Why doesn't position and momentum commute? I'm trying to understand the reason for Heisenberg's uncertainty principle which I read is a "fact of nature" rather than an experimental limitation. I found this  in which the accepted answer talks about quantum objects as being wavepackets which seems to satisfy the criteria for HUP and how position and momentum can be related by Fourier transforms. However we can reach this result without even considering quantum objects as wave-packets from commutator relations between position and momentum. I'm aware of how to calculate the commutator of position and momentum from their definition in Schrodinger's equation. What I am seeking is the reason why they can't commute. The introductory textbooks (Griffith's: Introduction to Quantum mechanics and Arthur Beiser's: Modern physics) I've read so far don't seem to mention the reason why they don't commute and take it as an axiom. I would like some justification for it.
Why quantum mechanics? 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.
Why cannot fermions have non-zero vacuum expectation value? In quantum field theory, scalar can take non-zero vacuum expectation value (vev). And this way they break symmetry of the Lagrangian. Now my question is what will happen if the fermions in the theory take non-zero vacuum expectation value? What forbids fermions to take vevs? Add a new repository to software sources When transfering data using TLS the browser and server agree the cipher suite to be used - so for example this could be chosen as AES-128 and is (probably) outside of my control. If I separately encrypt a file using AES-256 and then Alice downloads this via her browser (using AES 128) have I still effectively got AES-256 security?  My reasoning goes like this: even if the AES-128 encryption on the TLS link was broken by Eve then all she would get is the AES-256 encrypted file and she would then have to break that encryption to see the plaintext.  Of course the AES-256 key is sent to Alice over a separate channel. How to know if Google Sheets IMPORTDATA, IMPORTFEED, IMPORTHTML or IMPORTXML functions are able to get data from a resource hosted on a website? I want to know how can I achieve this effect in LaTeX:    Basically it looks like this: {F}IRST {L}ETTER, where the letters in the braces are a little bigger than the other ones but they all the letters are uppercase.  Any suggestions? What are the odd of a single coin toss after many consecutive ones? I have little knowledge of probability and have recently been thinking of a math problem I am sure you guys could answer  If for example I tossed a coin ten times and got all heads, would the odds of the next coin toss beeing heads be 50 50 or s very low as the odds of getting 11 in a row is tiny?   Thanks Resize multiple layers around their respective anchor points I have 50 or so layers (imported images as smart objects) that I would like to scale down, while they stay in place. Is there a way of doing this? I know After Effects has, is there a similar Scale property in photoshop? The sup norm on $C[0,1]$ is not equivalent to another one, induced by some inner product Let $\mathrm{C}[0,1]$ be the space of continuous functions $[0,1]\rightarrow \mathbb{R}$ endowed with the norm $||x||_{\infty}=\mathrm{max}_{t\in [0,1]}|x(t)|$. It is easy to verify that this norm is not induced by any inner product (really the parallelogram law fails for $x(t)=t$ and $y(t)=1$). Well, how to understand that this norm is not equivalent to anyone induced by an inner product? So, the norms induced by inner products should have some special properties... Possibility of recovering files from a dd zero-filled hard disk I have "zero filled" (complete wiped) an external hard disk using dd, and from what I have heard: people said you should at least "zero fill" 3 times to be sure that the data are really wiped and no one can recover anything.  So I decided to scan the disk once again after I've zero filled the disk. I was expecting the disk to still have some random binary left. It turned out that it has only a few sequential bytes in the very beginning. This is probably the file structure type and other headers stuff. Other than that, it's all zeros and nothing else.  So if we have to recover any file from a zero filled disk, ...how? From what I've heard, even you zero fill the disk, you should still have some data left. ...or could dd really completely annihilate all data? How vibration sensitive are hard disk drives? I've gotten my hands on a seriously cheap laptop grade 2.5 inch drive. I'm intending to use one on a desktop that doesn't have 2.5 inch bays as a boot drive. While I know there are adapters available, I know its fairly common with SSDs to simply velcro/tape the drive somewhere. I'm wondering if I can do the same with a laptop hard drive.  One issue I see here is SSDs are fairly insensitive to vibration, while hard drives are. Just how vibration sensitive are drives? Would a jerry rigged mount need to take this into account? Rationalize   $\dfrac{1}{\sqrt[3]{p^2}+\sqrt[3]{pq}+\sqrt[3]{q^2}}.$  How would I go about doing this without wading through lots of algebra?  Is there a trick similar to how you can multiply by $\dfrac{\sqrt a-\sqrt b}{\sqrt a-\sqrt b}$ with square roots?  Thanks in advance! Okay setting the scene. I have three tables, (Table1, Table2 and DataTable) and I want to insert into Table1 and Table2 using DataTable as source. So for every row in DataTable I want a row in Table1 and Table2, and Table2 needs to have the inserted id (PK) from Table1...  If I were to do this...  INSERT INTO Table1 SELECT A, B, C FROM MyTable INSERT INTO Table2 SELECT IDENTITY_INSERT(), D, E, F FROM MyTable   I'd get the ID of the last inserted record into Table1.  Is a CURSOR or WHILE loop the only ways to do this? Is there a database of Linux compatible hardware? I want to know if there is a centralised community site where all Hardware is listed in terms of compatability with Linux. Something like AppDB for Wine. Expected time to roll all 1 through 6 on a die What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die?  The order in which the numbers appear does not matter.  I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation.  We were given the answer $(1-(\frac56)^n)^6 = .5$ or $n = 12.152$  Can someone please explain this to me, possibly with a link to a general topic? I just noticed a weird thing in my reputation tab on StackOverflow.    That last question mentioned (but not linked, had to find it via search):   I must have amnesia as I have no memory of ever interacting with this post. I don't even know what xampp and pdftk are so I don't know why I would even be interested in such a question. 2 reputation points more or less is not a big deal, but the fact I got rep change from question I've never seen before is odd and I'm wondering what happened. Unknown Additional Driver: Processor microcode firmware for Intel CPUs for intel-microcode After upgrading to 15.04, I have one additional driver (it did not appear in 14.10). What is the purpose of this driver and should I enable it? How bias arises in using maximum likelihood to determine the variance of a Gaussian? While reading , I ran into the following explanation for why there is an error in estimating the variance of Gaussian distribution using Maximum Likelihood. On ,      Illustration of how bias arises in using maximum likelihood to   determine the variance of a Gaussian. The green curve shows the true   Gaussian distribution from which data is generated, and the three red   curves show the Gaussian distributions obtained by fitting to three   data sets, each consisting of two data points shown in blue, using the   maximum likelihood results (1.55) and (1.56). Averaged across the   three data sets, the mean is correct, but the variance is   systematically under-estimated because it is measured relative to the   sample mean and not relative to the true mean.   I'm pasting figure 1.15 and the required equations (1.55, 1.56) here for convenience.  $\mu_{ML} = \frac{1}{N}\sum_{n=1}^{N}x_n$   $\sigma_{ML}^2 = \frac{1}{N}\sum_{n=1}^{N}(x_n - \mu_{ML})^2$    I can imagine how maximum likelihood estimation would look pictorially. However, I'm not able to understand the figure or the explanation. I would appreciate if someone could label the figure or explain what points are considered in each part of the figure and why the curve could look like that in each case.  I looked up , and I'm able to understand why there is a factor of $\frac{N-1}{N}$ between the true $\sigma^2$ and $\sigma_{MLE}^2$.  ${\rm I\!E}[\sigma_{MLE}^2]= (\frac{N-1}{N})\sigma^2 $ C non-blocking keyboard input What blueprints become available during story? As I progress, I realize that some blueprints open up throughout the story and don't have to be researched. Are these random or are they always the same and is there a list over those? Seems you can easily waste research point if you don't know that. How to use siunitx with non numerical values?