The de Broglie’s Equation

Louis de Broglie is a French physicist who had made a remarkable proposal about the nature of matter in 1924. He proposed that, just like light, who has dual nature – wave and particle, electrons, which are particles, may also behave like wave. He based his thoughts on the symmetry of nature.

Since a wave has wavelength and frequency, if follows that a particle may also have a wavelength and a frequency if it behaves like a wave. Using the equation $$p = {h \over \lambda}$$, equation (1) below is derived which is then called as the de Broglie wavelength of a particle which is related to the particles momentum $$p = mv$$.

$$\lambda = {h \over p} = {h \over mv}$$     (1)

where

$$h$$ = Planck’s constant

$$p$$ = momentum of the particle

$$m$$ = mass of the particle

$$v$$ = speed of the particle

according to de Broglie, the frequency, $$f$$ is also related to the energy of the particle, expressed the same as the photon’s energy:

$$E = hf$$     (2)

De Broglie’s equation has no experimental evidence. American physicists Clinton Davisson and Lester Germer conducted an experiment to verify de Broglie’s claim about matter waves. Davisson and Germer are workers of the Bell Telephone Laboratories. They studied the surface of a piece of nickel by directing a beam of electrons at the surface and then they observed how many electrons bounced off at the different angles. An accident occurred during their experiment which allowed air to enter the vacuum chamber, and an oxide film formed on the metal surface. In order for the film to be removed, they baked the sample in a high-temperature oven. As an effect, large regions within the nickel are created with crystal planes that were continuous over the width of the electron beam. The sample looked like a single crystal of nickel, as viewed from the electrons.

Repeating the observation with the same sample, they obtained different results. Before the accident, they observed a smooth variation of intensity with angle, but, after the accident, a strong maxima in the intensity of the electron beam appeared at specific angles. The angular positions of the maxim are dependent on the accelerating voltage $$V_{ba}$$ which is used to produce the beam of electrons. In the process, both recognized that there is diffraction of the electron beam. This discovery of Davisson and Germer is a direct experimental confirmation of the wave hypothesis.

From the accelerating voltage, $$V_{ba}$$, they could determine the speeds of the electrons to compute the de Broglie wavelength. Since the work done on the electron $$eV_{ba}$$ is equal to its kinetic energy, $$K = ({1 \over 2})mv^2 = {p^2 \over 2m}$$. Thus, the work done for a nonrelativistic particle is given by

$$eV_{ba} = {p^2 \over 2m}$$

and since $$p = \sqrt {2meV_{ba}}$$, the de Broglie wavelength equation can be expressed as,

$$\lambda = {h \over p} = {h \over \sqrt {2meV_{ba}}}$$     (3)

where

$$m$$ = mass of the electron

$$e$$ = magnitude of the electron charge

$$V_{ba}$$ = accelerating voltage

Equation (3) shows that the accelerating voltage is inversely proportional to the wavelength of the electron. That is, the greater the accelerating voltage, the shorter the electron’s wavelength. The accident that happened on Davisson’s and Germer’s experiment was electron diffraction which became an evidence to confirm de Broglie’s experiment.