Wave Nature of Matter and De Broglie’s Equation

For many years, we have been learning that light behaves both as a wave and a particle. However, a particle remains stuck to a place. However, a wave spreads in space. 

On the basis of the above context, we conclude that the nature of light depends on the nature of our observation. Also, if you observe principles like interference, diffraction, or reflection, you notice that light is a wave. However, in phenomena like the photoelectric effect, you state that light has a particle character.

Now, a question arises, is light a wave or a particle? Well, light has a dual nature. Another question is, whether it is a specific property of light! Or does light have dual nature? Now, if we say that all the quantities in the universe have dual nature? How could we determine or prove that? 

Might be these questions stuck in the mind of Louis Victor de Broglie, for which he came up with one of the most influential equations in Physics named after him as the de Broglie equation.

Now, let us understand the dual nature of the wave with the derivation of De Broglie’s equation.

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Understanding Dual Wave Nature

The wave nature of matter is one of the most groundbreaking concepts in physics. A particle is confined at a place while a wave is spread in space. It can be said in this way that it is our observation on which the nature of light depends. While observing phenomena like interference, diffraction, or reflection we can find that the light is a wave if we are, but if we are looking at phenomena like the photoelectric effect, we can find that light has a particle character.

Light has a dual nature, i.e. it behaves as both wave and particle. Furthermore, one of the most revolutionary concepts in Physics is given by Louis Victor de Broglie, the de Broglie equation which answered many questions such as: Does only light have a dual nature? What if other quantities had dual nature? How could we realize that? He hypothesized that particles exhibit a wave nature. Experiments such as the Davisson-Germer experiment confirmed it.

Why Should Particles Behave as Waves?

In classical mechanics, radiation is considered as waves and particles are considered as hard billiard balls. It was realized that radiation can behave as both waves and particles. Both radiation and moving particles can provide energy and momentum to different objects. Motivated by the symmetry of nature, De Broglie hypothesized in 1924 that matter should also have dual nature. Particles are not localized in space. The dual nature of both radiation and matter laid the foundation for quantum theory. 

De Broglie’s Hypothesis

According to the hypothesis, particles behave as waves which are called matter waves. The wavelength  (De Broglie wavelength) of the matter-wave corresponding to a particle of momentum p is given by, 

\[\lambda = \frac{h}{p}\]

Here, he denotes Planck's constant. The De Broglie wavelength is inversely proportional to the momentum (hence mass) of a particle. For macroscopic objects, the wavelength is much smaller than the size of the object. The wave nature becomes prominent for microscopic objects e.g. electrons.

A photon having energy E has momentum:

\[p = \frac{E}{c}\]

Here, c denotes the speed of light in a vacuum.

According to Planck’s concept, the energy of a photon of frequency  and wavelength  is given by,

E = hv = \frac{hc}{\lambda}

The energies should be equal, suggesting:

\[\frac{hc}{\lambda} = pc\]

\[\lambda = \frac{h}{p}\]

De Broglie realized that the above relation should hold for particles also. A particle of mass m and velocity v has momentum p=mv. Therefore, it should have a wavelength given by,

\[\lambda = \frac{h}{p} = \frac{h}{mv}\]

Experimental Verification (Davisson-Germer experiment)

In 1927, Davisson and Germer performed an experiment on Nickel crystals. The crystal structure is such that the crystal behaves as a diffraction grating for an X-ray beam of wavelength 1.65 Å. 

The interplanar separation of the crystal is comparable with the X-ray wavelength. When the beam falls on the crystal, waves reflect from different planes with a constant phase relation. These reflected waves interfere to give a maximum intensity at a scattering angle = \[50^{0}\]. A similar setup (shown below) was considered with X-rays replaced by a beam of electrons having variable energy.

Experimental Setup

Davisson and Germer recorded the current of electrons at different scattering angles for fixed energy values. Around 54 eV electron-energy, they observed a pattern similar to that of the X-ray diffraction pattern. It had a peak at =50. 

Intensity Profile

According to the De Broglie equation, an electron with energy E and mass m has a wavelength:

λ = h/p

Substituting h = 6.626 x 10-34 Js, m = 9.1 x 10-31 kg, and E = 54 eV, the wavelength is obtained to be,

λ =1.67 Å

The theoretical prediction by de Broglie matched with the experimental results, thus verifying the hypothesis. 

Heisenberg’s Uncertainty Principle

The Uncertainty Principle states that the momentum and position of a particle cannot be measured with precision simultaneously. In fact, there is always some uncertainty Δx in position and Δp in momentum. The uncertainties are related by,

Δx Δp  ≤ h/2

If the momentum of a particle is measured accurately (i.e., p = 0), the uncertainty x in its position becomes infinite. A particle with a definite momentum should have a definite wavelength, according to de Broglie’s equation. Such a wave should extend to infinity, which is unphysical. Any particle should be represented by a localized wave (wave packet), which consists of multiple wavelengths. 

Did You Know?

The De Broglie wavelength of any particle is inversely proportional to its momentum. Everyday objects have large masses such that the corresponding matter waves have very small wavelengths. Hence, wavelike behavior is not observed at an ordinary scale and classical physics works fine.

The wave nature of electrons successfully explains the structure of an atom. According to Bohr’s theory, electrons can rotate around the nucleus at some fixed orbits only. At these orbits, the electron waves form a stationary wave such that their energy remains constant. The energy changes by emission or absorption of photons only during transitions from one orbit to another.

The wave nature of matter gives rise to interesting phenomena like quantum tunneling. Alpha decay of a heavy nucleus is the result of the wave nature of the nuclei.

In crystallography, electron diffraction is used to analyze different crystal structures.