We have never observed such small wavelengths.

Below is an electron microscope image of SARS-CoV-2 emanating from cultured cells. An electron microscope uses a beam of high speed electrons in order to image very small objects like cells and viruses.

“File:SARS-CoV-2 scanning electron microscope image.jpg” by National Institute of Allergy and Infectious Diseases (NIAID) is licensed under CC BY 2.0

Exercises

Exercise 1 What is the wavelength, in meters, of an electron (mass = 9.11 x 10^-31 kg) that is accelerated to 4.5% the speed of light, c?

Exercise 2 What is the wavelength of a butterfly with a mass of 0.500 g and a speed of 6.0 m/s? Can we observe this wavelength?

Exercise 3 What speed, ν, would an electron (mass = 9.11 x 10^-28 g) be traveling to have a de Broglie wavelength of 625 nm?

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