Can anyone give an mathematical elaboration of the following statement:     Quantization is a functor carrying the category of Hilbert space and linear maps to that of Symplectic manifolds satisfying some conditions specified by laws of Physics.
Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before.  How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try?  If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious.  Why should this be the case, or at least, why might we expect this to be the case?   Then there is the issue of measurement which is even more difficult to motivate.  In the usual formulation of quantum mechanics, we assume that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$).  How would you convince your students that this had to be the case?  I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be.  So, what do you guys think?  Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results?  Note that, at some point, you will have to make reference to experiment.  After all, this is the reason why we needed to develop quantum mechanics.  In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying.  I think we can do better.  Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory.  You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment.  Edit:  To clarify, I will try to make use of a much simpler example.  In an ideal gas, if you fix the volume, then the temperature is proportional to pressure.  So we may ask "Why?".  You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.".  If you choose the latter, you can then ask why that is true.  Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example).  The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment.  And in this particular case, the objective is to do this with quantum mechanics.
"Not with a whimper, but with a roar of triumph!" Looking for the title of an old sci-fi book, pretty sure it is a Baen or Del Rey imprint. Basically, mankind has reached the stars and expanded throughout the galaxy. Suddenly, an unknown race appears and begins a war of extermination against man. Finally, the galaxy is desolate, and one man remains. He is in a starship with an AI and becomes injured in the battle.  The ship's AI takes over and speeds out into the unknown, followed by the last remaining member of the unknown alien race. As all speeds are at tremendous multiples of the speed of light, the man cannot get away, but the alien cannot catch up.  Finally, communication is opened, at which point the man asks "why?" The alien tells him that man had discovered immortality, but that it required the death of a member of a race known as the Flower People for every human who achieved immortality. The alien says that Man would have found the answer eventually, but that genocide was so intolerable that the aliens came from another dimension, to which they could never return, to stop Man, which resulted in the war of extermination.  Finally, the light speeds become so great that the two remaining living beings in the universe approach the Big Crunch. As the Man begins to dive into the singularity, with the alien hot on his heels, he says that maybe there will be room in the next universe for both races to live in peace, and the final line is "not with a whimper, but with a roar of triumph"! ZIP Code (US Postal Code) validation I thought people would be working on little code projects together, but I don't see them, so here's an easy one:  Code that validates a valid US Zip Code. I know there are ZIP code databases out there, but there are still uses, like web pages, quick validation, and also the fact that zip codes keep getting issued, so you might want to use weak validation.  I wrote a little bit about zip codes in a side project on my wiki/blog:    There is also a new, weird type of zip code.     I can do the javascript code, but it would be interesting to see how many languages we can get here. How do I center a container in my HTML/CSS? Does Python SciPy need BLAS? Meaning of syntax $N(\mathbf{y} \mid \mathbf{0}, \mathbf{K})$ (multivariate normal distribution) I'm trying to deactivate/remove/uninstall a default Client Side Web-Part (the Yammer one) in modern pages and I don't find a way to do it:    I would like the Yammer option to be not visible in the menu. Any ideas?  Thank you very much in advance! I'm trying to find an anime movie I watched when I was younger; I believe I first watched it in 90/91. The story revolved around the recruitment and training of a group that would pilot a mecha that was similar to Voltron in that each pilot controlled a portion of the mecha that would merge into a bigger mecha. Points that I remember:   Started on Earth Program was government funded One of the pilots was related to the creator of the program, I think a female Mecha had chain weapon and a sword that was summoned from it's superhero-like chest icon. Killing blow was an m or w shape Ended with father/creator of mecha? being kidnapped by the aliens and used as human shield to deter attack on alien vessel An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? Where $(50)_5 = 50! / 45! $ and $50\choose 5$ $= 50!/(5!45!)$   I've narrowed it down to these two possibilities (though even these may be wrong), but I think I'm confusing myself as to which one I use. Can anyone suggest a good way of thinking of this problem that will make it apparent whether I need use use the falling factorial or the binary coefficient?   Edit: thank you for all the answers! Indeed, I meant $50!/45$ as the falling factorial. Also, the question doesn't specify whether it was meant to be ordered or unordered, but I think I understand my confusion now! The "Very Low Quality" (VLQ) flag reason is really annoying me.   It is the analog to the mommy mommy flag (which my 1.5-year-old child raises quite frequently). It provides no real clue as to what the flagger really wants us to do with the post.   Does she want us to close the question?  Does she want us to edit it into shape?  Does she want us to delete the question?  Does she want us to ban the user?  Does she want us to jump around and have a party?    Should we get rid of this flag reason?   Is it just me? Should and can we replace it with something better?     EDIT I accepted Jeff's answer. The tougher wording has heavily decreased the amount of very low quality flags. We now see only about 25% of the amount we used to and they are usually valid. Additionally, we now only allow very low quality on posts with a score of 0 or lower. Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$ Let $\phi: R \to S$ be a surjective homomorphism. Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$ for an ideal $J$ of $R.$  Obviously, $S \cong R/ \ker \phi$(first isomorphism theorem) and $\phi(J)$ is an ideal of $S$, since $\phi$ is surjective( $\forall s \in \phi(J), s = \phi(j)$ for some $j \in J, \forall s' \in S, s' = \phi(r)$ for some $r \in R$, so $ss' = \phi(j)\phi(r) = \phi(jr) \in \phi(J)$, and $ss' = \phi(r)\phi(j) = \phi(rj) \in \phi(J)$ ).  However, I'm having trouble with going on. If it's important, I mean a unitary ring by a ring. what is the difference between "yet" and "still"?   When we can use "still"? and when we can use "yet"?   Are they synonyms or not?  "The plan could yet succeed." Can I use "still" in that sentence instead of "yet"?  "I still don't understand." Can I use "yet" in that sentence too? This question exists as it fills a specific criterion.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will likely be closed per the . More information on the .   What kind of desktop environments or shells are there for Ubuntu users to install?  Please list a desktop environment or shell for each:   a description on why you like or suggest it (features, performance, etc.). a good screenshot, preferably of it running on Ubuntu and showing off some of its features. the minimal requirement for it to be usable, If there's any setting to lower it's requirement (Like kde's low-fat profile); how to enable the said setting some instructions on how to set it up if in the repositories, please provide a software centre link If I have a shapefile or geodatabase featureclass that has fields that I want to permanently reorder, is there a tool, script, or utility that will help me quickly accomplish this task? Generally I know you have to create a new field, then populate it with the old values, then delete the old field.  I am hoping there is something already created out there that will help with this. Assuming you don't have FME or another ETL tool available, what is the next best suggestion? This is NOT a view level change (ie- drag/drop the fields in order), but rather write out to a new shapefile or featureclass. I'm using custom post types, but they don't appear on tag pages along with regular posts and i'd like to include both standard and custom post types on tag or category pages. What would be required before the loop to include all posts? (including custom type). Updated: To add, I found this code , which worked perfectly for the first part: add_filter('pre_get_posts', 'query_post_type'); function query_post_type($query) {   if ( is_category() || is_tag() &amp;&amp; empty( $query-&gt;query_vars['suppress_filters'] ) ) {     $post_type = get_query_var('post_type');     if($post_type)         $post_type = $post_type;     else         $post_type = array('post','cpt'); // replace cpt to your custom post type     $query-&gt;set('post_type',$post_type);     return $query;     } }  Second question still remains though: Is there a way to allow readers to filter by post type? So either of the following would work..  example.com/tag/blue?post_type=custom example.com/tag/blue?post_type=post Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence? $$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$ Where $e$ is Euler's number.  Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus teacher told us that it diverges but he doesn't see any elementary way or simple test to prove it. He told us something about Euler's Gamma function and said that he knows it diverges because of Stirling's formula but this is not quite our level yet.  I tried to think about it myself but I established only that this  is some kind of an edge case. We could generalize the problem to series of $\frac{n!a^n}{n^n}$. Then, from the ratio test we know that it converges for $a &lt;e$ and diverges for $a&gt;e$. For $a=e$ the test is inconclusive. I've also tried to compare it with various divergent series, but nothing worked...  Does anybody have any idea for a simple solution? Manage SSH Users with MySQL backend What does "Could not find or load main class" mean? I created a custom module in drupal 8. And I am trying to add js and css file in  my_module.libraries.yml file using this   lrv_key:   version: 1.x   js:     js/lrv.js: {}        dependencies:       - core/jquery       - core/drupalSettings   css:     theme:       css/lrv.css: {}   But it doesnt work. Excessive water when cooking scallops This is partly a continuation of  question, since that one was already closed, but my question is about a specific problem encountered the last time I cooked scallops.  I got the pan up to a reasonable temperature and put the scallops in, and as they were cooking, TONS of water came out of the scallops and filled the pan.  I poured it out, continued cooking, and more water came out.  Several iterations later, I thought they were done, but they were still very raw inside, so we ended up tossing the batch (it definitely wasn't sushi grade).  Where did all the liquid come from, and is there any way to avoid that the next time? Word for Party *Steward*? Is there a word that describes a person who is usually not too drunk and takes care of the rest of the folks at a party?  Typically, they're the ones who drag the sloshed ones back to beds, clean up the house and generally maintain a bit of sanity.