I understand that if I have the Hamiltonian I solve it and I have the allowed energy values. The result of a measurement has to be one of them, true? Thus, where line broadening fits into the theory? Also, why the fundamental state is the only one without broadening? I ask for the 'theoretical' reason, not the usual simple explanations...
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There are many reasons why experimental measurements differ from the predictions of idealized models. Judging by the context of the question, you are asking about the broadening due to decay of the states, i.e. due to the relaxation to the lower energy states. Such relaxation happens due to interaction with the environment, which can be accounted via introducing imaginary components of energies: $$E_n \rightarrow E_n -\frac{i}{2}\Gamma,$$ where $\Gamma$ is the broadening (also called the level width). The observed density of states or the absorption line is then given by $$\rho(\hbar\omega)-\frac{1}{\pi}\Im\left[\frac{1}{\hbar\omega - E_n + \frac{i}{2}\Gamma}\right] = -\frac{\Gamma}{2\pi}\frac{1}{(\hbar\omega - E_n)^2 + \frac{\Gamma^2}{4}}.$$
The imaginary energies is a price to pay for considering only the system Hamiltonian, instead of the full Hamiltonian 'system + environment'. Since the ground state has the lowest energy, there is no decay from this state and it has zero width.
Roger V.
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