In normalising the complete wavefunction of a free partcile; $V(x)=0$  we arrive at  $\int_{-\infty}^{\infty}\Psi_{k} ^\dagger\Psi_{k}dx=|A|^{2}\int_{-\infty}^{\infty}dx=|A|^{2}\left(\infty\right)$  Which implies that this wave function is not normalisable.   Mathematically, it is not normalisable but that's as far as my understanding goes. I would like to see a further and more related explanation for which a non-normalisable wave function implies that a free particle cannot exists in a stationary state and also that such a free particle does not have a definite energy.
The wavefunction $\Psi(x,t)$ for a free particle is given by  $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$  This wavefunction is non-normalisable. Does this mean that free particles do not exist in nature?  Why then do we use free particles $\psi(\vec{x}) = e^{ikz}$, for example, in scattering theory?
Why do two bodies of different masses fall at the same rate (in the absence of air resistance)? I'm far from being a physics expert and figured this would be a good place to ask a beginner question that has been confusing me for some time.  According to Galileo, two bodies of different masses, dropped from the same height, will touch the floor at the same time in the absence of air resistance.  BUT Newton's second law states that $a = F/m$, with $a$ the acceleration of a particle, $m$ its mass and $F$ the sum of forces applied to it.  I understand that acceleration represents a variation of velocity and velocity represents a variation of position. I don't comprehend why the mass, which is seemingly affecting the acceleration, does not affect the "time of impact".  Can someone explain this to me? I feel pretty dumb right now :) Is the universe isomorphic to a universal turing machine? I often think about problems that require an understanding of the very essence of computation and its inherent limitations. So, my questions are as followed:   Is the universe isomorphic to a universal turing machine? Is the universe isomorphic to a universal quantum computer? What evidence do we have within our universe to support an answer to such a question? What corollaries follow from a universe that is or is not isomorphic to a universal turing machine (or universal quantum turing machine)? Is this question really answerable, and what is the current literature on this subject?   I'm mostly interested in the implications and corollaries that follow from a universe that is or is not isomorphic to a turing machine (or universal quantum computer) and the insights that it would provide us with, about the nature of computation. How can I resize a partition with Disk Utility? I have 2 partitions, I reduced the size of one and want to expand the other to fill the now available space.   How do I expand the Lion Partition into the empty grey space beneath Macintosh HD partition? So I am using cryptopp to encrypt a file in a project and I am using AES-GCM. I have a 256bit key and a 128bit ivec and a 128bit tag that I use to encrypt files.  However, I want to know where the authentication tag is stored? I notice if I specify a tag size of 0 then the encrypted file size is the same number of bytes as the original file.  Is the authentication tag simply appended to the end of the file? Is there a way to get trig functions without a calculator? In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?  Sometimes I don't feel right when I can't do things out myself and let a machine do it when I can't.  Or, if you could redirect me to a place that explains how to do it, please do so.  My dad said there isn't, but I just had to make sure.  Thanks. Is it possible to start windows with NUM Lock on? A password is always combination of characters &amp; numbers. I am so much used to num pad that every time windows start I end up entering wrong password just because NUM lock isn't ON by default. Since it's network login, it takes a bit long time to reset &amp; prompt for right password. Is there a way to enable NUM Lock on when Windows loads? This is a problem from Rudin's Functional Analysis:  Show that there is a sequence of functions $\{ f_n \}$ in $X$ (the vector space of all complex functions) such that $f_n \to 0$ pointwise, but if any sequence $\{ \gamma_n \}$ tends to $\infty$, $\gamma_n f_n \not\to 0$.  The hint says to use the fact that the collections of all complex sequences converging to $0$ has the same cardinality as $[0,1]$. Honestly, this confused me more than it helped me.  My main issue is through the method of showing existence: I am not sure if I should attempt to construct an explicit sequence $f_n$, or if I need to appeal to other nonconstructive means such as Zorn's Lemma, etc. I have been trying to construct clever choices of sequences for a good bit now, but it always seems like I will be able to find a sequence that diverges slowly enough to allow that $\gamma_n f_n \to 0$.   Thanks! warning message at command line `sudo` Everytime I try to sudo something at the terminal, I get a warning message:  alexandre@XPS-15Z $ sudo mount file.iso /mnt/ -o loop [sudo] password for alexandre:  no talloc stackframe at ../source3/param/loadparm.c:4864, leaking memory   It's just annoying to get that all the time. How do I solve that? I'm on Ubuntu 14.04 (upgraded from 13.10) Why must local variables have initial values? Is there any other size tires that will fit on a 27 x 1 1/4 rim? Specifying spaces between words How should we deal with duplicate answers? Duplicate questions are "solved" &#8212; it's clear what to do with them, there're tools to deal with them... but what happens when two users post the same answer? This happens a fair bit, particularly on questions with simple answers ("Click this button"-type ones).  Do we upvote both answers, since they are both correct? Do we upvote the first one (although if the answers were posted within a few minutes of each other, after an hour it's impossible to tell as they both show up as "answered 1 hour ago")? Do we upvote the currently highest voted one? Do we upvote one and downvote the other? Do we flag one (seems a little excessive)? Something else? Division ring if and only if it has no proper left ideals. Background image resize in Ubuntu 18.04.1 LTS Recently updated to 18.04.1 and now I cannot resize my background image from fill screen to center image.  How can I resize the image now that 18 doesnt have the quick resize feature? How are questions in the 'hot' tab on a site selected? I know questions on the "hot" tab of a site homepage are based on some sort of views/velocity criteria, but does anyone know the actual algorithm used?  Background: I was considering a feature-request about the "hot" tab, but I don't have enough information to speak about it intelligently. What is the name of this kind of character illustration? I have seen lots of UI designer nowadays including this kind of illustration i want to know the name of this style so that i can also learn and try this in my designs. Difference between static class and singleton pattern? Why the $(n+1)$th prime number cannot exceed the smallest prime divisor of $p_1\dotsc p_n + 1$ Let $p_n$ be the $n$th prime number in its natural order. We want to show that  $$p_{n+1} \leq p_1\dotsc p_n + 1 \tag{1}$$  I know $p_1\dotsc p_n + 1 &gt; 1$, so it has at least one prime divisor, but the text says that $p_{n+1}$ cannot exceed the smallest of these divisors, but I cannot understand why, especially when I imagine $p_{n+1}$ being very large. Although some examples confirm that this statement is true, I cannot make a connection as to why the $n+1$ prime number cannot exceed the smallest prime divisor. Once this has been shown to be true, $(1)$ immediately follows. The "I Get It" Reputation Problem I wrote a question about blocking certain characters in passwords. It became tremendously unpopular, and it seemed like everyone wanted to rain on my parade every day.  Eventually, I "got it," agreed completely with what people were saying to me, and just wanted to withdraw my question so that my reputation didn't slide even further downhill.  Unfortunately, the system is designed such that I can't withdraw my question or redeem myself. Instead, I just get daily reputation attacks. That's a terrible system, and it needs to be changed. Tell rate-limited users how long they have to wait before asking a new question New users are limited to one question per an hour, but once they'll hit this limit, all they see is You can only post once every 60 minutes. I think it would be very appreciate to change / extend this message, adding You need to wait next xx minutes. I saw similar solutions on many forums.  Would it be hard to implement something like that?  EDIT: It could be either added as extension to above mentioned warning message or as red-colored reminder above Title field. Or even as an explanation message, preventing user from clicking on a Ask Question button? Anything would be good, to not let user write the question (can be time consuming) only to hit the wall of limit. This can be a little bit annoying or frustrating, don't you think?