I'm working on a project covering Feynman's Path Integral Approach. I'm having trouble intuitively grasping what motivates the introduction of the expression $e^\frac{iS}{\hbar}$, where S is the action.   All the resources that I have seen (Feynman's book, 1948 article, Shankar's book) don't seem reason this assertion. They simply mention it as a postulate.   Can someone refer me to a good resource that explains the introduction of the term? I'm sure Feynman must have sound reasons to conclude that the complex number one is after has to be of this form!
In the book &quot;Quantum Mechanics and Path Integrals&quot; Feynman &amp; Hibbs state that  the probability $P(b,a)$ to go from point $x_a$ at time $t_a$ to the point $x_b$ at the time $t_b$ is $P(b,a) = \|K(b,a)\|^2$ of an amplitude $K(b,a)$ to go from $a$ to $b$. This amplitude is the sum of contributions $\phi[x(t)]$ from each path. $$ K(b,a) = \sum_{\text{paths from $a$ to $b$}} \phi[x(t)]$$ The contributions of a path has a phase proportional to the action $S$: $$ \phi[x(t)] = \text{const}\ e^{(i/\hbar)S[x(t)]}$$  Why must the contribution of a path be $\sim e^{(i/\hbar)S[x(t)]}$? Can this be somehow derived or explained? Why can't the contribution of a path be something else e.g. $\sim \frac{S}{\hbar}$, $\sim \cos(S/\hbar)$, $\log(S/\hbar)$ or $e^{- (S[x(t)]/\hbar)^2}$ ? Edit: I have to admit that in the first version of this question, I didn't exclude the possibility to derive the contribution of a path directly from Schrödinger's equation. So answers along this line are valid although not so interesting. I think when Feynman developed his formalism his goal was to find a way to quantize systems, which cannot be treated by Schrödinger's equation, because they cannot be described in terms of a Hamiltonian (e.g. the Wheeler-Feynman absorber theory). So I think a good answer would explain Feynman's Ansatz without referring to Schrödinger's equation, because I think Schrödinger's equation can only handle a specific subset of all the systems that can be treated by Feynman's more general principle.
Difference beween wiki page, application page, web part page and publishing page There are lots places in msdn sharepoint forums and blogs where these terms are used and I am confused between them. Can anybody clarify? How can I install Steam? I wish to travel to the US and think I am eligible to do so under the Visa Waiver Program. I found a lot of information out there but am still having some trouble understanding how this works.  What is the US Visa Waiver Program? What is an ESTA? 'this is' or 'these are'? I have the following sentences.        This is our first arrival in the UK.   These are our first arrival in the UK.      I am not sure, which sentence is correct? What is the meaning of double pound symbol (number sign, hash character) ##1 in an argument? How to remove film from stainless steel pan I have an unsightly film around my stainless pan as shown here:    What is causing this to happen?  And how can I remove the film? I have soaked it in soapy water, scraped it with a plastic scraper, and run it in the dishwasher several times, but I can't seem to get rid of the film.   UPDATE: the pan is an All-Clad stainless, fully-clad frypan: Stop Mathematica from giving imaginary solutions How could it possible to factorise $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$? How could it possible to factorize $x^8-1$ in product of irreducibles in the rings $(\mathbb{Z}/2\mathbb{Z})[x]$ and $(\mathbb{Z}/3\mathbb{Z})[x]$?  I'm having a hard time starting the problem. Could anyone help me at this point? Converting NumPy array into Python List structure? I need to convert some files to PDF and then attach them to an email.  I'm using Pear Mail for the email side of it and that's fine (mostly--still working out some issues) but as part of this I need to create temporary files.  Now I could use the  function but it sounds like it creates a file on the filesystem, which isn't what I want.  I just want a name in the temporary file system (using ) that won't clash with someone else running the same script of the same user invoking the script more than once.  Suggestions? Find the number of solutions to $ \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \ldots + \lfloor 32x \rfloor =12345$ Find the number of solutions of the equation    $$\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 8x \rfloor + \lfloor 16x \rfloor + \lfloor 32x \rfloor =12345,$$   where $\lfloor\,\cdot\,\rfloor$ represents the floor function.   My work: I use the fact that $$\lfloor nx \rfloor =\sum_{k=0}^{n-1} \left\lfloor x +\frac kn \right\rfloor.$$ So the equation becomes $$\lfloor x \rfloor +\sum_{k=0}^{1} \left\lfloor x +\frac k2 \right\rfloor +\sum_{k=0}^{3} \left\lfloor x +\frac k4 \right\rfloor +\sum_{k=0}^{7} \left\lfloor x +\frac k8 \right\rfloor   \\ \qquad {}+\sum_{k=0}^{15} \left\lfloor x +\frac k{16} \right\rfloor +\sum_{k=0}^{31} \left\lfloor x+\frac k{32} \right\rfloor   \\ = \lfloor x \rfloor + \left\lfloor x+\frac 12 \right\rfloor + \left\lfloor x+\frac 64 \right\rfloor + \left\lfloor x+\frac{28}{8} \right\rfloor   \\ \qquad {}+ \left\lfloor x+\frac{120}{16} \right\rfloor + \left\lfloor x+\frac{496}{32} \right\rfloor$$  What should I do next? Finding what's using all the space in *nix Which sequence of commands will tell me which files are the largest starting from a particular directory, including all sub directories? I want to know where all the space went.  Preferably just with regular unix'y commands.  If possible, try to keep it compatible with Sun OS 5.10 (perhaps in addition to regular Linux answers, not as a replacement, to keep it as general as possible.) So this is something I just found out about that I find odd:   Opportunity attacks—trigger off of movement that uses the creatures movement (action, move action, bonus action, reaction.)  Sentinel—Whenever a creature within 5 ft of you makes an attack at a creature other than you.   Booming blade—If the target willingly moves.   Mental Prison spell—if it moves, attacks through it, or reaches any part of body out of it.   So the “willingly” is different than other rulings for how effects related to movement works. However what exactly fits willing?   Cause fear—nope magic is having them move. Dissonant whispers—same thing. They have to immediately move.   So the creature has to want to move completely of its own accord.   Crown of Madness—Humanoid creature you choose must make a wisdom saving throw. On a fail it is charmed by you. (Next is flavor text so skipping.) While charmed in this way the target must make an attack on a creature other than itself that you mentally choose on each of its turns before moving.  It can act normally if you choose no creature or if none are within reach.   Crown of madness isn't making them move but doesnt specifically say willing.  And if the ruling for willing is movement not caused by magic what about magic causing situations where you take damage if you don't move. Would a creature shimmering with evocation energy be spared because he moved out of the magical blender that is cloud of daggers? If no, then what is the go-to rule for willing movement that I should be using?  In short, what movement counts as willing, and where can it be found? Is moving out of obvious danger caused by magic, willing? Is moving away from an ally so you don't hit them, willing?   If there isn't any actual ruling on willing, that's fine; I'll go back to running it with my interpretation, so that it has same wording as AOO. If there is an actual 5e term or explanation for willing (that isn't Sage Advice), please let me know. I just don't see why it was necessary to put a term in the spell that isn't used anywhere else. I understand it could have been overlooked since it was in a splatbook or has just been ignored since most people probably assume that there is "Forced " and "Willing" movement, with the willing being defined as: any movement from an action, move action, bonus action, or reaction. In same way AOO work. Make HttpURLConnection load web pages with images Revolution of Koch Snowflake Consider equations defined in $D_x = (0,\infty) \times (0,\infty)$  1) Find the maximal solution to  $$ \dot y = \frac{3t^2}{5y^4} $$ satisfying the initial value problem $y(1) = 2^{\frac{1}{5}}$  I have found the maximum solution to be $((0,\infty), (t^3+1)^\frac{1}{5})$  2) Find the maximal solution to   $$ \dot y = -\frac{2y}{5t}+\frac{3}{5y^4}$$  satisfying the IVP $y(1) = 2^{\frac{1}{5}}$.  (HINT: Use a transformation of the form $t^{\alpha}y(t)$.   I hope someone can explain how to use the transformation to find the maximal solution. What is the differences between &> and 2>&1 There are two forms of redirection the standard output and standard error into standard output. But which one is better? and why the &amp;&gt; is considered the perfect?  I can't find what is the differences so that many tutorials and even bash manual state that  &amp;&gt; is better!   So Why shall I use &amp;&gt; and not 2&gt;&amp;1  Mainly using bash shell    EDIT: Thanks for commentors   Only >&amp; works in csh or tcsh  In ksh only 2>&amp;1 works.  dash use >file 2>&amp;1 redirection only  Then which one to use to ensure my script is compatible with other systems whatever the used shells are! I'd like to install both Ubuntu 12.04 and Windows XP on a Dell laptop, and I was thinking about using a dual boot structure, and using the bulk of my hard drive as empty hard drive space to share files between the two operating systems (so choice of file system type is very important in this set-up).     The kind of partitioning structure I would like to use is  Partition 1 - Ubuntu 12.04 (root) (20GB)  Partition 2 - Ubuntu /home        (20GB)  Partition 3 : Free Space          (560GB)  Partition 4 : Windows XP          (35GB)  Partition 5 : SWAP                (3GB)  (Total Hardrive Capacity is ~640GB)    My question is; what is the best way to go about setting up this kind this system?   Should I install Windows XP first and setup the partitions, and then install Ubuntu which I believe will install the GRUB bootloader for OS booting choice   or   Do I install Ubuntu first, setting up the available partitions and then perform a WIndows install?  Please let me know if there is anything in this setup that I have left out and should know about, including things related to setting particular partitions as logical or primary, and whether the boot partition and the filesystem partition should actually be two separate partitions. I'm a bachelor student majoring in math, and pretty interested in physics. I would like a book to study for classical mechanics, that will prepare me to work through . What books would be good for a beginner? How is Russell’s Paradox equal to the universal set?