de Broglie Hypothesis for Photon and Electron

de Broglie relation and hypothesis pointed out that the light photon and electron have both particle and wave nature. This wave relationship uses to calculate and convert the wavelength and frequency of light. de Broglie’s suggestion of wavelength confirmation by Bohr’s theory and Davisson and Germer’s electron diffraction experiment and kinetic energy equation.

He suggested that electrons not be regarded simply corpuscles. A specific wave motion must also be assigned to them.

Therefore de Broglie hypothesis led from the consideration of quantum theory and theory of relativity by Einstein.

de Broglie wavelength frequency and energy relation for photon and electron

de Broglie Equation for Wavelength

de Broglie proposed an equation with the help of the Plank equation and Einstein mass-energy relation. He considered the particle and wave nature of photon with wavelength = λ, frequency = ν,  and energy = E.

    \[ \boxed{E = h\upsilon = h\frac{c}{\lambda }} \]

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.

    \[ \boxed{\lambda =\frac{h}{p}} \]

de Broglie relation extended light particle to the dynamics particle and calculate the mass, momentum, wavelength frequency and energy of an electron.

    \[ \boxed{\lambda = \frac{h}{p}=\frac{h}{mv}} \]

where m = total mass of electron
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.

Electron Diffraction Experiment

de Broglie’s relationship convert wavelength, frequency and energy of light photon and confirmation by Davisson and Germer’s electron diffraction experiment conclusively proves that electrons are not is not an ordinary particle.

From the evidence by the experiment of determination of mass and e/m electron has particle nature.

But the electron diffraction experiment by Davison and Gramer’s given the evidence of the wave nature of the electron.

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

    \[ E = Ve =\frac{1}{2}mv^{2} \]

where e is the charge of an electron.

Hence the relation between kinetic energy and wavelength

    \[ \boxed{\lambda =\frac{h}{\sqrt{2mVe}}} \]