We saw that in class how we can mesh up the fact of photon nature of light, that interaction of light with matter is always in terms of a discrete bundles of energy interacting with matter, with the smooth flow of radiation we are used to of seeing in everyday life, and which is characteristic of Maxwells theory, when the total energies involved are so large that we tend to ignore the small discrete packets. Let us see how this works out in case of Hydrogen atom. Just like an electron oscillating with a given frequency emits radiations of same frequency, Maxwells theory predicts that an electron orbiting in a circle with a certain frequency of revolution will emit light of the same frequency. As it does so, it loses energy and its orbit shortens, increasing the frequency of revolution and in next instant it emits radiation of a bit larger frequency in a continues fashion. Since we can test all of this for energies of larger scales, this picture should come out from our quantum formulas in those limits.
(a) Show that in Bohrs theory, the frequency of photon emitted as the electron makes a transition from an orbit with quantum number n to an orbit of quantum number n − 1 is given by,
For large orbits, as we encounter in everyday situations, n is very large. Take that limit in the above formula, ignoring additions of order 1 to the large number n.
(b) Now calculate frequency of revolution of the electron moving in an orbit quantized according to Bohrs formula (put Bohrs formula for electrons radius in n-th orbit) and show that it comes out exactly equal to the frequency of photon derived in part a.
