Bohr's Postulates and the Bohr Model of the Hydrogen Atom

The idea behind Bohr's model. In 1913, Niels Bohr set out to rescue Rutherford's nuclear atom (Section 1.3) from its stability and spectral problems. Rather than abandoning the nuclear picture, Bohr KEPT it -- a small central nucleus with electrons orbiting it under Coulomb attraction -- but combined it with an entirely new idea, borrowed from Max Planck's and Albert Einstein's recent quantum theory of radiation: certain physical quantities in an atom do not vary smoothly and continuously, as classical mechanics assumes, but exist only in fixed, discrete ("quantised") amounts. Restricting hydrogen's single electron to this restricted set of allowed possibilities, Bohr showed, is enough to explain both stability and the line spectrum at once. His model, though later superseded by full quantum mechanics, remains an excellent working description of the hydrogen atom and is built on three postulates.

Postulate 1 -- stationary (non-radiating) orbits. An electron can revolve around the nucleus only in certain special circular orbits, called stationary orbits, in which -- contrary to classical electromagnetic theory -- it does NOT radiate energy, however long it stays in that orbit. In any such orbit, the electrostatic (Coulomb) force of attraction between the nucleus (charge $+Ze$) and the electron (charge $-e$) provides exactly the centripetal force required to keep it moving in its circular path of radius $r$ and speed $v$:

This is simply Rutherford's force-balance condition (Section 1.3) carried over unchanged -- what is new is the claim that the electron, while obeying it, is exempt from classical radiation.

Postulate 2 -- quantisation of angular momentum. Of all the circular orbits that could in principle satisfy the force-balance condition above, only those are actually allowed for which the electron's orbital angular momentum is an exact integral multiple of $h/2\pi$ ($h$ = Planck's constant):

The whole number $n$ is called the principal quantum number. This postulate is what turns the continuous range of orbits Rutherford's model would classically allow into a discrete, countable SET of permitted orbits -- and Section 1.4.1 uses it, together with Postulate 1, to calculate exactly what those allowed radii are. (Though introduced here as an assumption, this quantisation condition is later shown, using de Broglie's matter-wave hypothesis, to be equivalent to requiring that the electron's orbit contain an exact whole number of its own de Broglie wavelengths -- a standing-wave condition, giving the postulate a deeper physical justification.)

Postulate 3 -- the frequency condition (Bohr's frequency rule). An atom radiates energy only when an electron makes a TRANSITION, jumping from one permitted stationary orbit of higher energy $E_i$ to another of lower energy $E_f$; the energy released is emitted as a single photon of frequency $\nu$ given by

$h\nu = E_i - E_f$ …