For example, a perfect measurement ($\hat A=\hat x$) on position:   $$\hat x\psi=x_0\psi$$  The eigenvalue $x_0$ is the result of measurement. The eigenfunction $\psi$ is the wavefunction after measurement. Here solve for the eigenfunction one could get $\psi=\delta(x-x_0)$, so we say that the wavefunction has collapsed to the Dirac Delta?  I am not sure if my understand is flawless.  In a non-ideal measurement, one can only get measurement results as a density distribution. In this case, how does the wavefunction collapse?
How does a wavefunction collapse into one state? More specifically, what conditions cause a wavefunction for a quantum particle to collapse?    Does this have to do with density matrices?    Please try to give a mathematical explanation that is simple and concise...
Windows License stored in UEFI/BIOS Since Windows 10 (or earlier?). It has become standard to store the license product key in the UEFI/BIOS.  I am in the market for buying a new (custom build, individual parts) PC with a new Windows 10 License.  However I was wondering (and can't find clear information on this subject). What the differences between the licenses are when it comes to OEM and Retail.  If in the far future I were to sell this PC:   Would the license (whether OEM or Retail), be stored in the UEFI/BIOS, where it could be used by the new owner? If so, can I remove the license key from the UEFI/BIOS?   I want to know this because I don't want to buy Windows 10 every single time I buy a new PC (since no new versions of Windows will be created, only updates). Error when updating texlive 2015 When I want to update packages of texlive 2015 I get this error? what is the problem?  tlmgr update --list  The locale codeset (cp720) isn't one that perl can decode, stopped at   C:\texlive \2015\tlpkg\tlperl\lib/Encode/Locale.pm line 123. Compilation  failed in require at C:\texlive\2015\tlpkg\tlperl\lib/LWP/UserAgent. pm  line 1001. Recently I paid a lot (almost one month of my salary) for repairing my car. I am wondering if there is a way to get deduction on federal or state taxes (USA, California). Potential difference across a zero resistance wire Oscillation of air particles and speed of sound wave A sound wave is essentially air particles oscillating parallel to the direction of travel of the wave.  We learnt that $v = f\lambda$, where $v$ is the speed of the wave, $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.  Suppose the air particles oscillate with a frequency of $f_{particle}$. Is there any relationship between $f_{particle}$ and $v, f, \lambda$ of the entire sound wave?   I thought about this because intuitively, if the air particles vibrated faster, then the wave should travel faster, but I am unable to come up with any formula that describe this relationship (if it is even true!). Gilbert, MacWilliams, and Sloane defined a solution to "perfect integrity" in , unconditionally secure message authentication, not vulnerable to any cryptanalysis on its own even by an adversary with unbounded computational power, provided you have a key chosen uniformly at random the same length as the message. It sounds counter-intuitive that a MAC could have perfect integrity, any hash function can be broken with enough computational power. How does their solution have "perfect integrity"? In Windows Vista, I have tried to set the setting 'the following number of lines at a time' to zero. I assumed my scroll wheel would not scroll anymore. But it still works.  I'm trying to disable the wheel because my RSI complaints worsen when using the scroll wheel often.  Does anyone have an idea on how to achieve this? Hello I'm writing a beamer presentation and I would like to change the colour of the text in the theorem block from one place and it will affect my whole presentation.  Is this possible?  My example:  \documentclass{beamer}   \usetheme{Madrid}   \usecolortheme[RGB={201,20,35}]{structure}   \setbeamercolor{background canvas}{bg=black}  \setbeamercolor{normal text}{fg=white}   \setbeamercolor{theorem text}{fg=black}    %\useoutertheme{shadow}  \useinnertheme{rounded}   \usepackage[utf8]{inputenc}  \usepackage{default}   \begin{document}   \begin{frame}{Primul Slide}   Bla bla bla    Bla bla bla   \begin{theorem}  This is a theorem it is very important!!  \end{theorem}   \end{frame}   \end{document} How to keep glasses dry in the rain? How to determine the value of a domain name? I'd like to register a second-level domain with the .com extension and I noticed that it has already been registered and the owner put it for sale and let the possibility to make an offer in their site or by contacting them.  I saw that domains are generally sold for extremely different prices, so I'm confused.  Are there any general criteria to decide how much I should offer to purchase a domain? Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$ What does the term $\mathcal O(\epsilon^2)$ mean? Does the refresh rate affect LCD screens? I have never really paid any attention to the refresh rate setting on computers with LCD screens.  I thought with LCD displays the pixels are either on or off, they don't flicker like in old CRT monitors.  Seems trivial to me, but my own graphics card allows me to select 59Hz or 60Hz - which leads me onto my question:  Does the refresh rate affect LCD screens?  Technical info to back up any answers would be great. How can a file be copied? How do I copy a file in Python?  I couldn't find anything under . How do I calculate the power consumed by a lightbulb? I'm studying a lightbulb and its variable resistance, given by the expression:  $R(T) = Ro[1 + α(T-T_0)]$, where $R_0$ is the resistance of the lamp at $T_0$.  In this case, $R$ is not given by Ohm's law ($V=Ri$). So, which expression can I use to calculate the power consumed by a lightbulb?   $P = R\cdot i^2$ $P = \frac{V^2}{R}$ $P = V\cdot i$ (I don't think I can use this one)   I have measured the current and the e.p.d. at the bulb using instruments. The number $2^{3^n}+1$ is divisible by $3^{n+1}$ and not divisible by $3^{n+2}$. Prove that for every $n\in \mathbb N$, $2^{3^n}+1$ is divisible by $3^{n+1}$ and not divisible by $3^{n+2}$.   I was able to prove that $2^{3^n}+1$ is divisible by $3^{n+1}$ using induction. First, I check whether the statement is true for $n=1$. The statement holds since: $$2^{3^1}+1=9$$ $$3^{1+1}=9$$ Second, I assume that the statement is true for every $n\in \mathbb N$. Finally, I take that $n=n+1$. From that, the original expression becomes $(2^{3^n})^3+1$, which is a sum of cubes, and the same as: $$(2^{3^n}+1)((2^{3^n})^2-2^{3^n}+1)$$ As I assumed that $2^{3^n}+1$ is divisible by $3^{n+1}$, and one of the factors is $2^{3^n}+1$, the entire expression must be divisible by $3^{n+1}$, and so the statement is true for every $n\in \mathbb N$.  Now, I don't know how to prove that $2^{3^n}+1$ isn't divisible by $3^{n+2}$. Please explain how I can do so step-by-step, without using complicated terms. How to log out user from web site using BASIC authentication? How to install updates via command line? When I log into my web server via SSH I see the information:  88 packages can be updated. 80 updates are security updates   I tried apt-get update then apt-get upgrade but each time I log in I still see the message about updates. How do I install them? Error: You are missing the following 32-bit libraries, and Steam may not run: libXtst.so.6 [...] I'm aware that many people are having problems on launching Steam (downloaded from Steam not the repos) on Ubuntu 16.04 64-bit with this error message:  svetlana@svetlana-desktop:~$ steam Running Steam on ubuntu 16.04 64-bit STEAM_RUNTIME is enabled automatically Error: You are missing the following 32-bit libraries, and Steam may not run: libXtst.so.6 libXrandr.so.2 libgobject-2.0.so.0 libglib-2.0.so.0 libgtk-x11-2.0.so.0 libpulse.so.0 libgdk_pixbuf-2.0.so.0 Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) Installing breakpad exception handler for appid(steam)/version(1463778979) [2016-05-22 16:33:15] Startup - updater built May 20 2016 18:57:58 [2016-05-22 16:33:15] Opted in to client beta 'publicbeta' via beta file You are in the 'publicbeta' client beta. Looks like steam didn't shutdown cleanly, scheduling immediate update check [2016-05-22 16:33:15] Checking for update on startup [2016-05-22 16:33:15] Checking for available updates... [2016-05-22 16:33:15] Download skipped by HTTP 304 Not Modified [2016-05-22 16:33:15] Nothing to do [2016-05-22 16:33:15] Verifying installation... [2016-05-22 16:33:15] Performing checksum verification of executable files [2016-05-22 16:33:15] Verification complete [2016-05-22 16:33:16] Shutdown   My question has anyone found a solution yet? Windows 10 April 2018 Update added an extra Language and I cannot remove it I have just updated to Windows 10 April 2018 Update (1803 / 17134.1)  I had my laptop configured completely as UK with only a UK keyboard layout  Since the update, a US keyboard has been added so I now see the switch icon in my task bar    It appears that all the keyboard language settings are now migrated to the new Settings app and no longer in the old Control Panel    However, as can be seen from the screenshot above, the Remove button under the newly added US layout is greyed out (as is the UK one)  How can I remove the US layout?  UPDATE 1: I just updated my desktop and it hasn't added a US keyboard!  UPDATE 2: I added a third language (French) and was able to remove it, the button was enabled. The US remove button remained disabled both after adding French and after removing it.  UPDATE 3: I added another English language (Australian) and this cannot be removed! Button is disabled  UPDATE 4: I followed this  except I removed the values from the keys for all users and also the values under the Substitutions key. The task bar switch icon is gone but I still have all 3 languages in Settings still