I am working through Shankar's Principles of Quantum Mechanics and I am confused about his derivation for the propagator of a free particle from the Schrodinger Equation (Section 5.1).  He states that the propagator is given by the following equation (in terms of an integral over $p$):  $$U(t) = \int_{-\infty}^\infty \lvert p\rangle\langle p\rvert \exp(-ip^2t/2m\hbar)dp\tag{5.1.9}$$  and in Exercise 5.1.1 he states that the propagator, in terms of an integral over $E$, is:  $$U(t) = \sum_{\alpha = \pm} \int_0^\infty \biggl[\frac{m}{\sqrt{2mE}}\biggr]\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}dE \tag{1}$$  However, from 4.3, the propagator appears to be defined as a sum over $E$ (the eigenvalues of the Hamiltonian):  $$U(t) = \sum_{E} \lvert E\rangle\langle E\rvert e^{-iEt/{\hbar}}\tag{4.3.13}$$  which, when $E$ is continuous, can presumably be written in integral form:  $$ U(t) = \int_{-\infty}^{\infty} \lvert E\rangle\langle E\rvert e^{-iEt/\hbar}dE \tag{2}$$  I am having difficulty reconciling these two forms of the propagator (1 and 2 above), as they appear to contradict. Surely, based on the definition given in 4.3, the propagator for a free particle should be given by:  $$U(t) = \sum_{\alpha = \pm} \int_0^\infty \lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}dE$$  and the extra $\bigl[\frac{m}{\sqrt{2mE}}\bigr]$ term should be present in the $p$ version of the equation?  Note 1:   I am aware of an existing question   that is similar. However, I find the accepted answer not very satisfactory. It's quite brief and doesn't seem to explain the whole story. It suggests there should be an additional term in the propagator integrand to account for a 'Density of States'.   So, what is happening here? Has Shankar glossed over this, in his earlier description of the propagator? If so, why would he do that? Does he address this point anywhere else in his book (as it seems fairly important)?  Can anyone explain more clearly why this Density of States is required and where it comes from? I thought the degeneracy of $H$ was being accounted for by summing over the +/- values of $p$. Why would an extra term be required?  Note 2:   Also, to clarify, I am aware that the extra term comes from changing the integral from $dp$ to $dE$:  $$p = \pm (2mE)^{1/2}$$ $$dp = \frac{m}{(2mE)^{1/2}}dE$$  However, I am confused as to why this factor appears in the integral over $E$, rather then the integral over $p$.
I am going through Shankar's Principles of Quantum Mechanics and am having trouble finding the free particle propagator $U(t)$ that satisfies $$\lvert\psi (t)\rangle = U(t)\lvert\psi (0)\rangle$$ due to the degeneracy of the $E$ eigenkets. Shankar says that if the Hamiltonian has degenerate eigenvalues, we change the propagator equation from $$U(t) = \sum_{E} \lvert E\rangle\langle E\rvert e^{-iEt/{\hbar}}\tag{4.3.13}$$ to $$U(t) = \sum_{E} \sum_\alpha \lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar} $$ where $\lvert E, \alpha \rangle $ are the orthonormal eigenkets for the $E$ eigenspace. This makes sense to me, because it seems like whenever we deal with degeneracy to use a projection. He also says for a Hamiltonian with no degeneracy to change the sum into an integral, so I imagine that for a Hamiltonian with a continuous, degenerate spectrum we would have this propagator: $$ U(t) = \sum_\alpha\int_\Re dE\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}$$  I tried applying this to the free particle and had some trouble. In the book, after solving for the eigenkets and eigenvalues, $\lvert E\rangle = \lvert p\rangle$ and $p = \pm\sqrt{2mE}$, he chooses the propagator as a function of the momentum eigenvalues $p$ so that $$U(t) = \int_\Re dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m).\tag{5.1.9}$$ I understand how he got to this, but I'm having some trouble because this doesn't appear to be the same expression that I have above. Splitting the integral up, we have $$U(t) = \int_{-\infty}^0dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ The left integral can changed to an integral over $-p$, and the bounds switch from $0$ to $\infty$ such that: $$U(t) =  \int_{0}^\infty d(-p)\lvert -p\rangle\langle -p\rvert \exp(-ip^2t/2m) + \int_0^\infty dp\lvert p\rangle\langle p\rvert \exp(-ip^2t/2m)$$ Now we can substitute and interchange $\lvert p\rangle = \lvert E, + \rangle$,  $\lvert -p\rangle = \lvert E, - \rangle$, $E = p^2/2m$, and $dp = \pm mdE/\sqrt{2mE}$ to get: $$U(t) = \sum_{\alpha = \pm} \int_0^\infty dEm/\sqrt{2mE}\lvert E, \alpha\rangle\langle E, \alpha\rvert e^{-iEt/\hbar}. $$ This is the result that we are asked to prove in Exercise 5.1.1. Why is this not the same as the first expression for the propagator? Where did the extra $ m/\sqrt{2mE} $ come from? Am I just wrong in assuming that that is the general expression for a Hamiltonian with continuous and degenerate eigenvalues?
Your favourite application of the Baire Category Theorem Reference style links validating as improperly formatted code Here is an interesting piece of code that my fellow team members were just having a slightly heated discussion about...    Dim fred As Integer    If True Then fred = 5 : fred = 3 : fred = 6 Else fred = 4 : fred = 2 : fred = 1   After executing the above code snippet, what is the value of fred?  Try not to cheat and debug the code.  This is a highly contrived code example that started out as an example of using the colon with an If statement, but then someone decided to take it upon themselves to proffer a result for fred.  UPDATE: I would not normally write code like this and this snippet only serves as an example. As it so happens, this question originated from a discussion involving the creation of a coding standards document for our team. How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$? How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$ ? i know that its "1", but why? How can i calculate this?  Thank you very very much =) Ever since I upgraded to 15.10, fdisk -l reports 16 ram disks (/dev/ram0 ... /dev/ram15). I'm a bit unsure what those are needed for. Is it safe to delete them? If not, how can I get rid of that fdisk output? Prove equality between binomial coefficients. Sounds of supersonic objects Would you, as a stationary observer, hear a sound in reverse when the source of the sound travels with twice the speed of sound? Of course, he wouldn't hear anything at all before the airplane passed by and the sound reaches him, but what about after? Does it reverse due to the Doppler effect? Uniform distribution on the surface of unit sphere write some AMPscript to skip a record if its empty I have a data extension with a CustomerId(PK) that has multiple Productid's as Productid1, ProductId2, ProductId3...ProductId10.   Some of the Productid's are empty for some Customers and I'm trying to write some AMPscript if the productIdx is empty to show the next one.  I'm trying to display 6 products in an email and at the moment when I preview the email, if from the first six Products one doesn't have a value(is empty) I get the error "The subscriber preview failed to generate. Review the details, correct all issues, and try again. Product1 not found"  I want to skip the product that is NULL and go to the next one available.  %%[ SET @customerId = [customerId]  SET @customer_rows = LookupRows("PRL","customerId",@customerId) IF Rowcount(@customer_rows) &gt; 0 THEN SET @customer_row = ROW(@customer_rows, 1)  SET @product_count = 0  SET @productId1 = FIELD(@customer_row, "productId1")  IF NOT EMPTY(@productId1) THEN SET @product_rows = LookupOrderedRows("PI_UK", 1,"Normal_SalePrice Desc", "ProductCode", @productId1)  IF RowCount(@product_rows) &gt; 0 THEN  SET @product_count = ADD(@product_count,1) ELSE  SET @product_count = ADD(@product_count,0) ENDIF ELSE SET @product_count = ADD(@product_count,0) ENDIF    SET @productId2 = FIELD(@customer_row, "productId2") IF NOT EMPTY(@productId2) THEN SET @product_rows = LookupOrderedRows("PI_UK", 1,"Normal_SalePrice Desc", "ProductCode", @productId2) IF RowCount(@product_rows) &gt; 0 THEN SET @product_count = ADD(@product_count,1) ELSE SET @product_count = ADD(@product_count,0) ENDIF ELSE SET @product_count = ADD(@product_count,0) ENDIF  SET @productId3 = FIELD(@customer_row, "productId3") IF NOT EMPTY(@productId3) THEN SET @product_rows = LookupOrderedRows("PI_UK", 1,"Normal_SalePrice Desc", "ProductCode", @productId3) IF RowCount(@product_rows) &gt; 0 THEN SET @product_count = ADD(@product_count,1) ELSE SET @product_count = ADD(@product_count,0) ENDIF ELSE SET @product_count = ADD(@product_count,0) ENDIF  and so on till ProducId10...   I have uploaded a photo with the PRL Data extension The total number of subsets is $2^n$ for $n$ elements In my probability book, it says that to count the total number of subsets of n elements is a process of $n$ stages with binary choice of either adding this element to the subset or not to add it. Therefore, the total number is $$2^n$$  But, for instance, we have 3 elements, according to this formula, there are 2 to the power of 3 elements, namely 8, which are $${\emptyset},A,B,C, AB, AC, BC, ABC$$  However, I have a hard time of imagining the process or N stages binary choice that form this many subsets. Can anyone explain/help me to understand it? I mean, ABC, if we are making the choice of A, put it in or do not put it in, exactly which subset are we choosing to put in or not? Thank you. Is a constant condition more costly than switching shaders? The Riemann zeta function $ζ(s)$ is defined for all complex numbers $s ≠ 1$ with a simple pole at $s = 1$. It has zeros at the negative even integers, i.e., at $s = −2, −4, −6, ...$.   My question: How one can obtain these roots. I am wondering if my proof is correct? Thank you for whoever willing to take a look at it for me.     Proof $\liminf E_k \subset \limsup E_k $   If $\{E_k\}_{k=1}^\infty$ is a sequence of sets, we define \begin{align*} \limsup E_k &amp; = \bigcap_{j=1}^\infty\left(\bigcup_{k=j}^\infty E_k\right)\\ &amp;= \bigcap_{j=1}^\infty(E_j \cup E_{j+1} \cup \cdots)\\ &amp;= (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots. \end{align*}  Therefore, $\limsup E_k$ consists of those points in $\mathbb{R}^n$ which belong to infinitely many $E_k$. \begin{align*} \liminf E_k &amp; = \bigcup_{j=1}^\infty\left(\bigcap_{k=j}^\infty E_k\right)\\ &amp;= \bigcup_{j=1}^\infty(E_j \cap E_{j+1} \cap \cdots)\\ &amp;= (E_1 \cap E_2 \cap \cdots) \cup (E_2 \cap E_3 \cap \cdots) \cup \cdots. \end{align*} Therefore, $\liminf E_k$ consists of those points in $\mathbb{R}^n$ which belong to all $E_k$ for $k \geq k_0$.  Now consider $x \in \liminf E_k$, then $x \in E_1 \cap E_2 \cap \cdots$, or $x \in E_2 \cap E_3 \cap \cdots$ and so on, that is to say, $x \in E_{k_0} \cup E_{k_0+1} \cup \dots$ for some $k_0$. More specifically, $x \in E_k \forall k \geq k_0$. Therefore, $x \in (E_1 \cup E_2 \cup \cdots) \cap (E_2 \cup E_3 \cup \cdots) \cap \cdots$, so we showed $x \in \limsup E_k$. Projecting sp objects in R I have a number of shapefiles in different CRSs (mostly WGS84 lat/lon) that I'd like to transform into a common projection (likely Albers Equal Area Conic, but I may ask for help on choosing in another question once my problem gets better-defined).  I spent a few months doing spatial stats stuff in R, but it was 5 years ago.  For the life of me, I cannot remember how to transform an sp object (e.g. SpatialPolygonsDataFrame) from one projection to another.  Example code:  P4S.latlon &lt;- CRS("+proj=longlat +datum=WGS84") hrr.shp &lt;- readShapePoly("HRR_Bdry"), verbose=TRUE, proj4string=P4S.latlon)  # Shapefile available at  #   http://www.dartmouthatlas.org/downloads/geography/hrr_bdry.zip  #   but you must rename all the filenames to have the same  #   capitalization for it to work in R   Now I have a SpatialPolygonsDataFrame with appropriate projection information, but I'd like to transform it to the desired projection.  I recall there being a somewhat unintuitively-named function for this, but I can't remember what it is.  Note that I do not want just to change the CRS but to change the coordinates to match ("reproject", "transform", etc.).  Edit  Excluding AK/HI which are annoyingly placed in Mexico for this shapefile:  library(taRifx.geo) hrr.shp &lt;-    subset(hrr.shp, !(grepl( "AK-" , hrr.shp@data$HRRCITY ) |                                      grepl( "HI-" , hrr.shp@data$HRRCITY )) ) proj4string(hrr.shp) &lt;- P4S.latlon Is the proposal creator guaranteed moderator status? When an Area 51 site goes into public beta is the person who created the proposal one of the pro tem moderators or not?  They are talking on Area 51 about combining all the sports proposals into , including one I proposed. Would I still be one of the pro tem moderators on the merged site?  Is the proposal creator afforded any elevated status or perk? different font while using url I'm having troubles while using the url command in my bibliography.  I mean:  1) \bibitem{[8]} {\url{http://projects.webappsec.org/w/page/13246989/Web20Application20Security20Statistics\#Tableofcontent}}  2) \bibitem{[5]} Justin~Clarke ``SQL Injection Attacks and Defense'', Syngress Publishing, Inc.2009   Unfortunately font 1) is different from 2).  What should I do in oder to get the same font for both items?   I'd like the font used in 1)  to be identical to the one displayed in 2) Expected number of games played in a game with 'win by two' rule. Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability p of winning each game (independently). They play with a “win by two”” rule: the first player to win two games more than his opponent wins the match. What is the expected number of games played? My attempt: I use Markov chains. Let $S$ denote that starting state, $W$ and $L$ denote the states where Calvin and Hobbes win respectively. Then, I have $$\mu_S = 1+ p\mu_W+q\mu_L $$ $$\mu_W = 1+ q\mu_S $$ $$\mu_L = 1+ p\mu_S $$ where $\mu_j$ denote the expected number of games from state $j$. Solving the system, I get the solution $\mu_S = \frac{2}{1-2pq}$ which I think is correct. I however would like to know how to approach it with geometric distribution instead of Markov chains. I read one solution where the number of games is given as $1+\text{Geom}(p^2+q^2)$ but I do not understand it. My understanding is that for the game to stop, the required probability is $p^2 + q^2$ since this is the probability that either Calvin or Hobbes win two games in a row. How to use this information in a Geometric random variable? Thanks. How to append something to an array? How do I append an object (such as a string or number) to an array in JavaScript? I have the following $n\times n$ matrix:  $$A=\begin{bmatrix} a &amp; b &amp; \ldots &amp; b\\ b &amp; a &amp; \ldots &amp; b\\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ b &amp; b &amp; \ldots &amp; a\end{bmatrix}$$  where $0 &lt; b &lt; a$.  I am interested in the expression for the determinant $\det[A]$ in terms of $a$, $b$ and $n$.  This seems like a trivial problem, as the matrix $A$ has such a nice structure, but my linear algebra skills are pretty rusty and I can't figure it out.  Any help would be appreciated. How do I recover my password in Pokemon Go?