Is a state with definite momentum possible in presence of an infinite square-well potential? The momentum operator is usually defined as  $$\hat p = -i \hbar \frac d {dx}$$  The eigenfunctions of this operator are plane waves  $$\tag{1} \label{1} \psi_p(x) = A \exp\left(\frac{i p x}{\hbar}\right)$$  and they are not square-integrable, i.e.  $$\int_{\mathbb R} | \psi_p(x)|^2 dx = \infty$$  therefore, it is usually said that a free particle with a well-defined momentum is a physical impossibility.  But we have considered a case in which the wavefunction $\psi$ can extend from $-\infty$ to $\infty$. In general, the wavefunction $\psi$ should satisfy the following conditions:  $$\tag{2} \label{2} \psi \in C^0 (\mathbb R)$$ $$\tag{3} \label{3} \int_\mathbb{R} | \psi(x)|^2 dx = 1$$  In the case of an infinite square-well potential  $$V(x) = 0 \ \ \ \text{if} \ x \in [0,L]$$ $$V(x) = \infty \ \ \ \text{if} \ x \notin [0,L]$$  we usually require that   $$\tag{4} \label{4} \psi(x) = 0 \ \ \ \text{if} \ x \notin [0,L]$$  so that \ref{2} implies  $$\tag{5} \label{5} \psi(L)=\psi(0)=0$$  But by applying \ref{5} to \ref{1}, we obtain  $$\tag{6} \label{6} e^{ipL/\hbar}=1=0$$  which is clearly absurd and cannot be valid.  However, in  the authors introduce a self-adjoint extension of the operator $\hat p$ (which they call $P_\theta$) by replacing \ref{5} with  $$\tag{7}\label{7} \psi(L)=e^{i \theta} \psi(0)$$  from which they obtain the "momentum eigenfunctions"  $$\tag{8} \label{8} \phi_n(x,\theta) = \frac 1 {\sqrt{L}} \exp \left(2 i \pi \nu \frac x L \right), \ \ \ \nu = n + \frac{\theta}{2 \pi}, \ \ \ n \in \mathbb Z$$  They conclude with the remark     For a particle in a box, it is often argued that the “physical” wave   function should continuously vanish on the walls $x = 0$ and $x = L$,   ensuring that the presence probability vanishes continuously for $x ≤0$   and for $x ≥ L$. One should realize that the continuity of the   measurable quantity      $$ Pr(0 ≤ x ≤ u) = \int_0^u |\phi(x)|^2 dx, \ \ \ u \in[0,L] $$ is   ensured as soon as the integral $\int_0^L |\phi(x)|^2 dx $ does   converge and does not require any continuity property of $\phi(x)$.   Specializing this remark to the eigenfunctions of $P_\theta$ we   observe that $|\phi_n(x,\theta)|^2$ does not vanish continuously at $x=0$   but nevertheless the physical quantity $$Pr(0 ≤ x ≤ u) = \frac{u}{L}$$ vanishes continuously, as it should, for $u \to 0$.   They also say that the Heisenberg inequlity is not valid for the "momentum eigenfunctions" of Eq. \ref{8}, and indeed we would obtain  $$\sigma_p = 0$$ $$\sigma_x = \frac{L}{\sqrt {12}}$$   From a mathematical point of view, their reasoning is more or less clear to me: if we replace \ref{5} with \ref{7} and restrict ourselves to the interval $[0,L]$, it is possible to extend the momentum operator, making it self-adjoint, and to introduce eigenfunctions for this extended version. But is this reasoning physically meaningful?   In particular, what about the value of the wavefunction outside the interval $[0,L]$? Also, what about the Heisenberg inequality? Is the fact that it is not valid acceptable from a physical point of view?    Update  I don't understand if the authors are implying that the wave function is set to $0$ for $x \notin [0,L]$; if this was the case, shouldn't we have the problem that $Pr(0\leq x \leq u)$ is discontinuous at $u=L$?
How can I solve this quantum mechanical "paradox"? Let a (free) particle move in $[0,a]$ with cyclic boundary condition $\psi(0)=\psi(a)$. The solution of the Schrödinger-equation can be put in the form of a plane wave. In this state the standard deviation of momentum is $0$, but $\sigma_x$ must be finite. So we find that $\sigma_x\sigma_p=0$. Is something wrong with the uncertainty principle?
Default Class Accessibility in C# by default is a class:   private ? internal ?  sealed ? Blender 2.79b is installed (from .exe) on my computer, currently. And I just downloaded the Blender 2.8 release candidate 2 (.zip).  When I create something in Blender 2.8, after I save it, I cannot open it again because the computer tries to open the blend file in version 2.79b. And manually selecting the 2.8 version from zip file does not work.  How can I open my blender 2.8 files? Do I have to delete Blender 2.79b? I want to give a client access to my server, but I want to limit those users to their home directories. I will bind-mount in any files I want them to be able to see.  I've created a user called bob and added him to a new group called sftponly. They have a home directory at /home/bob. I've changed their shell to /bin/false to stop SSH logins. Here is their /etc/passwd line:  bob:x:1001:1002::/home/bob:/bin/false   I've also changed the /etc/ssh/sshd_config to include the following:  Match Group sftponly         ChrootDirectory /home/%u         ForceCommand internal-sftp         AllowTcpForwarding no   When I try to log in as them, here's what I see   $ sftp bob@server bob@server's password:  Write failed: Broken pipe Couldn't read packet: Connection reset by peer   If I comment out the ChrootDirectory line I can SFTP in but then they have free rein over the server. I have found that ChrootDirectory /home works, but it still gives them access to any home directory. I have explicitly tried ChrootDirectory /home/bob but that doesn't work either.  What am I doing wrong? How can I limit bob to /home/bob/?  ----EDIT-----  Okay so I just had a look at /var/log/auth.log and saw this:  May  9 14:45:48 nj sshd[5074]: pam_unix(sshd:session): session opened for user bob by (uid=0) May  9 14:45:48 nj sshd[5091]: fatal: bad ownership or modes for chroot directory component "/home/bob/" May  9 14:45:48 nj sshd[5074]: pam_unix(sshd:session): session closed for user bob   I'm not entirely sure what's going on there,  but it suggests something is wrong with the user directory. Here is the ls -h /home output:  drwxr-xr-x 26 oli      oli      4096 2012-01-19 17:19 oli drwxr-xr-x  3 bob      bob      4096 2012-05-09 14:11 bob Notation for intervals I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed . I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall someone calling the notation with $[a,b]$ and $]a,b[$ as French notation.   What are the origins of the two notations?  Is the name French notation correct? Are they used frequently in France? Or were they perhaps prevalent in French mathematical community at some point? (In this  Bourbaki is mentioned in connection with the notation $]a,b[$.)      Since several answerers have already mentioned that they have never seen $\langle a,b \rangle$ to be used for closed intervals, I have tried to look for some occurrences for this. The best I can come up with is the article on , where these notations are called Czech notation and French notation. Using $(a,b)$ and $[a,b]$ is called English notation in that article. (I am from Central Europe, too, so it is perhaps not that surprising that I have seen this notation in lectures.) I also tried to google for  or . Surprisingly, this lead me to a  where this notation is used for open interval. How to hide file extensions In the Upcoming Halloween Update Posted By SOOPUR SELL (supercell), there have been sneak peeks released showing new skeleton traps for TH 8+. I also remember the Santa Strike from one of the Christmas Updates. My question is when these holiday events finish, do we lose the money we spent on the Traps or do we keep them forever? Western Digital drive making beeping noise PostgreSQL Shapefile import failed I am very new to PostGIS/PostgreSQL. In fact, I downloaded it because of the answers I got to this question:   I installed both PostgreSQL and PostGIS. But I am not able to import a shapefile to the database? It gives an error, "Shapefile import failed". What am I doing wrong? Does Lin's  assume that the 2 data sets have linear or monotonic tendencies? Or, can I measure the concordance between 2 data sets that have a sinusoidal tendency? Brace expansion not working in a script How do I fix gaps around my lean-to greenhouse? List partition labels from the command line Is there a command that will list all partitions along with their labels? sudo fdisk -l and sudo parted -l don't show labels by default.  EDIT: (as per comment below) I'm talking about ext2 labels - those that you can set in gparted upon partitioning.  EDIT2: The intent is to list unmounted partitions (so I know which one to mount). Who can use magic scrolls? I'm confused about who can use scrolls in 5e. In the DMG, p139 under Magic Item Categories: Scrolls, it states that:     Any creature that can understand a written language can read the   arcane script on a scroll and attempt to activate it.   This implies that anyone can try to cast a spell using a scroll. However, on p.200 in the Spell Scroll entry it states that:     If the spell is on your class's spell list, you can use an action to read the scroll and cast its spell without having to provide any of the spell's components. Otherwise, the scroll is unintelligible.   Then there is the bit about a DC 10 + scroll spell level ability check if the spell is higher than a spell slot that you can cast.     If the spell is on your class's spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + spell's level. On a failed check, the spell disappears from the scroll with no other effect.    So, which is it? Can anyone who can read a language use a scroll (like a fighter) if they make an arcana check, or just a PC with that specific spell in their spell list (only spell casters)? On every boot up I get this:    For about 10 seconds (or more) before the "normal" Ubuntu Gnome loading screen and the login screen comes up.  I've read some questions and a bug report about this already, but this problem was dismissed as a minor inconvenience or there were no real solutions (that worked for me). For me, this problem leads to an unacceptable slow boot compared to Ubuntu (Unity) or Windows.  Does anybody know how to fix this or is this a "feature" of Ubuntu Gnome and not a bug? What canonical momenta are the "right" ones? I'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the conjugate momenta $\pi^\mu_{\ \ \ \nu} = \partial \mathscr{L} / \partial(\partial_\mu A^\nu)$, I can use two methods.  First method: directly apply this to $\mathscr{L}$. We get a a factor of $2$ since there are two $F$'s, and another factor of $2$ since each $F$ contains two $\partial_\mu A_\nu$ terms, giving $$\pi^\mu_{\ \ \ \nu} = -F^\mu_{\ \ \ \nu}.$$  Second method: get $\mathscr{L}$ in terms of $A$ by expanding and integrating by parts, yielding $$\mathscr{L} = \frac{1}{2}(\partial_\mu A^\mu)^2 - \frac{1}{2}(\partial_\mu A^\nu)^2.$$ Differentiating this gets factors of $2$ and gives $$\pi^\mu_{\ \ \ \nu} = \partial_\rho A^\rho \delta^\mu_\nu - \partial^\mu A_\nu.$$  These two answers are different! (They do give the same equations of motion, at least.) I guess that means doing the integration by parts changed the canonical momenta.  Is this something I should be worried about? In particular, I have another exercise that wants me to show that one of the canonical momenta vanishes -- this isn't true for the ones I get from the second method! Plus, my stress-energy tensor is changed too. When a problem asks for "the" canonical momenta, am I forbidden from integrating by parts? Is there a way to configure your Stack Overflow account so that you receive notification emails when posts are made to specific tags?    Or do you simply have to log on to see new posts? how to count dates entries monthwise from a resultset Misplaced/Overlapping Navigation Item Text on iOS 10 The bug displayed in the attached screenshot occurs consistently for me while running the developer beta of iOS 10 using v1.5.2 of the Stack Exchange app on an iPhone 6.  I was able to recreate this by simply tapping on any question or answer in my feed with a long title. This doesn't appear to be a bug with iOS 10 itself since I haven't seen it in other apps, but I could be wrong. Blocking web sites with Windows Firewall Is it possible to use Windows Firewall to block specific web sites for all browsers? List extending strange behaviour