Why should an equation () in which first time derivative is related to second space derivative have a solution that contains $i$?The wave function is supposed to be complex, but I am unable to understand why it can also be assumed to be complex directly from my previous statement as stated in Quantum Physics by . Can anyone help me develop an intuition?
1.  Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an explanation that I need some help with: that the de Broglie wavelength and the wavelength of an elastic wave do not show similar properties under a Galilean transformation. He basically says that both are equivalent under a gauge transform and also, separately by Lorentz transforms. This, accompanied with the observation that $\psi$ is not observable, so there is no "reason for it being real". Can someone give me an intuitive prelude by what is a gauge transform and why does it give the same result as a Lorentz tranformation in a non-relativistic setting? And eventually how in this "grand scheme" the complex nature of the wave function becomes evident.. in a way that a dummy like me can understand.  2.   A wavefunction can be thought of as a scalar field (has a scalar value in every point ($r,t$) given by $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{C}$ and also as a ray in Hilbert space (a vector). How are these two perspectives the same (this is possibly something elementary that I am missing out, or getting confused by definitions and terminology, if that is the case I am desperate for help ;)   3.   One way I have thought about the above question is that the wave function can be equivalently written in $\psi:\mathbb{R^3}\times \mathbb{R}\rightarrow \mathbb{R}^2 $ i.e, Since a wave function is complex, the Schroedinger equation could in principle be written equivalently as coupled differential equations in two real functions which staisfy the Cauchy-Riemann conditions. ie, if $$\psi(x,t) = u(x,t) + i v(x,t)$$ and $u_x=v_t$ ; $u_t = -v_x$ and we get $$\hbar \partial_t u  = -\frac{\hbar^2}{2m} \partial_x^2v + V v$$ $$\hbar \partial_t v  = \frac{\hbar^2}{2m} \partial_x^2u - V u$$ (..in 1-D) If this is correct what are the interpretations of the $u,v$.. and why isn't it useful. (I am assuming that physical problems always have an analytic $\psi(r,t)$).
I recently changed my username to Undo. I was amazed when I pressed Submit and saw that it actually went through - surely someone somewhere was using this same username.  Out of curiosity, I then search the SO user list for Undo - and found  with Undo as thier name. They have 50 rep. I have around 1,500.   Is there some kind of pecking order, I.e. a high-rep user can 'steal' a low-rep user's name? Or is this a bug?  Or can two users have the same username without causing the SO servers to go down in a burning ring of fire? Need a matrix to seventh power which is the identity but the original is not or the negative identity The question on the homework wants a $2\times2$ matrix $R$, but $R$ not equal to positive or negative identity, that when you raise to seventh power you get the identity.  $R^7=\operatorname{Id}$.  I thought it would have to be some ones and negative ones,  it no luck, then tried some ones and some $.5$.  Just using an online matrix calculator to do brute force.  We just started talking about eigenvalues with visual example, so assuming wouldn't need that since we have not talked about how to find them yet.  Any hints or ideas to get me back on track? Can a monk make a bonus unarmed attack after using shocking grasp? I'm playing a 5e Wizard/ Monk build. My DM and I are questioning the viability of the cantrip Shocking Grasp. As a monk I can do another unarmed strike as a bonus action after I attack with a melee attack. Being a touch spell, can I punch (touch) with Shocking Grasp and then do my monk's bonus action? How can I know which radio button is selected via jQuery? Are $10\times 10$ matrices spanned by powers of a single matrix? I don't know how to answer this question:     Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the vector space of $10 \times 10$ matrices over $\mathbb{F}$?   I think there is a connection to Jordan normal form. Mixing C Structs and Objective-C Properties Electrodynamics textbooks that emphasize applications Please recommend undergraduate-level textbooks on electrodynamics which emphasize practical applications and real life examples. Please describe the book's level and contents and its intended audience in as much detail as possible.   Please provide both applications from everyday life for curious students as well as engineering applications from a physics perspective. What happens if I unsubscribe from Apple Music, then resubscribe at a later date? I want to make a JavaScript application that's not open source, and thus I wish to learn how to can obfuscate my JS code? Is this possible? I was about to add an extra signal handler to an app we have here and I noticed that the author had used sigaction() to set up the other signal handlers. I was going to use signal(). To follow convention I should use sigaction() but if I was writing from scratch, which should I choose? Left pedal coming loose On my bike, which I just built out of spare parts, the left pedal seems to constantly be coming loose. It seems to come out about 5m every time I ride ~10km.  It's an old cotter pin crank, and the pedal is a left pedal (there's an L stamped into the end of the screw that goes into the crank). The pedal has a crack in one side of the axle compartment, but I don't think that would affect it.  I thought that these pedals had left-hand threads so that the action of pedalling would work to tighten the screw? If so, why would this be happening, and is there anything I can do about it short of replacing the crank and pedal? Give moderators the ability to mark reviews as "bad" From what I've seen, many SO moderators trawl completed reviews and watch out for robo reviewers; and manual banning them when they see enough (>= 3?) bad reviews. This is inefficient because:   It doesn't integrate into the audit system. There's no "memory" to it. If mod A finds that user X has made two (any number less than the threshhold) bad reviews, he may make a mental note but there probably won't be any permanent/shared memory of this. If moderator B finds this user at a later date, he will have to look through the entire history, including stuff that mod A has already checked.    What I propose is that moderators get the ability to mark individual reviews as "invalid" from the completed review page. This is only to be used for reviews where any reviewer who is paying attention (and knows the rules) would choose the opposite reason (spam and other low quality stuff).   These bad review flags are treated like audits for a user -- if the number of failed audits and bad reviews is enough, the user gets autobanned. The nice thing is that this system should (probably) easily be able to , fixing the "Why was I manual banned?" problem.  This could also be integrated well with . I have noticed there is a Stack Overflow in language differs from English which is global and mostly used by programmers.  I think there is no need for a site in other languages. Otherwise we will end up with 127 StackOverflow sites. I am afraid that we will lose the amount of people sharing data. Given that the sequence $(v,Av,A^2v,...,A^nv)$ is linearly dependent (1), $A_{n \times n}$, over $\mathbb{F^n}$ prove that scalars $c_0,c_1,..,c_n$ exist (such that not all of them are zeroes) such that the matrix $M=c_0I+c_1A+..+c_nA^n$ is singular.    First I'm not sure how (*) can be used here exactly. We can say that the sequence $(c_0I,c_1A,..,c_nA^n)$ resembles the sequence (1) where $v=I$. But here we have scalar $c_i$ so it's not exactly the same sequence.   Maybe I could say that the sequence $(c_0I,c_1A,..,c_nA^n)$ is linearly dependent because it has $n+1$ elements while over $dim\mathbb{F^n}=n$ $\quad$(2).  Intuitively I understand that suppose we sum all the matrices $c_iA^i$ and then in the last sum operation we get to some $c_jA^j$ and if it's a linear combination of other matrices then we can have such a $c_j$ that: $$ c_oI+\sum_{i=1}^{n-1} c_iA^i+c_jA^j=0 $$ But I don't think this is enough to prove. Currently selected indicator's colour in flag summary and help pages, can be the same as the site's colour code Some of the Stack Exchange sites has it's own colour code. Like Meta Stack Exchange has #17a7db, ELU has #AE3737, Travel has #277789 for the currently selected item's indicators in main page and the user activity page. But in the flag summary section and help pages of those sites, the indicator has the colour of #F48024 which is same as the Stack Overflow site's colour code. So instead of applying the Stack Overflow's colour code to all the sites, shall we apply the site's specific colour code in flag summary and help pages too. Selected indicator in flag summary section of MSE, ELU and Travel:      Selected indicator in help page: Remove Meta Privilege for MSE (only) What I am asking here is to please remove the privilege to participate in Meta for MSE.  This privilege is literally useless. You can't really have a meta site for a meta site. There's no Meta.Meta.SE here or any other SE site out there: Meta.SE covers for every site in SE, including itself.     When users find out that you don't really get anything new, it's kinda sad for those people. There is no use for it except just to be there and fill up the space between the first and third privilege. Privileges should be shown only if you earn something new, and this privilege to participate in the meta isn't new because it's where you report problems in all of SE, including Meta.SE.    So please remove the privilege to participate in meta just for Meta.SE.    This is not a dupe as this is a feature-request to remove it, not support on why it is there. .NET DateTime.Now returns incorrect time when time zone is changed Hom is a left-exact functor In German, we often use "Elfhundert" (literally, "eleven hundred") for 1100 or "neunzehnhundert" ("nineteen hundred") for 1900; but is this correct in English? What harm can be done with a copy of one's passport? In one of the questions on this site, it was suggested the use of  to save a backup copy of your passport, in case you need to proof who you are if your original document is lost.   I like the idea, but what if someone gets access to your Dropbox folders and is able to acquire the copy of your passport.  The same goes for when you leave a copy of your passport at a hotel receptions desk. Once someone has the info printed on that copy, would that make it easier to steal your identity?