What do you mean by wave nature of an electron? How was quantisation of angular momentum of the orbiting electron in Bohr's model of hydrogen atom explained by de-Broglie hypothesis ?

Explanation

As per de-Broglie concept of matter waves, even electron particles also exhibit wave nature and the wavelength `lamda` associated with an electron of mass m moving with a speed v is given as :
`lamda=h/(mv)=h/p=h/(sqrt(2mK))`
where p is the momentum and K is the kinetic energy of electron. If an electron is accelerated by a potential of V volt, then its de-Broglie wavelength is given as :
`lamda=h/(sqrt(2meV))=(1.227)/(sqrtV)` nm
As per de-Broglie.s hypothesis in terms of electron wave, only those stationary Bohr orbits are possible in which total distance (i.e., the circumference of the orbit) contains a definite numberof these electron waves. Mathematically,
total path length of orbit = n `(lamda_("electron wave"))`
where n is an integer and may have values 1, 2, 3, 4, ....... etc.
From de-Broglie concept of matter waves, we know that wavelength of electron waves may be expressed as
`lamda=h/(p_(n))=h/(mv_(n))`
where m = mass of electron and `v_(n)` = velocity of electron in state corresponding to n. Substituting the value of A in de-Broglie.s quantum condition, we get
`2pir_(n)=n.h/(mv_(n))rArrmv_(n)r_(n)=nh/(2pi)`
which is Bohr.s quantum condition ( on the second postulate for quantisation of angular momentum.