The Bohr Model of the Hydrogen Atom — Question 8

Question 8

The high-index photon frequency approaches the classical orbital frequency

For hydrogen write $E_n=-B/n^2$. In the Bohr model the orbital frequency is $f_{{\mathrm{orb}},n}=2B/(hn^3)$. Consider a photon emitted in the transition $n+1\to n$.

1. Derive the ratio of photon frequency to orbital frequency at the lower orbit.

2. Find the large-n limit and evaluate the ratio at $n=20$.

Question 8 - Solution

The high-index photon frequency approaches the classical orbital frequency

(a) Method and calculation

$$f_\gamma=\frac Bh\left(\frac1{n^2}-\frac1{(n+1)^2}\right)=\frac Bh\frac{2n+1}{n^2(n+1)^2}.$$$$\boxed{\frac{f_\gamma}{f_{{\mathrm{orb}},n}}=\frac{n(2n+1)}{2(n+1)^2}}.$$

(b) Interpretation and further calculation

$$n\to\infty:\quad\boxed{f_\gamma/f_{{\mathrm{orb}},n}\to1}.$$$$n=20:\quad\boxed{f_\gamma/f_{{\mathrm{orb}},20}=820/882=.929705}.$$

Physical check

Agreement is asymptotic. At low quantum number, the emitted frequency is an energy-difference frequency and need not equal the orbital revolution frequency.