The Bohr Model of the Hydrogen Atom — Question 10

Question 10

Classical radiation would destabilize an orbit without the stationary-state postulate

In a hypothetical slowly shrinking circular Coulomb orbit, let $E(r)=-\mathcal K/(2r)$ with $\mathcal K=k_ee^2$. Suppose the radiated power is $P(r)=P_0(r_0/r)^4$, starting at radius $r_0$. Use $dE/dt=-P(r)$ and treat this classical model formally down to zero radius.

1. Derive the radial contraction law.

2. Integrate to obtain the formal collapse time and express it using initial binding $B_0=\mathcal K/(2r_0)$.

Question 10 - Solution

Classical radiation would destabilize an orbit without the stationary-state postulate

(a) Method and calculation

$$\frac{dE}{dr}=\frac{\mathcal K}{2r^2}\Rightarrow\boxed{\frac{dr}{dt}=-\frac{2P_0r_0^4}{\mathcal K r^2}}.$$$$r^2dr=-\frac{2P_0r_0^4}{\mathcal K}dt\Rightarrow\frac{r^3-r_0^3}{3}=-\frac{2P_0r_0^4}{\mathcal K}t.$$

(b) Interpretation and further calculation

$$r=0:\quad\boxed{t_{\mathrm{collapse}}=\frac{\mathcal K}{6P_0r_0}}.$$$$\boxed{t_{\mathrm{collapse}}=B_0/(3P_0)}.$$

Physical check

The classical approximation fails before zero radius, but its loss of stability motivates the need for quantum stationary states. Bohr postulated nonradiating stationary orbits rather than deriving them from classical radiation theory.