# Bohr's model: Spectrum of hydrogen-like species

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?

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.