The Bohr Model of the Hydrogen Atom — Question 7

Question 7

A subthreshold photon is absorbed resonantly only if it matches a discrete gap

An ideal hydrogen atom begins in its ground state. Compare photon energies $12.75\,\mathrm{eV}$ and $12.00\,\mathrm{eV}$. Assume negligible linewidth, recoil and multiphoton processes. Use the nonrelativistic Bohr model with an infinitely heavy nucleus unless stated otherwise: $E_n=-13.6Z^2/n^2\,\mathrm{eV}$, $r_n=a_0n^2/Z$, $a_0=.0529\,\mathrm{nm}$, and $hc=1240\,\mathrm{eV\,nm}$. Here Z is the nuclear charge number.

1. Find the upper principal quantum number resonant with the first photon.

2. Test whether the second photon gives a bound-state resonance or ionization.

Question 7 - Solution

A subthreshold photon is absorbed resonantly only if it matches a discrete gap

(a) Method and calculation

$$12.75=13.6(1-1/n^2)\Rightarrow1/n^2=.850/13.6=1/16.$$$$\boxed{n=4}.$$

(b) Interpretation and further calculation

$$12.00=13.6(1-1/n^2)\Rightarrow\boxed{n^2=8.50\text{, not an integer square}}.$$$$12.00<13.6\Rightarrow\boxed{\text{neither an ideal bound-state resonance nor ionization}}.$$

Physical check

This conclusion is about the stated ideal absorption model. Real linewidths, collisions and other interactions can change the response away from a discrete resonance.