Q.Calculate the wavelength of the photon emitted when an electron in a hydrogen atom falls from $n = 4$ to $n = 2$. (Rydberg constant $R_H = 1.097 \times 10^{7}\ \text{m}^{-1}$.)

Niels Bohr's 1913 model for the hydrogen atom rests on four postulates, drawing on Planck's quantum theory and hydrogen's line emission spectrum to fix the instability that plagued Rutherford's model. (1) The electron can move only in one of several fixed circular orbits (stationary states) of definite radius and energy, arranged concentrically in increasing order of energy. (2) The electron's energy in a given orbit does not change with time; it moves to a higher orbit only by absorbing, and to a lower orbit only by emitting, exactly the required energy, always as a discontinuous jump. (3) Bohr's frequency rule: a transition between two states differing in energy by ΔE emits/absorbs radiation of frequency $\nu = (E_2-E_1)/h$. (4) The electron's angular momentum is quantised to integral multiples of $h/2\pi$: $mvr = n\,h/2\pi$. Solving these gives orbit radii $r_n = n^2a_0$ (Bohr radius $a_0=52.9$ pm) and energies $E_n = -R_H/n^2$ ($R_H=2.18\times10^{-18}$ J for hydrogen), extended to hydrogen-like species (He⁺, Li²⁺, …) via $E_n=-2.18\times10^{-18}(Z^2/n^2)$ J and $r_n=52.9(n^2/Z)$ pm. This theory successfully re-derives the empirical Rydberg equation for hydrogen's spectrum from first principles.