The Photon, the Quantum of Light — Question 7

Question 7

An ideal two-level absorber selects an energy difference rather than a lower bound

An ideal isolated two-level system has level separation $\Delta E=2.00\,\mathrm{eV}$. Assume negligible recoil and linewidth, and no other levels, ionization or multiphoton channels. Use $hc=1240\,\mathrm{eV\,nm}$ and $1\,\mathrm{eV}=1.60\times10^{-19}\,\mathrm J$.

1. Find the resonant one-photon wavelength.

2. Decide whether a 310 nm photon can drive this transition and quantify its energy mismatch.

Question 7 - Solution

An ideal two-level absorber selects an energy difference rather than a lower bound

(a) Method and calculation

$$E_\gamma=\Delta E\quad\Rightarrow\quad\boxed{\lambda_{\mathrm{res}}=hc/\Delta E=620\,\mathrm{nm}}.$$$$E_{310}=1240/310=4.00\,\mathrm{eV}.$$

(b) Interpretation and further calculation

$$\boxed{E_{310}-\Delta E=2.00\,\mathrm{eV}\ne0}.$$$$\boxed{\text{No allowed one-photon absorption into the specified upper level}}.$$

Physical check

A discrete transition requires energy matching under these assumptions. Having more photon energy is not by itself sufficient.