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de Broglie Relation

de Broglie Hypothesis for Photon and Electron

de Broglie wavelengths relation pointed out that just as photon light has both particle and wave nature, electrons have also these duel properties of matter. de Broglie’s hypothesis suggested that electron travels in waves with the definite wavelength, frequency. Therefore, the de Broglie equation given from mass energy relation and plank quantum theory used to calculate and convert the wavelength and frequency of electromagnetic radiation. This wavelength relation tested by Davisson Grammar experiments and Bohr’s theory of hydrogen atom. Hence de Broglie’s hypothesis led from the consideration of quantum theory and theory of relativity by Einstein in physics.

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

de Broglie Equation for Wavelength

de Broglie proposed an equation with the help of the Plank equation and Einstein energy mass law of matter. He considered the mass of the photon and wave nature of photon light quanta with wavelength = λ, frequency = ν, and energy = E.

E = hν = hc/λ
where c = velocity of light
h = Plank 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 particle to the dynamics particle of matter and calculate the mass, momentum, wavelength frequency, and energy of electron.

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

Bohr’s Theory and de Broglie Relation

Angular momentum of moving electrons in Bohr’s model

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
λ = h/mv
or, mv = h/λ
where λ = wavelength of the moving electron.
Combining de Broglie equation and Bohr’s theory
2πr = nλ

Therefore, a standing wave is fixed within the space allowed to it and does not travel beyond the allowed space. It does not travel beyond the allowed space.

Electron Diffraction Experiment

Within the few years of de Broglie’s hypothesis, Davisson and Germer tested these predictions by diffracting electrons by crystals. The diffraction is similar to the diffraction of x-ray radiation. Therefore, they established that a beam of electrons really has wave properties along with particle properties. The wavelength associated with electrons of known momentum value was exactly the same that predicted by the de Broglie equation, λ = h/p. This relationship applies to all particles but for heavy particles, the wavelengths are too small to be observed experimentally.

Kinetic Energy and Wavelength

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

Thus the energy of an electron
E = Ve = ½ mv2
where e is the charge of an electron.
λ = h/√2mVe

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