How is angular momentum an integral multiple of nh/2pie?

Dear Student,

• Derivation of de-Broglie Equation

The wavelength of the wave associated with any material particle was calculated by analogy with photon.
In case of photon, if it is assumed to have wave character, its energy is given by
E = hv   …(i)
(According to the Planck’s quantum theory)
Where nth frequency of the wave and ‘h’ is is Planck’s constant
If the photon is supposed to have particle character, its energy is given by
E = mc²   ….… (ii)
(according to Einstein’s equation)
where ‘m’ is the mass of photon, ‘c’ is the velocity of light.
By equating (i) and (ii)
 
hv = mc²
But v = c/λ
or, h c/λ   = mc²
(or) λ = h /mc
The above equation is applicable to material particle if the mass and velocity of photon is replaced by the mass and velocity of material particle. Thus for any material particle like electron.
λ = h/mv   or λ =  where mv = p is the momentum of the particle.

• Derivation of Angular Momentum from de Broglie Equation

According to Bohr’s model, the electron revolves around the nucleus in circular orbits. According to de Broglie concept, the electron is not only a particle but has a wave character also.
If the wave is completely in phase, the circumference of the orbit must be equal to an integral multiple of wave length (λ)
Therefore 2πr = nλ
where ‘n’ is an integer and ‘r’ is the radius of the orbit
But λ = h/mv
∴ 2πr = nh /mv   or mvr = n h/2π
 


Regards