What is Wave Function? Well, what is the meaning of wave function? What does it represent? In Schrodinger's equation, we find the value of Ψ. But what is Ψ exactly? Max Born only gave an explanation of what $Ψ^2$ (the probability of particle with low mass enough to have an observable wavelength to be in a position in space). But no one knows what Ψ is.
Is there a direct physical interpretation for the complex wavefunction? The Schrödinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated Hamiltonian. And this wavefunction is, in general, complex, and its modulus squared yields the probabilities observed experimentally. Though, perhaps, this question has been asked many times, I am wondering if there is a direct physical interpretation - something that physically corresponds to - the wavefunction. Or is it just an intermediate calculational tool to arrive at the appropriate predictions for experimental outcomes, and nothing more? Of course, things like superposition and interference effects follow from the complex nature of the probability amplitude. So there must be something physical about it. What is it? Or are we not supposed to ask that question? Is it because the probability amplitude is complex that we have difficulty in relating it to something physical? Can we do quantum mechanics without complex numbers?
What are the ways to position things absolutely on the page? I know that there is tikz, but also eso-pic and textpos.  I'd like to see a list of possible ways (not necessarily packages) for absolute positioning of things (which I believe amounts to "boxes") on the page, together with their advantages and disadvantages (for instance: tikz's current page node is very easy to use, but usually requires two LaTeX runs, and overhead from loading tikz if you don't need it for any other things is considerable).  Both LaTeX and ConTeXt solutions are welcome Nonlinear function continuous but not bounded I would like an example of a map $f:H\rightarrow R$, where $H$ is a (infinite dimensional) Hilbert space, and $R$ is the real numbers, such that $f$ is continuous, but $f$ is not bounded on the close unit ball $\{ x\in H : \|x\| \leq 1\}$.  Actually, $H$ could be replaced by any Banach space (but not just a normed space-- that's too easy).  My motivation is that if $f$ is linear, this is impossible; but I have next to no intuition about non-linear functions.  Edit: Here's an example for $c_0$ which is even differentiable (disclaimer: I found it here: ).  Define $f:c_0\rightarrow F$ (where F is your field, real or complex) by $$ f(x) = \sum_{n=1}^\infty x_n^n \qquad (x=(x_n)). $$  You can estimate the sum by a geometric progression, so it does converge.  A bit of checking shows that f is Frechet differentible (so certainly continuous).  But $f(1,1,\cdots,1,0,\cdots)=n$ (if there are $n$ ones) so $f$ is not bounded on the closed unit ball.  What I don't immediately see is how to adapt this to $\ell^2$, say. Normal subgroup of prime index Generalizing the case $p=2$ we would like to know if the statement below is true.  Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal. What is the safest way to clean up /boot partition? I have 200 MB assigned for the /boot partition.  Whenever I try to update the kernel, I receive an error message that basically states /boot is full.  What can I do to cleanup /boot and remove/backup the older kernels? Absolute zero and Heisenberg uncertainty principle I got to read Volume I of Feynmann's lectures. It said that at absolute zero, molecular motion doesn't cease at all, because if that happens, we will be able to make precise determination of position and momentum of the atom. We do know that Heisenberg uncertainty principle holds for microscopic particles in motion. But what then is wrong to consider that all molecular motion ceases at absolute zero? In other words, does the uncertainty principle not hold when there is no motion?  Need some help! What is the "En passant" rule in chess? How useful is that rule and how many times we can use that in one game?  Under what conditions can the en passant rule be used?  How often does it come up and is it a good idea to use it? Firefox running with rare arguments I got this output from ps aux|grep firefox command:  $ ps aux|grep firefox auraham   5443 17.3 11.0 2600628 426484 ?      Sl   18:10   0:54 /usr/lib/firefox/firefox auraham   5496 24.0 17.9 2558108 691700 ?      Sl   18:10   1:15 /usr/lib/firefox/firefox -contentproc -childID 1 -isForBrowser -intPrefs 5:50|6:-1|18:0|28:1000|33:20|34:10|43:128|44:10000|49:0|51:400|52:1|53:0|54:0|59:0|60:120|61:120|91:2|92:1|106:5000|117:0|119:0|130:10000|155:24|156:32768|158:0|159:0|167:5|171:1048576|172:100|173:5000|175:600|176:4|177:1|186:2|200:60000| -boolPrefs 1:0|2:0|4:0|26:1|27:1|30:0|35:1|36:0|37:0|38:0|41:1|42:1|45:0|46:0|47:0|48:0|50:0|55:1|56:1|57:0|58:1|62:1|63:1|64:0|65:1|66:1|67:0|68:1|71:0|72:0|75:1|76:1|80:1|81:1|82:1|83:0|85:0|86:0|87:1|88:0|93:1|94:0|100:0|105:0|108:1|109:0|111:1|112:1|114:1|118:0|120:0|122:0|124:1|125:1|131:0|132:0|133:1|135:0|146:0|153:0|154:0|157:1|160:0|162:1|164:1|165:0|170:0|174:1|179:0|180:0|181:0|182:1|183:0|184:0|185:1|188:1|192:0|193:0|194:1|195:1|196:0|197:1|198:1|199:1|201:0|202:0|204:0|212:1|213:1|214:0|215:0|216:0| -stringPrefs 3:7;release|134:3;1.0|151:332;  ¼½¾ǃː??։֊׃״؉؊٪۔܁܂܃܄ᅟᅠ᜵           ???‐’․‧??????? ‹›⁁⁄⁒ ⅓⅔⅕⅖⅗⅘⅙⅚?⅜⅝⅞⅟∕∶⎮╱⧶⧸⫻⫽⿰⿱⿲⿳⿴⿵⿶⿷⿸⿹⿺⿻　。〔〕〳゠ㅤ㈝㈞㎮㎯㏆㏟꞉︔︕︿﹝﹞?．／｡ﾠ???￼�|152:8;moderate| -greomni /usr/lib/firefox/omni.ja -appomni /usr/lib/firefox/browser/omni.ja -appdir /usr/lib/firefox/browser 5443 true tab auraham   5694  0.0  0.0  15444  1092 pts/2    S+   18:15   0:00 grep --color=auto firefox   According to , this error occurs after firefox crashes. However, every time I run Firefox again, that large line of arguments appears too.  I already try these things:   Run Firefox with addons disabled Reinstall Firefox (sudo apt-get purge firefox; sudo apt-get install firefox)   However, it did not work.  Is it possible to remove such a line of arguments? What regulations should a European driver know when driving in the US? There are the obvious ones. The US has the keep-your-lane system, where as in Europe you are expected to keep to the right (left for UK) as long as possible.  On my latest travel to the US, I was confronted with the possibility to go through a red light if you turn right, unless it is explicitly not allowed.  What should a european driver know when driving in the US? What are the do's and don'ts compared to driving in Europe. I'm trying to follow the  (on my Ubuntu 10.04 VM).  The first step is:  sudo add-apt-repository ppa:linaro-maintainers/toolchain   But when I run that I get:  Error reading https://launchpad.net/api/1.0/~linaro-maintainers/+archive/toolchain: &lt;urlopen error [Errno 110] Connection timed out&gt;   I'm guessing the problem is with my configuration for my employer's proxy server, particularly for HTTPS.  When I opened  in Firefox, I got "The connection has timed out".  I then went into Firefox Preferences -> Advanced -> Network -> Settings... and set HTTP Proxy to "gateway" and Port to 8080, and checked "Use this proxy server for all protocols".  Then the page loaded.  This supports my theory.  I've tried setting the http_proxy and https_proxy environment variables (both lower and upper case):  $ set | grep -i proxy HTTPS_PROXY=https://gateway:8080 HTTP_PROXY=http://gateway:8080 http_proxy=http://gateway:8080 https_proxy=https://gateway:8080   I also tried changing them to:  $ set | grep -i proxy HTTPS_PROXY=gateway:8080 HTTP_PROXY=gateway:8080 http_proxy=gateway:8080 https_proxy=gateway:8080   And I tried adding this second line to /etc/apt/apt.conf:  Acquire::http::proxy "http://gateway:8080/"; Acquire::https::proxy "https://gateway:8080/";   But I continue to get the same error.  How do I troubleshoot this?  Update: I followed the instructions in the  to  and got further:  $ sudo add-apt-repository ppa:linaro-maintainers/toolchain Executing: gpg --ignore-time-conflict --no-options --no-default-keyring --secret-keyring /etc/apt/secring.gpg --trustdb-name /etc/apt/trustdb.gpg --keyring /etc/apt/trusted.gpg --primary-keyring /etc/apt/trusted.gpg --keyserver keyserver.ubuntu.com --recv 1DD749B890A6F66D050D985CF1FCBACA7BE1F97B gpg: requesting key 7BE1F97B from hkp server keyserver.ubuntu.com gpgkeys: HTTP fetch error 7: couldn't connect to host gpg: no valid OpenPGP data found. gpg: Total number processed: 0   I wonder if the "HTTP fetch error" means I also need to add "Defaults env_keep = http_proxy" to /etc/sudoers...  Update #2: I added:  Defaults env_keep="http_proxy" Defaults env_keep="https_proxy"   ...as in , but I still get the same "gpgkeys: HTTP fetch error 7: couldn't connect to host" error.  And this is curious:  $ sudo env | grep proxy https_proxy=https://gateway:8080   ...because http_proxy isn't there.  Any advice is appreciated.  Update #3: Since the host of my VM is a laptop, I took it home and tried there (with no proxies):  $ sudo add-apt-repository ppa:linaro-maintainers/toolchain Executing: gpg --ignore-time-conflict --no-options --no-default-keyring --secret-keyring /etc/apt/secring.gpg --trustdb-name /etc/apt/trustdb.gpg --keyring /etc/apt/trusted.gpg --primary-keyring /etc/apt/trusted.gpg --keyserver keyserver.ubuntu.com --recv 1DD749B890A6F66D050D985CF1FCBACA7BE1F97B gpg: requesting key 7BE1F97B from hkp server keyserver.ubuntu.com gpg: key 7BE1F97B: public key "Launchpad Linaro Overlay PPA" imported gpg: Total number processed: 1 gpg:               imported: 1  (RSA: 1)   It worked.  So problem is related to proxy configuration.  But I suppose I have a workaround. I want to kill all running processes of a particular user from either a shell script or native code on a Linux system.  Do I have to read the /proc directory and look for these?  Any ideas? Is there a dynamic mapping of the pids under UIDs in Linux? Isn't this in the proc?  If not, then where is this list maintained? Should I read from it? Also where is the static list of all UIDs in the system so I can validate this this user exists and then proceed to kill all processes running under it? Cooking steak in frying pan, problem with oil splatter I was using canola oil and put the stove top setting on "high - 9". I read plenty of guides suggesting there will infact be oil splatter, but it was pretty excessive and burning my arm with one drop getting near my eye before I stopped. I turned the heat setting down to "medium - 4", where the oil splatter seemed to drop substantially. Before I try to make steak again I'd like to get some clarification  So my questions are:   Which heat setting am I supposed to use? Is that type of oil splatter normal or should I try using a different oil, and if so, what oil? Should oiling the meat instead of the pan help the problem? Unable to install any updates through update manager/apt-get upgrade How closely can a game legally resemble another? Most games build on other games' successes and many are downright clones. Where is the limit of imitation before legal issues come into play? Is it down to literary or graphic work like characters and storyline that cause legal problems, or can someone actually claim to own gameplay mechanics?  There are so many similar clone games out there that the rules are probably very slack or nonexistent, but I'd like to hear the views of more experienced developers / designers. I have Google Chrome Version 70.0.3538.77 (Official Build) (64-bit).  Today, after running sudo apt update and sudo apt full-upgrade, I saw:  E: Repository 'http://dl.google.com/linux/chrome/deb stable Release' changed its 'Origin' value from 'Google, Inc.' to 'Google LLC' N: This must be accepted explicitly before updates for this repository can be applied. See apt-secure(8) manpage for details.   I looked at the man page for apt-secure but it's pretty technical and difficult, at least for me, to follow.  So how do I fix this issue?    Edit: for now, I've removed the existing Google Chrome ppa from Software Sources and then ran sudo apt update and sudo apt full-upgrade successfully with the following being installed/upgraded successfully:  The following NEW packages will be installed:   python3-netifaces The following packages will be upgraded:   netplan.io nplan ppp 3 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. Why do Americans have to file taxes, even if their only source of income is from a regular job? Trying to run a Ubuntu server for the first time... But everything is in a terminal. Is there a way to switch to a GUI? Proof that $a\mid b \land b\mid c \Rightarrow a\mid c $ I am trying to prove:     $a\mid b \land b\mid c \Rightarrow a\mid c $   $a\mid b$ means that a divides b if there is an integer k, that $b=k\cdot a$  Please give me a hint on how to start, because I have no idea. How can I deduce/identify an app from a notification icon? I'm  trying to create a FPS game in blender game engine. When I do KeyBoard[Spacebar] sensor and hook it up to a simple motion of z +1 or something like that, the guys pops up really fast as if he didn't have any motion between being on the ground and being in the air. Is there a way to make a more realistic, more slow jump in BGE? Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$