(i) The energy of electrons in atoms is quantised
(ii) The existence of quantized electronic energy levels is a direct result of the wave like properties of electrons. The solutions of Schrodinger wave equation gives the allowed energy levels (orbits).
(iii) According to Heisenberg uncertainty principle, the exact position and momentum of an electron can not be determined with absolute accuracy. As a consequence, quantum mechanics introduced the concept of orbital. Orbital is a three dimensional space in which the probability of finding the electron is maximum.
(iv) The solution of  Schrodinger wave equation for the allowed energies of an atom gives the wave function ψ,which represents an atomic orbital. The wave nature of electron present in an orbital can be well defined by the wave function ψ.
(v) The wave function ψ itself has no physical meaning.However, the probability of finding the electron in a small volume dxdydz around a point (x,y,z) is proportional to |ψ(x,y,x)|² dxdydz |ψ(x,y,x)|² is known as probability density and is always positive.

de-Broglie combined the following two equations of energy of which one represents wave character (hu) and the other represents the particle nature(mc²).(i) Planck's quantum hypothesis: E = hv
(ii) Einstein's mass - energy relationship :
E = mc2
From (i) and (ii)
hv = mc²
hc/λ =mc²
λ = h/mc
(a) The equation represents the wavelength of photons whose momentum is given by mv (photons have zero rest mass)
(b) For a particle of matter with mass m and moving with a velocity v, the equation can be written as  λ = h/mv
(c) This is valid only when the particle travels at speeds much less than the speed of light

Limitation of Bohr's atom model:
(a) The Bohr's atom model is applicable only to species having one electron such as hydrogen, Li²⁺ etc ... and not applicable to multi electron atoms.
(b) It was unable to explain the splitting of spectral lines in the presence of magnetic field (Zeeman effect) or an electric field (Stark effect).
(c) Bohr's theory was unable to explain why the electron is restricted to revolve around the nucleus in a fixed orbit in which the angular momentum of the electron is equal to nh/2π

Applying Bohr's postulates to a hydrogen like atom (one electron species such as H, He⁺ and Li²⁺ etc...)the radius of the n^th orbit and the energy of the electron revolving in the n^th orbit were derived. The results are as follows:
r_n = \(\frac { (0.529){ n }^{ 2 } }{ x } \mathring { A } \)  ...(1)
E_n = \(\frac { -13.6({ z }^{ 2 }) }{ { n }^{ 2 } } ev\quad { atom }^{ -1 }\)   or ...  (2) or
E_n = \(\frac { (-1312.8){ z }^{ 2 } }{ { n }^{ 2 } } kJ\quad { mol }^{ -1 }\) .....(3)

Bohr's atom is based on the following assumptions:
(a) The energies of electrons are quantised
(b) The electron is revolving around the nucleus in a certain fixed circular path called stationary orbit.
(c) Electron can revolve, only in those orbits in which the angular momentum (mvr) of the electron must be equal to an integral multiple of h/2π i.e mvr =nh/2π
where n = 1,2,3 ...etc...
As long as an electron revolves in the fixed stationary orbit, it doesn't lose its energy.
However, when an electron jumps from higher energy state (E₂)to a lower energy state (E₁)the excess energy is emitted as radiation. The frequency of the emitted radiation is
E2 = E1 = hv and 
\(v=\frac { ({ E }_{ 2 }-{ E }_{ 1 }) }{ h } \)
Conversely, when suitable energy is supplied to an electron, it will jump from lower energy orbit to a higher energy orbit.