Function Definition and 1000 Threads

The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value? (A) 2x/L^2. (B) 2x^2 / L^2. (C) 2x^2 /L^3. (D) 3x^2 / L^3. (E) 3x^3 / L^3 Ans : D

I started out by rewriting the function as (f(x^2))^(1/2). I then did chain rule to get 1/2(f(x^2))^(-1/2) *(f'(x^2). - I think I need to go further because it is an x^2 in the function, but not sure.

Hallo at all! I'm learning statistic in python and I have a problem to show you. I have this parametric function: $$P(S|t, \gamma, \beta)=\langle s(t) \rangle \left( \frac{\gamma-\beta}{\gamma\langle s(t) \rangle -\beta}\right)^2\left( 1- \frac{\gamma-\beta}{\gamma\langle s(t) \rangle...

I have a problem where I am given the density of states for a Fermion gas in terms of momentum: ##D(p)dp##. I need to express it in terms of the energy of the energy levels, ##D(\varepsilon)d\varepsilon##, knowing that the gas is relativistic and thus ##\varepsilon=cp##. Replacing ##p## by...

In QFT by peskin scroeder page 30 the action of Klein Gordon Operator on propagator (∂2+m2)DR(x-y)=∂2θ(x0-y0)... how to compute this ∂2θ(x0-y0)?

Integral \int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3} Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?

I say the answer is A.

This is more of a conceptual question. To find the horizontal velocity as a function of time for the above wave function, you take its partial t derivative and insert x=4. In other words the function would be -2.4sin(1-12t). Im wondering why you take the partial t derivative and not to partial...

Hey! :giggle: Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$, $f(x):=x^2$. Show that : (i) $f$ is well-defined. (ii) $f(1)=1$, $f(3)=9$ (iii) $f$ does not satisfy the intermediate value theorem (e.g. not on $[1,3]\cap \mathbb{Q}$) For (i) do we just say that $f$ is well-defined from $\mathbb{Q}$...

First off let me say I am a bit confused by this question. Searching for some references I found the following related to the KG propagator, given by (P&S, chapter 2 pages 29, 30) Then they Fourier-transformed the KG propagator Is this what is aimed with this exercise? If yes, could you...

I am having a trouble to understand why the helium's wave function (in which we are ignoring the electric interaction between the electrons, as well the motion and problems that arise in considering the nucleus in the wave function) can be written as the product of the wave function of both...

Solution attempt: We first write ##u(x)=\frac{1}{2}||x||^2## as ##u(x)=\frac{1}{2}(x_1^2+x_2^2+...+x_n^2)## Operating on ##u(x)## with ##\Delta##, we have ##u(x)=\frac{1}{2}(2+2+...+2)## adding 2 to itself ##n## times. So ##\Delta u(x)=n## and the function satisfies the first condition...

Hi everyone. Using the Green function, I want to obtain the density of states of a one-dimensional (linear) lattice. Depending on the problem conditions, we will have an iterative loop with 4,000 data for the energy component and a iteration loop with 2,000 data for the wave number component. In...

As a starting point I immediately thought about the equation: ##\frac{dp^\mu}{d\tau}=qF^{\mu\nu}v_\nu## From this I proceed component by component: ##\frac{dp^0}{d\tau}=qF^{0\nu}v_\nu=q\gamma E_yv_y## ##\frac{dp^1}{d\tau}=qF^{1\nu}v_\nu=q\gamma v_yB_z##...

Hello, I have a question regarding the Taylor expansion of an unknown function and I would be tanksful to have your comments on that. Suppose we want to find an analytical estimate for an unknown function. The available information for this function is; its exact value at x0 (f0) and first...

Hi I just moved into Chapter 16 of Gaddis on Exception, Templates and STL. Seems like it jump a chapter. This is a partial sample from the book: I have to complete the program as shown below but compiler doesn't like it. #include <iostream> #include <functional> using namespace std; int...

I know how to work through this problem but I have a question on the initial separation of the wave function. Assuming ##\psi(\rho, \phi) = R(\rho)\Phi(\phi)## then for the azimuthal part of the wavefunction we have ##\Phi(\phi)=B\left(\frac \rho\Delta cos\phi+sin\phi\right)##, but this function...

Let suppose I have three positive integers ##a, b,c## and one unknown ##x## (##x## is also a positive integer). Solve for the smallest ##x## that satisfies this equation$$\left \lfloor\frac{x}{a} \right\rfloor + \left \lfloor \frac{x}{b} \right\rfloor \geq c$$ where ##\lfloor x\rfloor## is the...

Hi, I have question about finding marginal distributions from 2d marginal pdfs that lead to the probabilities being greater than 1. Question: If we have the joint probability distribution ## f(x, y) = k \text{ for} |x| \leq 0.5 , |y| \leq 0.5 ## and 0 otherwise. I have tried to define a square...

Hi, I am interested in understanding the relationships between Fourier series and Fourier transform better. My goal is 1) Start with a set of ordered numbers representing Fourier coefficients. I chose to create 70 coefficients and set the first 30 to the value 1 and the remaining to zero. 2)...

Hi PF! I have the given data points here data = {{1.92, 0.74}, {2.32, 1.36}, {2.44, 1.88}, {2.52, 2.08}, {2.68, 1.92}, {2.64, 1.4}, {2.46, 0.78}}; and the following plots the correct interpolation Show[{ListLinePlot[{data}, InterpolationOrder -> 3], ListPlot[\[Lambda]cplx1]}] but...

y = 2sin(x) -1≤ sin(x) ≤ 1 -2 ≤ 2sin(x) ≤ 2 so -2 and 2 are the max/min limits but the domain is -π < x ≤ π Do I find the values of x that outputs -2 and 2 and show that they are within the domain ?

Hi, I have a quick question about something which I have read regarding the use of dirac delta functions to represent conditional pdfs. I have heard the word 'mask' thrown around, but I am not sure whether that is related or not. The source I am reading from states: p(x) = \lim_{\sigma \to...

This is my first attempt ...

The assignment states: Pipe A takes 20 hours longer to fill swimming pool than pipe B. Together, pipe A and pipe B can fill the swimming pool in 30 hours. The assignment question I am stuck on is to write a function R(x) that models the combined rate of the two pipes in relation to the time it...

Hello, I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane. This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend...

my thinking was to have everything changed to a function that has cosine only... ##\int_0^{0.5π} \frac {1-cos^2x}{sin x + cos x}dx## ##\int_0^{0.5π} \frac {(1-cos x)(1+cos x)}{(1-cos^2x)^{0.5} + cos x}dx## ... first of all is this integration possible? if so then let me know if i am on the...

It is a rather simple question: In my textbook it writes something like: $$\frac {\partial \Psi} {\partial t}= \frac{i\hbar}{2m}\frac {\partial^2 \Psi} {\partial x^2}- \frac{i}{\hbar}V\Psi$$ $$\frac {\partial \Psi^*} {\partial t}= -\frac{i\hbar}{2m}\frac {\partial^2 \Psi^*} {\partial...

I am trying to call a function declared in a .hpp file and defined in the corresponding .cpp file, from my main.cpp file, but I keep getting an error. From what I have googled it seems as if I am doing this the right way, so I was hoping you guys could help out. Here's my code: #ifndef CHAP_HPP...

At 02:08, this video shows a function that grows from exactly 0 at input x = 0+, up to 1 at ##x=\infty##. Its value and all its derivatives approach 0 as x -> 0. The function is Exp(-1 / x^2). www.youtube.com/watch?v=Wwg_15a0DJo&t=146s Q. : What function would have its value and all...

Let's consider the Taylor power series of a function on real numbers. Some of them represent elementary functions, and some of them represent special functions. The special functions cannot be expressed via finite combination of elementary functions on real or complex numbers. Now, take some...

First thank you for taking your time to take a look at this simple question. And sorry for the informal math language and equations, I hope you guys can understand it. So, depending on the case, I have 2 or 8 simple quadratic functions f(a), f(b), f(c),… f(z). Each a,b,c,…,z have a different...

Hi everyone, We've been looking at Fourier series and related topics in online class, touching upon odd and even periodic extensions. However, we haven't looked at what this translates to for sine and cosine functions - only sawtooth and line examples. So, I'm trying to do my own investigation...

The answer in the textbook writes: $$ f(x) = \frac{1}{4} +\frac{1}{\pi}(\frac{\cos(x)}{1}-\frac{\cos(3x)}{3}+\frac{\cos(5x)}{5} \dots) + \frac{1}{\pi}(\frac{\sin(x)}{1}-\frac{2\sin(2x)}{2}+\frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}\dots)$$ I am ok with the two trigonometric series in the answer...

Hi, I have a question about a homework problem: I am not sure why I do not seem to get the same answers when using different methods. Question: Given transfer functions G(s) = \frac{s - 1}{s + 4} and C(s) = \frac{1}{s - 1} , find the state space models for those systems. Then find the...

Here is the solution I have been given: But I really don't understand this solution. Why can I just add these two exponential factors (adding two individual partition...

For ##x=-1## to be an *horizontal* inflection point, the first derivative ##y'## in ##-1## must be zero; and this gives the first condition: ##a=\frac{2}{3}b##. Now, I believe I should "use" the second derivative to obtain the second condition to solve the two-variables-system, but how? Since...

Hello all, Is this statement true ? Is every increasing monotonic function in a closed interval also continuous ? How do you prove such a thing ? Thank you !

Dear all, I am trying to figure out if a non continuous function is also not bounded. I know that a continuous function in an interval, closed interval, is also bounded. Is a non continuous function in a closed interval not bounded ? I think not, it makes no sense. How do you prove it ? Thank...

Perhaps use the definition of continuity, partial differentiability?

Hi, PF This is the quote: "If ##m## is an integer and ##n## is a positive integer, then 6. Limit of a power: ## \displaystyle\lim_{x \to{a}}{\left[f(x)\right]^{m/n}} ## whenever ##L>0## if ##n## is even, and ##L\neq{0}## if ##m<0##" What do I understand? -whenever ##L>0## if ##n## is even: ##m##...

Let $a$ and $b$ be two positive integers. Prove that the integer $ a^2+\Bigl\lceil \dfrac{4a^2}{b}\Bigr\rceil$ is not a square. (Here $\lceil z \rceil$ denotes the least integer greater than or equal to $z$.)

We know that the non-relativistic propagator describes the probability for a particle to go from one spatial point at certain time to a different one at a later time. I came across an expression (lecture notes) relating ##\Psi(x,t)##, an initial wave function and the propagator. Applying the...

Reading the classical Feynman lectures, I encounter the formula(19.53) that gives the radial component of the wave function: $$ F_{n,l}(\rho)=\frac{e^{-\alpha\rho}}{\rho}\sum_{k=l+1}^n a_k \rho^k $$ that, for ##n=l+1## becomes $$ F_{n,l}=\frac{e^{-\rho/n}}{\rho}a_n\rho^n $$ To find ##a_n## I...

for reference you can see JS in formulajs.info/functions. lognormdist use by call : formulajs.LOGNORMDIST(value, mean, stdev, true) logpearsondist use by call : formulajs.NORMSDIST(z, true) anybody can help?

Hi all, I am trying to figure out a way to simplify this problem to give the image of the map. I have not seen this function before and I am having trouble figuring out what the image should come out as. I have tried graphing u and v separately as a function of x and y in R3 and both surfaces...

Suppose I have a function $$f(x) = \lim_{\eta \rightarrow 0} \int_{-\infty}^{\infty} d \zeta \frac {g(\zeta)}{x - \zeta + i \eta}$$ and suppose ##g(\zeta)## is a continuous (maybe even differentiable) function. Can ##f(x)## have complex poles of the form ##a + ib## with ##b## not an...

Hi I need help I'm lost Assignment: A function given by a logical expression Y = A.B.D + A.not (C) .D + A.not (B) .C.D + A.D write in the truth table.

for example, I want to know velocity of a person when time is equal to t, that person start running from 0m/s (t=0s) to max velocity of 1m/s (t=1s). I am thinking that this is like rain droplet that affected by gravity and drag force, where force is directly proportional to its velocity, to make...

I am currently trying to compute the Green's function matrix of an infinite lattice with a periodicity in 1 dimension in the tight binding model. I have matrix ##V## that describes the hopping of electrons within each unit cell, and a matrix ##W## that describes the hopping between unit cells...