De Broglie relation

Louis de Broglie, a French physicist, described the theory of the atomic structure in 1923. By using a sequence of alternatives, de Broglie hypothesised particles to hold properties of waves. After some years, de Broglie’s hypothesis was rechecked by scientists firing electrons and rays of light through slits. Scientists discovered that the electron stream acted the same way as light, proving de Broglie correct.

De Broglie equation states that matter can behave as waves more like light and radiation, behaving as waves and particles. The equation further explains that a beam of electrons can also be emitted, just like a beam of light. In quintessence, the De Broglie equation helps us understand the idea of matter having a wavelength.

Therefore, if we look at every moving particle, whether a microscopic particle or a macroscopic particle, it will have a wavelength.

De Broglie relation

The De Broglie relation is one of the frequently used equations to define the wave properties of the matter. It defines the wave nature of the electron.

Electromagnetic radiation exhibits the dual nature of a particle and wave (which are expressed in frequency wavelength). Microscopic particles as electrons also proved to possess this dual nature property.

Louis De Broglie, in his thesis, advised that any moving particle, whether it is microscopic or macroscopic, will be related to a wave character. Hence, it was called ‘Matter Waves.’ He further suggested a relation between the velocity and momentum of a particle with the wavelength if the particle had to behave as a wave.

The particle and wave nature of matter, however, looked contrasting as it was not possible to prove the existence of both properties in any single experiment. This is because every experiment typically depends on some principle, and results related to the principle are only reflected in that experiment and not the other. 

However, both the properties are necessary to understand or describe the matter entirely. Hence, particles and the wave nature of matter are interdependent. Both of these don’t need to be present simultaneously, though. 

Significance of de Broglie relationship

The importance of the de Broglie relation is that it is more useful for microscopic and fundamental particles like electrons.

De Broglie’s equation helps us understand the idea of matter having a wavelength. Therefore, if we look at every moving particle, whether microscopic or macroscopic, it will have a wavelength. In macroscopic objects, the wave nature of matter can be detected, or it is visible.

De Broglie set out the following relation between wavelength (λ) & momentum (p) of a material particle. 

λ= h/mv = h/p

Where,

          λ = wavelength,

          p = the momentum

Important Note

• De Broglie’s prediction was certified experimentally when it was found that an electron beam goes through diffraction, a phenomenon characteristic of waves.
• Every object in motion has a wave character. The wavelengths corresponding to the ordinary objects are so short (because of their large masses) that their wave properties cannot be detected.

Derive de Broglie relationship

We have Einstein’s mass-energy relation equation for a photon of light of frequency.

E = pc

Where,

            E = energy

            p = momentum

            c = speed of light

Light behaves as a wave when it undergoes interference, diffraction, etc., and is completely described by Maxwell’s equations. But then, the wave nature of electromagnetic radiation is questioned when it is embroiled in blackbody radiation, photoelectric effect, etc. Einstein forwarded his idea of the photon, a bundle of quantized radiant energy localised in a small volume, as a way to describe the particle-like nature of light. The energy and momentum of such a photon was suggested to be,

Now from Planck’s equation of wave nature of light, we can write as:

E = hν= hc/λ

Where

          h= Planck constant

          E=energy

          c=speed of light

Now, as per de-Broglie, both these energies should be equal

hc/λ = pc

λ = h/p

De Broglie advised the above equation is a general one that applies to material particles and photons. Now the momentum of the particle of mass m and velocity v is p=mv, so its de-Broglie wavelength is:

λ = h/mv

Drawback

It is applicable to microscopic particles like electrons, protons, and neutrons. Still, it fails in the case of large-size objects because they have more mass and their wavelength becomes smaller, which is not an easy task to measure.

Conclusion

De Broglie concluded that most particles are too heavy to observe their wave properties. If the mass of an object is very small, however, the wave properties can be explained experimentally. De Broglie anticipated that the mass of an electron was small enough to express the properties of both particles and waves. All matter expresses wave-like behaviour. For example, a beam of electrons can be emitted the same as that of a beam of light or a water wave.

In most cases, however, the wavelength is too small to have a practical effect on day-to-day activities. Hence in our daily lives with objects the size of tennis balls and people, matter waves are not relevant. The de Broglie equations also describe the relationship between the wavelength λ and the momentum p, frequency f, and the total energy E of a free particle.