Understanding De Broglie Equation Derivation

According to the de Broglie equation/hypothesis, “a matter particle in motion is also related to waves.” In other words, a wave character will be applied to any moving microscopic or macroscopic particle. These waves are also referred to as de Broglie waves or matter waves. The de Broglie equation is used to describe matter/electron wave characteristics. As a consequence, matter particle-like electrons have a dual character, acting both as a particle and as a wave.

De Broglie Hypothesis

The wavelength relation of de Broglie demonstrated that, like photons or light, electrons have both particle and wave aspects of matter. In 1924, French physicist Louis de Broglie proposed it. The hypothesis is referred to as de Broglie’s hypothesis. He postulated that electrons flow in waves, similar to how light travels in waves with a specific wavelength or frequency. The de Broglie principle could be used to determine the Bohr model’s stated relationship.

The de Broglie relation is derived in physics or chemistry using a combination of Einstein’s mass energy formula and Plank’s quantum theory. It’s employed to figure out how to calculate the wavelength and frequency of electromagnetic radiation. Davisson Grammar experiments and the Bohr hypothesis of hydrogen atoms were used to test this wavelength relationship.

De – Brogile Wavelength

Any moving particle, according to Louis de Broglie, will act like a wave. Clinton Davisson and Lester Germer later replicated this experiment in 1927. Matter waves are waves which have something to do with matter. De Broglie waves are another name for them.

Except for photons, particles including electrons and protons have a distinct de Broglie wavelength formula. The momentum of a particle at non-relativistic speeds is equal to its rest mass m multiplied by the velocity v.

De Brogile Wavelength Formula

The De Broglie Wavelength Formula relates the nature of a wave to the nature of a particle. Light can act either as a wave or as a particle, as per numerous investigations. Photons are the particles which comprise light. In 1924, a French physicist named Louis de Broglie derive a formula to describe light’s dual nature as a wave and a particle. This formula can also be applied to electrons and protons.

As per De Broglie, any moving object’s wavelength, that is denoted by (λ), is determined by

λ=h ⁄ p

Or

λ=h ⁄ mv

Here, 

λ stands for de Broglie wavelength in metres.

h stands for Planck’s constant.

 p denotes momentum of a particle is measured in kg per second.

 m stands for mass of particles in kg.

 v stands for velocity of particle in ms-¹.

Understanding De Broglie Equation Derivation

De Broglie Equation Derivation

We can conclude from Einstein’s theory of relativity that

E=mc²      → Equation (1)

Here,

E denotes energy of the particle

m denotes mass of the particle

c denotes speed of light.

As per Planck’s hypothesis, each quantum in a wave has its own quantity of energy, resulting in the given equation.

E=hf  →   Equation (2)

Here,

E is the particle’s energy, 

h is Planck’s constant,

f is frequency.

The Wavelength Equation of De Broglie implies that particles and waves behave the same way. By simplifying equations (1) and (2), Einstein equated the energy relation for both the particle and the wave.

mc²=hf     Equation (3)

The frequency will be f=v ⁄  λ where is the wave’s wavelength.

When we replace this in equation (3), we get

mv²=hv ⁄ λ

λ=h ⁄ mv

or

λ=h ⁄ p

where p denotes the particle’s momentum.

Significance of De Broglie Equation

All moving objects, according to de Broglie, have a particle character. A moving car or a moving ball, on the other hand, do not appear to have particle nature. To better explain it, de Broglie calculated the wavelengths of the electrons in a cricket ball.

wave-particle duality

The Wave Particle Duality hypothesis states that matter and light are both waves and particles. This behaviour has been demonstrated for both elementary and compound particles, including atoms and molecules. Several phenomena assumed different natures of light to describe the notion, such as the photoelectric effect of light, which assumed light to be particles, while interference and diffraction assumed light to be waves. The phenomenon may be explained in its entirety using wave-particle duality.

Quantum physics is based on the concept of wave-particle duality. As per this theory, matter and light have both wave and particle qualities. It ‘s as though it behaves like a wave at one point and then like a particle at another.

Momentum

The product of a particle’s mass and velocity defines its momentum. The magnitude and direction of momentum are both present.  As a consequence, it is classified as a vector quantity. The second law of motion, as stated by Newton, is that “the rate of change of momentum is directly proportional to the force acting on the part.”

The formula is as follows:

P=m*v

Here,

The symbols P, m, and v stand for momentum, mass, and velocity, respectively.

Conclusion

De Broglie’s wavelength relation revealed that electrons, including photons or light, have both particle and wave elements of matter. Louis de Broglie, a French physicist, proposed it in 1924. De Broglie’s hypothesis is the term given to this hypothesis. Many particles are too heavy to examine their wave characteristics, according to De Broglie. Furthermore, whenever an object’s mass is extremely small, the wave characteristics can be observed experimentally. De Broglie anticipated that an electron’s mass would be tiny enough for it to have particle and wave qualities.