De Broglie Waves

The de Broglie waves or matter waves refer to the properties of matter that vary in time or space while showing a wave-like behaviour. They are named after the French physicist Louis de Broglie. In 1924, for the first time, Louis de Broglie proposed the wave-like nature of the electron and said that all matter exhibits properties similar to that of a wave. 

For example, the diffraction of a water wave or a beam of light is similar to the diffraction of a beam of electrons.  

Objects in our everyday life have a much smaller computed wavelength when compared to electrons. Hence, wave properties of these objects have never been detected. So, everyday objects only show particle behaviour. The de Broglie waves are usually only applicable in the realm of subatomic particles. The de Broglie waves combine the particle character of matter to its wave character. The de Broglie waves form the basis of quantum physics. 

The de Broglie wavelength 

The de Broglie wavelength,  λ (for a particle that has a certain mass) is related to its momentum, p, through the Planck’s constant, h, in the equation: 

                λ= h/p = h/mv

According to the above formula, the de Broglie wavelength is inversely proportional to momentum.  

Derivation of the de Broglie wavelength 

The wavelength of any massive particle is calculated in comparison with photons. It is as follows: 

According to Planck’s quantum theory, the energy of a photon, while it exhibits wave-like character, its energy can be said to be

                         E = hv 

(i) where v is the frequency of the wave and h is Planck’s constant.

On the other hand, according to Einstein’s equation, if the photon is assumed to have particle character, then its energy is

                         E = mc2 

(ii) where m is the mass of a photon and c is the velocity of light.

From equations (i) and (ii), we get

hν = mc2

But ν = c / λ

h * c / λ = mc2

or  λ = h / mc

According to de Broglie, this equation applies to any particle. 

If the velocity of a photon, c, is replaced by the velocity of a particle v and the mass of the photon is replaced by the mass of the particle, then, for any particle such as an electron, the de Broglie wavelength can be written as 

           λ = h / mv or λ = h / p

Where mv is the momentum p of the particle. 

The de Broglie waves exist in a closed loop.

Around any closed loop, de Broglie waves would be associated with electrons, circling the nuclei in atoms. They can only exist as standing waves that fit evenly around a closed loop, or else they will cancel themselves out. This is why electrons in atoms have a particular configuration, or state, among other available states. 

Conclusion 

Matter waves form the foundation of quantum physics. They are based on the fact that everything in this universe exhibits both wave and particle nature. This concept was first given by French physicist Louis de Broglie in 1924. Therefore matter waves are also referred to as the de Broglie waves. They help in relating the particle character with the wave character of matter. The de Broglie wavelength is represented by the following relationship: 

λ= h/p = h/mv

This de Broglie wavelength is only calculated for a massive particle, i.e. a particle with a certain mass instead of a massless particle. The de Broglie wavelength of any massive particle can be calculated by comparing it with a photon. The de Broglie waves are only applicable in the realm of subatomic particles. This is because everyday objects have wavelengths too small to be detected.