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The de Broglie wavelength is an important concept in the study of quantum mechanics. The wavelength (λ) associated with an object relative to its momentum and mass is termed as the de Broglie wavelength. The de Broglie wavelength of a particle is inversely proportional to its force.
De Broglie Waves
Matter is said to have a dual nature of waves and particles. De Broglie waves, which was named after discoverer Louis de Broglie, are the property of a body which changes in time or space while behaving in a wave-like manner. They are also considered as matter waves. It is very similar to the dual nature of light, which behaves as a particle and a wave, which has been experimentally proven.
De Broglie Wavelength and Hypothesis
Louis de Broglie in1923 explained the concept of de Broglie waves. In his doctoral thesis, he proposed that any moving particle, whether microscopic or macroscopic, would be associated with a wave character. Davisson and Germer later experimented and demonstrated in 1927. The waves which are associated with matter were considered as “matter waves.” These waves determine the nature of the wave related to the particle. We know that electromagnetic radiation has the dual nature of a particle (with momentum) and a wave (given in frequency, wavelength). He also proposed a relationship between velocity (or speed) and momentum with wavelength when the particle behaves like a wave.
Particles such as electrons and protons, except photons, have different de Broglie wavelength formulas. At non-relativistic velocities, the momentum of a particle would be equal to the product of its rest mass m velocity v.
De Broglie Wavelength Formula
The de Broglie wavelength formula is a formula which defines the nature of a wave as that of a particle. Many experiments show that light can behave as a wave and also as a particle. The light particles are called photons. A French physicist Louis de Broglie in 1924 defined a formula to determine the dual nature of light as wave nature and particle nature. This formula is also used for electrons and protons.
According to the De Broglie, the wavelength is given as
And also
Here,
h = Planck’s constant
p = momentum
m = mass
v = velocity
De Broglie Equation
De Broglie equation is derived from Einstein’s theory.
As we know from Einstein’s theory
————- (1)
Here,
E = energy of particle
m = mass of particle
c = speed of light
According to Planck’s theory, all quantum of a wave has a separate amount of energy which is related to it. The equation of Planck’s theory is given as
E=hf—————– (2)
Here,
E = energy of particle
h = Planck’s constant
f = frequency
From equation (1) and equation (2), we get
mc2=hf—————- (3)
When 
f=v/
Put f in equation (3), then we get
mc2=hv
And
=h/mv
Therefore,
=h/p
And also
p=h ————- (4)
De Broglie Wavelength and Kinetic Energy
Kinetic energy of a body is given as
K=1/2mv2
mK=1/2(mv)2
As we know,
p=mv
Therefore,
mK=1/2p2
From equation (4)
mK=1/2h2
=h/√2mK
Thermal De Broglie Wavelength
There is a relationship between the de Broglie equation and the temperature of given molecules of gas, and which is provided by thermal de Broglie wavelength. The thermal de Broglie equation shows the average de Broglie wavelength of gas particles at a certain temperature in an ideal gas.
At temperature T, the thermal de Broglie wavelength is given as
Here,
h = Planck’s constant
m = mass of gas particle
= Boltzmann constant
Bohr’s model for Hydrogen
Electrons move in atoms in circular orbits around the nucleus. Electrons are in the form of disc-shaped clouds. For the hydrogen atom, the minimum-energy ground-state electron can be represented by a rotating disk whose inner edge has radius and outer edge has radius, where is the Bohr radius.
If we consider that the orbit of the electron in the atom contains ‘n’ numbers of de Broglie wavelengths, then in the case of a circular orbit with radius, for circle perimeter and angular momentum L of electron, we get
2πr=n
L=rp=nh/2π
=h/p
This is exactly what the Bohr model postulates for the hydrogen atom. According to the postulate, the angular momentum of a hydrogen atom is quantized and directly proportional to the orbit number ‘n’ and Planck’s constant.
Conclusion
Matter is said to have a dual nature of waves and particles.
The waves which are associated with matter were considered as “matter waves.”
Particles such as electrons and protons, except photons, have different de Broglie wavelength formulas. At non-relativistic velocities, the momentum of a particle would be equal to the product of its rest mass m velocity v.
The De Broglie Wavelength formula is given as
The relation between de Broglie wavelength and kinetic energy is
=h/√2mK
At temperature T, the thermal de Broglie wavelength is given as
