Discussion
Match the physical effects and order of magnitude of their energy scales given below, where \(\alpha = \frac{e^{2}}{4\pi\epsilon_{0}\hbar c}\) is fine structure constant; \(m_{e}\) and \(m_{p}\) are electron and proton mass, respectively.
| Group I | Group II |
|---|---|
| P: Lamb shift | 1: \(\sim O(\alpha^{2}m_{e}c^{2})\) |
| Q: Fine structure | 2: \(\sim O(\alpha^{4}m_{e}c^{2})\) |
| R: Bohr energy | 3: \(\sim O(\alpha^{4}m_{e}^{2}c^{2}/m_{p})\) |
| S: Hyperfine structure | 4: \(\sim O(\alpha^{5}m_{e}c^{2})\) |
Answer: C
The correct matching is P-4, Q-2, R-1, S-3.
- The Bohr energy, \(\tfrac{1}{2}\alpha^{2}m_{e}c^{2} = 13.6\) eV, is of order \(\alpha^{2}m_{e}c^{2}\) (R-1).
- Fine structure, from relativistic and spin-orbit corrections, is smaller by a further \(\alpha^{2}\): order \(\alpha^{4}m_{e}c^{2}\) (Q-2).
- The Lamb shift, a quantum electrodynamics correction, is smaller again by about \(\alpha\): order \(\alpha^{5}m_{e}c^{2}\) (P-4).
- Hyperfine structure comes from the nuclear magnetic moment, which is smaller than the electron’s by \(m_{e}/m_{p}\): order \(\alpha^{4}m_{e}^{2}c^{2}/m_{p}\) (S-3).
The energy scales therefore fall in the order Bohr, fine structure, Lamb shift, hyperfine. Options giving the Bohr energy \(\alpha^{4}\) or putting fine structure at \(\alpha^{2}\) reverse the hierarchy.
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