Interpretation of $\langle \phi | A | \psi \rangle$ If the current state of some quantum system is $| \psi \rangle$, what is the physical interpretation of  $$ \langle \phi | A | \psi \rangle $$ where $|\phi\rangle$ is some other -maybe the same- quantum state, and $A$ is a generic self-adjoint operator?
Meaning of $\int \phi^\dagger \hat A \psi \:\mathrm dx$ While analysing a problem in quantum Mechanics, I realized that I don't fully understand the physical meanings of certain integrals. I have been interpreting:    $\int \phi^\dagger \hat A \psi \:\mathrm dx$  as "(square root of) probability that a particle with state $|\psi\rangle$ will collapse to a state $|\phi\rangle$ when one tries to observe the observable corresponding to $\hat A$" $\int \phi^\dagger  \psi \:\mathrm dx$ as "(square root of) probability that a particle with state $|\psi\rangle$ will collapse to a state $|\phi\rangle$".   All integrals are over all space here, and all $\phi$s and $\psi$s are normalized.  Now, I realize that  interpretation of the first integral doesn't really make sense when put side by side with the second one.   For example, when $|\phi\rangle$ is an eigenstate of $\hat A$, I get two different expressions for "the square root of probability that one will get the corresponding eigenvalue of $\psi$ when one tries to observe the observable corresponding to $\hat A$". As far as I can tell, $\int \phi^\dagger \hat A \psi \:\mathrm dx \neq\int \phi^\dagger  \psi \:\mathrm dx $, even if $|\phi\rangle$ is an eigenstate of $\hat A$. I am inclined to believe that the second integral is the correct answer here (it comes naturally when you split $|\psi\rangle$ into a linear combination of basis vectors). But I am at a loss as to the interpretation of the first integral.  So, my question is, what are the physical interpretations of $\int \phi^\dagger \hat A \psi \:\mathrm dx$ and $\int \phi^\dagger  \psi \:\mathrm dx$?  (While I am familiar with the bra-ket notation, I have not used it in this question as I don't want to confuse myself further. Feel free to use it in your answer, though)
Including Interaction Terms in Random Forest Suppose we have a response Y and predictors X1,....,Xn. If we were to try to fit Y via a linear model of X1,....,Xn, and it just so happened that the true relationship between Y and X1,...,Xn wasn't linear, we might be able to fix the model by transforming the X's somehow and then fitting the model. Moreover, if it just so happened that the X1,...,XN didn't affect y independent of the other features, we also might be able to improve the model by including interaction terms, x1*x3 or x1*x4*x7 or something of the like. So in the linear case, interaction terms might bring value by fixing non-linearity or independence violations between the response and the features.  However, Random Forests don't really make these assumptions. Is including interaction terms important when fitting a Random Forest? Or will just including the individual terms and choosing appropriate parameters allow Random Forests to capture these relationships? How do I get my Mac to boot from an Ubuntu USB key? If you select "USB" and "Mac" on , it gives a series of command line instructions to make a USB key which the MacBook will boot into Ubuntu from.  I've followed them to the letter two or three times on different USB keys, and it doesn't work. There's a very great deal of technical discussion about EFI etc. but this set of instructions seems to suggest it should just work, but it doesn't.  Help? I'm increasingly unhappy with the more locked-down approach Apple is taking, and I'd quite like to start using Linux with a view to transitioning over to using it as my main operating system, but booting from the CD takes forever, runs slowly and I'm really hoping to get it moving off USB.  Can anybody help me? I've updated my Ubuntu to 19.10 couple of weeks ago and on installation system suggests to change or add shortcut for input source selection despite one was already defined previously. I've added ctrl+shift shortcut and just realized that it somehow overlapped any other ctrl+shift+something shortcuts globally, i.e. when I want to open new window in vscode, it opens new tab and it looks like system doesn't see the shift button press.   I tried to search this combination in shortcuts settings, but it's just not there. How can I create a zip archive of a whole directory via terminal without hidden files? I have a project with lots of hidden folders / files in it. I want to create a zip-archive of it, but in the archive shouldn't be  any hidden folders / files. If files in a hidden folder are not hidden, they should also not be included.  I know that I can create a zip archive of a directory like this:  zip -r zipfile.zip directory   I also know that I can exclude files with the -x option, so I thought this might work:  zip -r zipfile.zip directory -x .*   It didn't work. All hidden directories were still in the zip-file. Show/Hide CSS Ignored When Inside an SLDS Grid Column What are sentences like "the longer X, the more Y" called and can they be used in formal written English? What is the type of sentence exemplified below called? Is it appropriate to use it in a scientific paper and formal written English in general?     1. The more pronounced the variation, the   more the waste.      2. The more you give, the more you get.      3. The longer I live, the more I realize the impact of attitude on life. Share Wireless connection with Wired Ethernet Port How to show an alert when user navigates to another page without filling the form? I have a form created by a custom module. Now I want to show an alert like ("Please fil the complete form before navigating otherwise lost all the data") if the user navigate to another page without filling all required fields.  How can I do so? In the movie, there's one scene that seems very strange to me.  First, the point:  When the team gets in the first level, they find out that Fischer has a militarized subconscious. After they freak out, and argue about it, they also find out that they can't die, otherwise they'd get stuck in limbo.  So, when Eames says that he won't go further, Cobb states that, since the first level sedative will last a week they would be killed. They needed to go on to the next levels as fast as possible, so the combined multi-level kick would make them wake up.  Here's the point: The plan was the multi-level kick would wake them up before the sedative was off     Ariadne: ...is ten years! Who would wanna be stuck in a dream for ten years?   Yusuf: Depends on the dream.    Arthur: So, once we've made the plant, how do we go out? Hope you have something more elegant in mind than shooting me in the head?    Cobb: A kick.    ...   Arthur: Are we going to feel a kick with this kind of sedation?   Yusuf: Well, that's the clever part. I customize the sedative to leave inner ear function unimpaired. That way, however deep the sleep, the sleeper still feels falling, or tipping.     Now, the question:  When they finish the job, they perform the kicks   The van hits the water Arthur detonates the elevator Eames explodes the fortress   But they don't wake up. They are back in the drowned van. So, as far as I could understand, they still had to sit and wait for the first level's week to end.  (We see Fischer talking to his godfather)  Why didn't they wake up with the kicks, as it was planned? And how did they managed not to get killed by Fischer's projections after they got out of the Van? Compiling the following code and got the error of type illegal.  int main() {     // Compilation error - switch expression of type illegal     switch(std::string("raj"))     {     case"sda":     } }   You cannot use string in either switch or case. Why? Is there any solution that works nicely to support logic similar to switch on strings? Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem Let $f(z)$ be entire function. Consider the functions $e^{if(z)}$ and $e^{−if(z)}$ and applying the Maximum Modulus Theorem, show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function.  (We take $f(z)=u(z)+iv(z)$)  I am confused as so far I have $|g(z)|=|e^{if(z)}|=|e^{-v(z)}|$ and then since $f(z)$ is real, $f(z)=u(z)$ and $v(z)=0$ so I assumed it would follow that $|g(z)|=|e^{v(z)}|=1$.  Similarly, $|g(z)|=|e^{-if(z)}|=|e^{v(z)}|=1$.  Using Liouville I assumed one could say that both $g(z)$ and $h(z)$ are bounded entire functions, they are constant and so it follows that $v(z)$ is constant, meaning that both its partial derivatives are equal to 0 and, due to Cauchy Riemann, both of the partial derivatives of $u(z)$ are equal to zero.  It would then follow that $f(z)$ is constant.  I don't know how to go about the question using the Maximum Modulus Theorem, also I feel I am overlooking the importance of $|z|=1$ perhaps?  Any help would be much appreciated!! I used the inbuilt dataset stackloss here. I used following R code for creating the multiple regression model.  &gt; head(stackloss)   Air.Flow Water.Temp Acid.Conc. stack.loss   1        80         27         89         42   2        80         27         88         37   3        75         25         90         37   4        62         24         87         28   5        62         22         87         18   6        62         23         87         18  &gt; fit &lt;- lm(stack.loss ~ .,data=stackloss) &gt; plot(fit)   And when I plot the model, the four following graphs are displayed.      How to interpret these graphs? What are the significance of these graphs? How to use these graphs? When, if ever, should voting override answer acceptance for answer sorting? The accepted answer always appears at the top of the list, regardless of sort order. This can be misleading if the question author accepted an invalid solution and a better solution was proposed later, and eventually got more votes. I have seen somewhat obscure questions where the accepted answer has a negative score and the next answer has 10 or more up-votes. jQuery Datepicker with text input that doesn't allow user input How do I use the jQuery Datepicker with a textbox input:  $("#my_txtbox").datepicker({   // options });   that doesn't allow the user to input random text in the textbox. I want the Datepicker to pop up when the textbox gains focus or the user clicks on it, but I want the textbox to ignore any user input using the keyboard (copy &amp; paste, or any other). I want to fill the textbox exclusively from the Datepicker calendar.  Is this possible?  jQuery 1.2.6 Datepicker 1.5.2 "He needn't say" / "He doesn't need to say" When I studied English at school (in Italy, more than 35 years ago), we were taught a rule according to which one should say, e. g., "He needn't (need not) say", but NOT "He doesn't need to say".  If I'm not mistaken, it was explained to us that the rule holds good with the following limitations:  1) only in the negative form: thus, "He needn't say", but "He needs to say", and "Does he need to say?";  2) only in the present tense: thus, "He needn't say", but "He didn't (won't, wouldn't, etc.) need to say";  3) only when an infinitive follows: thus, "He needn't see a doctor", but "He doesn't need a doctor".  But, if I google "doesn't need to", I find 125,000,000 occurrences.  I was wondering: does the rule hold good more for British than for American English?  Also, perhaps it isn't an absolute rule, and both constructions ("He needn't say" and "He doesn't need to say") are possible.  Is the rule at least a good description of the circumstances when we CAN (but needn't necessarily) use this construction?  Thank you very much, for this and the other answers. How to align a  to the middle (horizontally/width) of the page Why do we have OS names like Mountain Lion, Jellybean, and Quantal Quetzal instead of numbers? Less of a pressing question and more a "what happened here"?   At some point in the past decade (most) OS's have switched over to these codenames for their versions with names like Mountain Lion, Jellybean, etc. After a heated discussion with my Dad (who prefers version numbers) about why the names would be done like this, it dawned on me to ask if there was a (valid) UX reason for it vs. just a marketing/trademark one.  I don't think this idea has been explored all that well, just because we have such scattered implementations of it -- Android and Ubuntu are the only ones who use alphabetized naming conventions as a reinforcement of version numbering, Apple uses it for OSX (Snow Leopard, Lion...) but not for iOS (iOS4, iOS5...), and Windows isn't consistent about this stuff at all (XP, then Vista, then 7, then 8...)  Is there any rhyme / reason to all of this? How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$ How to derive this formula?   $$n^2=\sum_{k=1}^n(2k-1).$$ How to horizontally center an element How can I horizontally center a &lt;div&gt; within another &lt;div&gt; using CSS? &lt;div id=&quot;outer&quot;&gt;   &lt;div id=&quot;inner&quot;&gt;Foo foo&lt;/div&gt; &lt;/div&gt; If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?