Conservation of energy (or lack thereof) in Doppler cooling I did not find my question answered elsewhere, so here it is.  I'm doing a project in my first optics course, and we are reading a bit about Doppler cooling. I understand that a laser is tuned to a frequency where the photon energy $E_p = h \nu$ corresponds to an energy slightly below the energy $E_{abs}$ required to excite the gas atoms, and that an atom moving towards the laser at a high speed (relative to rest of the gas) will experience a blueshift that allows the atom to absorb the photon, thus slowing down the atom.  I also understand that the photon is re-emitted in a random direction, which of course means that the atom's velocity in the direction of the laser will be decreased. What I'm not sure I understand is how the re-emission of the photon doesn't just bring the atom back to the same kinetic energy (but with a different direction of motion).  I have not been able to find a source explaining this, but here's my own attempt at an explanation (which I am not sure is correct, which is why I am asking here): The absorbed photon has an energy (in the frame of the laboratory) of $E_1 = h \nu_1$ which is slightly less than the absorption energy $E_{abs}$. But after the absorption the atom's speed is reduced, thus the photon will be emitted with a different frequency $\nu_2$ (in the frame of the laboratory). Since the speed of the atom (relative to the frame of the laboratory) has been reduced by the absorption, the energy $E_2$ of the emitted photon will be closer to $E_{abs}$, thus $|E_{abs} - E_2| &gt; |E_{abs} - E_1|$ (the last step of the explanation only works if the atom's speed perpendicular to the laser is not greater than the speed parallel to the laser for most atoms, but since the laser is tuned to only hit the fast moving atoms, this seems improbable enough to not invalidate the explanation).  Is this explanation correct or have I made mistake? I have taken a special relativity course but no quantum mechanics yet, so please keep the quantum mechanics to a minimum if possible.
When lasers are used to cool atoms or ions, etc where does the heat go? According to the first law of thermodynamics, sourced from wikipedia "In any process in an isolated system, the total energy remains the same."  So when lasers are used for cooling in traps, similar to the description here:  where is the heat transferred?    From what I gather of a cursory reading on traps, whether laser, magnetic, etc the general idea is to isolate the target, then transfer heat from it, thereby cooling it.  I don't understand how sending photons at a(n) atom(s) can cause that structure to shed energy, and this mechanism seems to be the key to these systems.
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How do I integrate the cardinal sine? $$ \int_{-\infty}^{\infty} \frac{\sin(a)}{a} \, {\rm d} a $$ The particle number on my settings appear in gray, I don't think i'm able to change the number even thought i need it, how can i make it white again? $n$-th digit in the sequence of natural numbers What's the $n$-th digit in the sequence $S$ of numbers formed by the natural numbers, i.e.,  $n$-th digit in the sequence 1 2 3 4 5 6 7 8 9 10 11 12... ? For example 11th digit in the sequence is 0. How can I get a complete list of fields for a standard object? Boot to console (standard method fails) I'm trying to get Ubuntu to boot directly to the console. I don't want to completely remove the GUI because I still kind of need it for training wheels sometimes, but I don't want to deal with it if I don't need it. In the past I have been able to achieve this pretty easily in previous versions by editing /etc/default/grub, but for some reason it's not working on 15.10. 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However, the only thing that seems to make a difference is the quiet boot. When I add and remove the hash from that line it affects the behavior when the machine boots, so I know grub is updating when I tell it to. But no matter what I do it still loads the GUI immediately. If you provide 0 as the dayValue in Date.setFullYear you get the last day of the previous month:  d = new Date(); d.setFullYear(2008, 11, 0); //  Sun Nov 30 2008   There is reference to this behaviour at . Is this a reliable cross-browser feature or should I look at alternative methods? Real Size Image Printing How to modify a Wiki Site Page  layout We are looking to see if we can leverage a WIKI to advertise sections of our governance plan to the end user community.   Right now with a WIKI we get the option to add web parts to the bottom of the layout.   The desired design would be to design a layout that would have a right column that would be a zone of other Web Parts.  For instance, a LVWP that would contain contacts and/or FAQS, an image web part etc.   We just want more flexibility with layout.  I am thinking we could add zones with Designer etc but wanted to see if there was something more obvious.  Thanks,  Paul How can I run a different command but with the same arguments? For example, I edit and then run the same source.   vim arbit.py python arbit.py   I know I could run it from within vim, but I want a general solution on Bash. I keep finding situations where I reuse arguments.  How can I do that? Illogical twoside margins Shouldn't the twoside option produce a wider margin at the binding/spine edge, rather than the other way around? I'm using \documentclass[twoside]{report}.