Sufficient condition for square integrability The necessary condition for $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ to be integrable is that $\psi(x)\rightarrow 0$ as $x\rightarrow\pm\infty$. But this is not the sufficient condition. For example, $\delta(x)$ vanishes as $x\rightarrow \pm \infty$, but it is not square-integrable. What is the sufficient criterion for $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ to converge? In the previous question $\psi(x)$ was an arbitrary function. If now $\psi(x,t)$ be an arbitrary solution (not necessarily a stationary state) of the time-dependent Schrodinger equation what is the sufficiency condition for $\int\limits_{-\infty}^{+\infty}|\psi(x,t)|^2dx$ to be finite? Does this conditions remain valid in three-dimensions?   EDIT: Can we have a continuous function $\psi(x)$ which does not go to zero as $x\rightarrow \pm\infty$ and yet the integral $\int\limits_{-\infty}^{+\infty}|\psi(x)|^2dx$ converges?
Necessary condition for square integrable functions? I'm studying Quantum Mechanics and I came across this which I don't quite understand: For a vector space of functions $f(x)$ to be square integrable (i.e $\int{|f(x)|^2dx &lt; \infty)}$, the necessary condition is $\lim_{|x| \to \infty}f(x) \to 0$. Can someone help me understand why that must be the case? I sort of understand that it must not blow up at the infinities but can a (continuous or piece-wise continuous) function have a non-zero finite value at the infinities and still be square integrable?
Poisson Distribution of sum of two random independent variables $X$, $Y$ What does this triangle/diamond schematic symbol represent? Could someone please tell me what the symbol ringed in red represents?     I know it has ON Switch pointing to it, but the symbol has other lables in different parts of the document. In Peter Jackson's film The Fellowship of the Ring, when the Council of Elrond takes place it shows a total of 22 members:    We all know the names of the ones who were part of the Fellowship, but I want to know who are the rest of the people in the council. Who are the elves that came with Legolas? The men next to the dwarves? The three old men sitting next to Boromir? Or the two elves next to Aragorn?  Starting from left to right:   Frodo Gandalf Elf Elf Legolas Elf Old Man Old Man Dwalin Gloin Gimli Nori Old Man Boromir Old Man Old Man Elf Elf Aragorn Elladan Elrond Elrohir I would like to use  with a video.  With ffmpeg I can export some (.jpeg) images from a video. 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I want to remove all the files from different directories, and want to keep only latest files 20 files. is this the correct command to do this?  ls -t1 /mnt/dwh/ftp/dwh_ftp_cbs/ARLOGS/ | tail -n +22 | xargs rm -f I have what seems to me a very simple proof that $A_{n}$ is the only subgroup of $S_{n}$ of index 2.  Since I've seen other people prove it with what feel like really complicated methods (), I'm wondering if I've overlooked something.  Proof: Let $H$ be any subgroup of index 2 in $S_{n}$.  Then $H\cap A_{n}$ is a normal subgroup of $A_{n}$ and since $A_{n}$ is simple, $H$ is either the trivial or the improper subgroup of $A_{n}$.  If it's the trivial subgroup it doesn't have index 2 in $S_{n}$, and otherwise the theorem is proved. I have Windows 8 (final, MSDN) on my Macbook Pro Retina.  Windows 8's ongoing, continuous tantrum of insisting that it's on a tablet device is causing it to always have a touch keyboard 'icon' on  the taskbar.  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When Cantor's Intersection theorem won't work with closed sets Give an example to show that Cantor's Intersection Theorem would not be true if compact sets were replaced by closed sets.    Compact set is closed and bounded, so what I'm going to find is something that is closed but not bounded.  By the Cantor theorem, which says that a decreasing sequence of non-empty compact subset will have a non-empty intersection.  What I was trying to approach is how two sets just "touched" and as if they have not bounded, two sets "touched" nothing. But I cant give an example for that I know that IList is the interface and List is the concrete type but I still don't know when to use each one. What I'm doing now is if I don't need the Sort or FindAll methods I use the interface. Am I right? Is there a better way to decide when to use the interface or the concrete type? How do I update my TeX distribution? Generating random results by weight in PHP? I know how to generate a random number in PHP but lets say I want a random number between 1-10 but I want more 3,4,5's then 8,9,10's. How is this possible? I would post what I have tried but honestly, I don't even know where to start. I`m using Firefox Sync on a bunch of computers.   I would like to manage my account, by removing some computers. I can't find the place to do this, it should be on the Firefox Sync page, but the official page only has the privacy and download information. Commandline shortcut for current directory similar to ~ for home directory? Where did the age go on the network profile? Ever since the profile redo, the age was shown in the network profile, but not in the per-site profiles. So if someone wanted to see the age, they had to go to the network profile. However.. It's vanished! The field is still there in the profile editing, however. It doesn't appear as if this was implemented:   So... Where is it? I'm using a common charger for both of my laptops.  The charger is specified at 20 Volts and 3.25 Amperes. My other laptop has a specification of 19 V and 3.42 A.  Is it safe to charge my laptop with chargers whose specifications exceed or are not equal to the mentioned specifications?  I have read somewhere that it may cause some damage if the voltage or amperage is not matching the specs.