From Wikipedia article :  Spectral lines are the result of interaction between a quantum system (usually atoms, but sometimes molecules or atomic nuclei) and a single photon. When a photon has about the right amount of energy (which is connected to its frequency) to allow a change in the energy state of the system (in the case of an atom this is usually an electron changing orbitals), the photon is absorbed. Then it will be spontaneously re-emitted, either in the same frequency as the original or in a cascade, where the sum of the energies of the photons emitted will be equal to the energy of the one absorbed (assuming the system returns to its original state).   A spectral line extends over a range of frequencies, not a single frequency (i.e., it has a nonzero linewidth).   The lifetime of excited states results in natural broadening, also known as lifetime broadening. The uncertainty principle relates the lifetime of an excited state (...) with the uncertainty of its energy.  Can you provide proof for the last statement? (Maybe without using QED, if possible)
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Consider the atomic spectrum (absorption) of hydrogen.     The Bohr's model postulates that there are only certain fixed orbits allowed in the atom. An atom will only be excited to a higher orbit, if it is supplied with light that precisely matches the difference in energies between the two orbits.   But how precise does 'precisely' mean. Of course, if we need energy $E$ to excite the electron to a higher energy level, and I supply a photon with just $E/2$ I would expect nothing to happen (since the electron cannot occupy an orbit between the allowed ones). But what if I supplied a photon with energy $0.99E$, or $1.0001E$ or some such number. What will happen then?  I think that the electron should still undergo excitation precisely because the lines we observe in the line spectrum have some thickness. Which means that for a given transition, the atom absorbs frequencies in a certain range.   Is my reasoning correct? If not, why? How does Bohr's model explain this? How about modern theory?   If I'm right, what is the range of values that an atom can 'accept' for a given transition?
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Yesterday I read that we can affect the path and the 'form' (particle or wave) of a photon after the fact ().  Part of what is puzzling me is the beam-splitter. Are the individual photons actually being split into two new photons of lesser energy?  This  implies that you cannot split a photon but it seems that beam splitters do exactly that.
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These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by the author, who merely states that the non-normalizability is linked to the fact that such eigenfunctions do not tend to zero at infinity.  Not a very satisfying answer. What I'm really after is an explanation based in functional analysis. I believe there is a generalized result about inner products being finite for discrete spectra but infinite for continuous spectra.  Can anyone shed some light on this?
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When an atom “lases” it always gives up its energy in the same direction and phase as the incoming light. Why does this happen? How can this be explained? How does the photon generated because of stimulated emission, know which direction to take? What are the factors leading to this?
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I read that stars show both absorption and emission lines. While how we see absorption lines is clear to me, I don't understand how (and when) we see the emission lines. Based on what I read in several of your threads such as  we should not be seeing emission lines at all, since all the emissions would come from the hotter interior but those can't reach the outside (unless I misunderstood this point), and the only emission we see is blackbody radiation. But then how do we see the few emission lines such as in the spectrum shown ?
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Let's say we have a emitter, emitting light that has frequency f, less than the threshold frequency of a metal.  If you leave light shining onto that metal, for long enough, does the energy of the individual photons accumulate, on the electrons, so eventually they will ionize, or does this not happen? What am I missing?
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Why is it that the Lyman series for hydrogen atom is available for both absorption and emission, but the Balmer series is only available for emission?   The book I'm reading doesn't contain any detail about such a discrepancy. Our professor didn't teach this either. I looked  this up on the internet but I did not come across any valid answer.   Any help would be appreciated.
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Suppose I have an atom in its ground state $|g⟩$, and it has an excited state $|e⟩$ sitting at an energy $E_a=\hbar\omega_0$ above it. To excite the atom, one generally uses a photon of frequency $\omega$ equal (or sufficiently close to) the transition frequency $\omega_0$, and this will stimulate a transition.  One thing that is often left by the roadside,* though, is the fact that the incoming photon has momentum as well as energy, and that if the atom wants to swallow the energy it also needs to swallow the momentum. So, in the nuts and bolts of states and operators,     how does one describe the transfer of momentum during an atomic transition?   In addition to this, the fact that this recoil momentum is rarely mentioned is a good indication that it is also rarely an issue. Why is it that in most circumstances we can safely ignore the photon's momentum when describing electronic transitions?  *Apart from treatments of Doppler cooling, which simply take the momentum transfers for granted and do not explain how and why they happen.
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The way I learned it from practicing Fourier analysis and signal processing besides quantum mechanics, is that Energy conservation cannot be achieved in short time scales, and that limits energy conservation in Quantum mechanics.  In other words: Energy conservation is limited by the Heisenberg uncertainty principle in our universe.  $$\Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$  As I posted this answer somewhere, someone said this is wrong. So I posted this issue here, for maybe I don't understand it properly and you guys could tell me why this is wrong.  Example:  A beta decay produces a W-boson, which is ~85 times larger in mass than the initial particles (which decays eventually to neutrino and some beta particle), and this is possible due to the very short time scale, during which the W-boson is produced. I.e, energy is not conserved in very short time scales.  Do I understand this correctly? Thank you.
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My textbook and the following extract from feynman's lectures present the same idea regarding wavetrains and uncertainty in their wavelengths. Why is it that a wavetrain confined to some space has an uncertainty in its wavelength or the wave number? Is not a confined wave-train equivalent to a burst of successive pulses which can have a definite wavelength according to their origin or nature of origin.   Next, i understand that the De Broglie relation relates the uncertainty in wavelength to the uncertainty in momentum but what links the finiteness of the wave-train to the uncertainty in position?
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When an electron absorbs energy and jumps to the another excited state by absorbing the photon and why it is always said that the electron will come back to $\hbar\omega/2$?  Why doesn't the electron sit in the excited state ?   What is the actual reason for it to come back to ground state?
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For a system has both continuous and discrete spectrum, is it possible that a physical states is something like:  $$\psi(x)=\sum_{n=n_1}^{n=n2}c_n\psi_n(x)+\int_{E_0-\Delta_E}^{E_0+\Delta_E}\psi_E(x)\mathrm{d}E$$  I am asking this because of some comments under my  to this Phys.SE question:
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If you emit a single photon, wait an interval, emit a second photon, wait the same interval, and keep repeating, is the resultant stream of photons describable by the frequency at which you emitted each one? Or does each photon have its own "frequency", independent of the rate of emission? I thought the former description explained the Doppler effect.
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The question pretty much sums up what I need to know. Why is it that light only gets dispersed into a spectrum when travelling through two non-parallel sides(like a prism) and not through something like a glass slide with parallel slides.  And how does this experiment(an experiment conducted by Newton) explain this? Because this is what is in my school textbook.
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I'm aware that there're some questions posted here with respect to this subject on this site, but I still want to make sure, is frequency quantized? Do very fine discontinuities exist in a continuous spectrum like the black body spectrum?
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Are photons , quantum waves, or both?  If I subdivide an electromagnetic field into smaller , should I eventually find an electromagnetic wave of a ?   How can individual quantum waves combine to form the macroscopic observable of an electromagnetic field?
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Is it theoretically possible to have a photon of any energy/wavelength? I have a vague memory in my mind of something like there being a minimum energy level for a photon and then the possible energy levels jump up in discrete but ever decreasing amounts, so that near e.g. the visible spectrum photon energies are for practical purposes continuous.