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de Broglie Relation

de Broglie Hypothesis

de Broglie wavelength relation pointed out that just like photons of light, electron have both particle and wave properties of matter. It was proposed by French physicist Louis de Broglie in 1924. It is called de Broglie’s hypothesis.

de Broglie wavelengths, frequency and energy relation for light photon or electron

He suggested that electrons travel in waves, analogous to light with a definite wavelength or frequency. The de Broglie hypothesis could be fitted to derive the relation that the Bohr model proposed.

In physics or chemistry, the combination of Einstein’s mass energy formula and Planck’s quantum theory is used to derive the de Broglie relation. It is used for calculating the wavelength and frequency of electromagnetic spectrum radiation. This wavelength relation was tested by Davisson Grammar experiments and Bohr’s theory of hydrogen atoms.

de Broglie Relation Derivation

de Broglie proposed the wavelength relation with the help of the Planck equation and Einstein’s mass energy formula of matter.

The mass of the photon, the smaller the light quanta given by mass energy equation of Einstein. If the wavelength of light = λ, frequency = ν, and energy = E.

From Planck equation, E = hν = hc/λ
where c = velocity of light
h = Planck constant = 6.627× 10−27

Theory of relativity from Einstien,
E = m c2

Combining these two relations,
mc = h/λ
or, p = h/λ
where, p = mc = mv = momentum
∴ Wavelength (λ) = h/p

de Broglie relation extended light particles to the dynamics particles of matter and calculated the mass, momentum, wavelength frequency, and energy of an electron.

λ = h/p = h/mv
where m = total mass of electron
v = velocity

de Broglie Wavelength and Bohr Model

According to de Broglie, the electron is not a solid particle revolving around the nucleus of an atom. It is a standing wave extending around the nucleus of a circular orbit. Angular momentum of moving electrons from Bohr atomic structure,

mvr = nh/2π
or, mv = nh/2πr
where m = mass of an electron,
n = principle quantum number = 1, 2, 3, 4, …
r = radius of the orbital of an atom.

According to de Broglie’s equation
λ = h/mv
or, mv = h/λ
where λ = wavelength of the moving electron.

Combining de Broglie equation and Bohr’s theory
2πr = nλ

From the above formula, it should be suggested that there will be an integral number of wavelengths that must fit into a circular orbit of Bohr. Such an integral number of wavelengths must generate a standing wave.

A standing wave produces a stationary pattern or fixed profile. It does not travel beyond the allowed space.

Davisson and Germer Experiment

Experimental verification of the de Broglie equation was obtained by Davisson and Germer in 1927. The success comes by diffracting a beam of electrons from the nickel surface. The pattern of electron diffraction obtained by them is similar to that of x-ray diffraction.

The wavelength of electrons obtained from the experiment is identical to the calculated wavelength by the de Broglie relation. In this way, the dual nature of an electron and the quantized nature of the de Broglie relation were established.

Kinetic Energy Formula with Wavelength

When the particle-like light photon or electron is subjected to the potential difference V, it acquires a velocity v  and generates two types of energies, potential and kinetic energy.

The energy of an electron,
E = Ve = ½ mv2
where e is the charge of an electron.

Again, λ = h/mv

From the above two relations,
wavelength (λ) = h/√2mVe

This is the derivation between emission kinetic energy and de Broglie wavelengths relation of photon or electron.