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Suppose some quantum system has two allowed energy levels $E_1$ and $E_2$. The system is initially in the lower energy state $E_1$. Then it absorbs energy from outside and gets excited to level $E_2$. What is the time lag for the system to return to previous energy state? What factors does it depend upon? Does of depend on $E_2$ - $E_1$? Is it even necessary that it returns to the original state?

This leads to a specific confusion regarding spectrums of hydrogen-like atoms. If the electrons gets excited by absorbing a specific wavelength, the emitted light is missing this wavelength (absorption spectrum). What if it returns to original state even before we can measure the emitted spectrum? No wavelength would be missing. Energy would neither have been absorbed nor released. Does this happen?

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For any transition where an atom is allowed to absorb radiation, it is also allowed to emit via spontaneous emission. The details of that re-emission are complicated and they require a full quantum mechanical treatment of the atom as well as the radiation field, along the lines drawn in this Wikipedia page. The re-emission may or may not include radiation on the absorbed frequency - it depends on the situation.

The important thing about spontaneous emission, though, is that it is completely isotropic: the emission is spread evenly over the full $4\pi$ of solid angle. This means that, even if all of the absorbed energy is pumped right back into light of the same frequency, you will still see a strong absorption peak from a collimated source, because only a small fraction of the re-emission will go along the same direction as the original beam of light.

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  • $\begingroup$ Need a more elaborate answer. Up voting anyways. $\endgroup$ – PhyEnthusiast Apr 11 '17 at 14:03
  • $\begingroup$ Is the probability for emission the same as that of absorption? $\endgroup$ – PhyEnthusiast Apr 11 '17 at 14:08
  • $\begingroup$ If you need a more elaborate answer, then you shouldn't pack two completely separate questions into the same post. If you edit the question into something more focused, with the other part in a separate question, then I can elaborate further (if I find the time). On a separate tack, if you're going to ask about probabilities of emission, you should distinguish between spontaneous and stimulated emission. $\endgroup$ – Emilio Pisanty Apr 11 '17 at 14:58
  • $\begingroup$ I thought the questions were connected since one led me to another. I don't know about spontaneous and stipulated emission. $\endgroup$ – PhyEnthusiast Apr 11 '17 at 15:01
  • $\begingroup$ If you have yet to learn the difference between those two, you should give serious thought to the possibility that "a more elaborate answer" will involve a good deal of material that you have yet to see; without that material, the answer is simply "it depends". In any case, the current post as phrased is too broad; "one led me to the other" is irrelevant. So: if you want more details, ask a more focused question. $\endgroup$ – Emilio Pisanty Apr 11 '17 at 15:58

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