I've decided to work through the assignments on https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/ where they have problems and posted solutions. Of course I am not enrolled in any course, just doing this for my own edification.

I am looking at:

I started off with trying to compute the energy lost per revolution and started thusly:

\(\displaystyle \frac{m_e v^2}{r} = \frac{k}{r^2} = m_e a\: \textrm{where} \: k=\frac{1}{4 \pi \epsilon_0} \: \textrm{and} \: m_e \: \textrm{is the mass of the electron}\)

If I solve for "a" (as a function of r) and substitute into the above differential equation I am no better off as r is not independent of time.

Initially I thought I could use conservation of angular momentum to my advantage only to finally realise that probably doesn't hold due to the radiation of energy.

What am I missing for the second step? I kind of figure I must be missing something obvious. Is there some other law I can substitute in there to make this a straightforward differential equation I can solve by separation of variables?

I am looking at:

At the moment I am working on (a). Now I'm not quite sure how to proceed.Suppose the world was actually governed by classical mechanics. In such a classical universe, we might try to build a Hydrogen atom by placing an electron in a circular orbit around a proton. However, we know from 8.03 that a non-relativistic, accelerating electric charge radiates energy at a rate given by the Larmor formula,

\(\displaystyle \frac{dE}{dt} = -\frac{2}{3}\frac{q^2 a^2}{c^3} \)

(in cgs units) where q is the electric charge and a is the magnitude of the acceleration. So the classical atom has a stability problem. How big is this effect?

(a) Show that the energy lost per revolution is small compared to the electron’s kinetic energy. Hence, it is an excellent approximation to regard the orbit as circular at any instant, even though the electron eventually spirals into the proton.

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(b) Using the typical size of an atom (1A) and a nucleus (1 fm), calculate how long it would take for the electron to spiral into the proton.

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(c)

Compare the velocity of the electron (assuming an orbital radius of 0.5 A) to the speed of light – will relativistic corrections materially alter your conclusions?

(d)

As the electron approaches the proton, what happens to its energy? Is there a minimum value of the energy the electron can have?

I started off with trying to compute the energy lost per revolution and started thusly:

\(\displaystyle \frac{m_e v^2}{r} = \frac{k}{r^2} = m_e a\: \textrm{where} \: k=\frac{1}{4 \pi \epsilon_0} \: \textrm{and} \: m_e \: \textrm{is the mass of the electron}\)

If I solve for "a" (as a function of r) and substitute into the above differential equation I am no better off as r is not independent of time.

Initially I thought I could use conservation of angular momentum to my advantage only to finally realise that probably doesn't hold due to the radiation of energy.

What am I missing for the second step? I kind of figure I must be missing something obvious. Is there some other law I can substitute in there to make this a straightforward differential equation I can solve by separation of variables?

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