(a) Explain in detail what would one observe (about the radiation) if an electron were revolving around

Question

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) Explain in detail what would one observe (about the radiation) if an electron were revolving around a proton in a macroscopic sized radius and how it matches with Maxwells predictions. Do we need to use quantum theory for motion of particles in macroscopic orbits? 

(b) Find the frequency of photons emitted by an electron moving around a proton in a radius of 1 cm.

Answer 1

(a) One would observe a continuous radiation because the levels would be placed so close, that distinguishing of each packet of radiation from another would be impossible. 

No, we do not need quantum theory. The electron would still emit discrete photons but they are not detectable due to closely packed energy levels as n → ∞. 

(b) Now we are given that, 

Radius of electron’s orbit = r = 1 cm = 10⁻² m. 

We want to calculate the frequency of the photon emitted. As we have already calculated in part, frequency of emitted photon is given by,

Let’s at first calculate n by using the equation for Bohr’s radius,

Thus frequency of the emitted photon will be,

= 2.54 × 10³ Hz 

= 2.54 kHz.