Function Definition and 1000 Threads

Instinct tells me to just plug in the number, say the limit is zero, and be done with it. But at the same time, while reading the statement from the "Relevant equations" section of this post, I cannot feel but feel some doubt as to whether or not this is the right approach. I mean, only the...

Recently in college, we have started learning python. I found that the print statement can accept multiple variables. For example, if I have variables x1, x2 and x3, I can write print(x1, x2, x3) in python. Something similar exists in Matlab as well. I have learned Java for nearly four years...

Fermions such as the electron and proton can be described by wave function in momentum and in position, and it is possible to get the momentum wavefunction from space wave function and vice versa by Fourier Transform. what about photons? can photons be described by position wave function? If...

217 $f(x)=\begin{cases} \dfrac{(2x+1)(x-2)}{x-2}, &\text{for } x\ne 2 \\[3pt] k, &\text{for } x=2 \\ \end{cases}$$(A)\,0\quad(B)\,1\quad(C)\,2\quad(D)\,3\quad(E)\,5\quad$

Probability density function plays fundamental role in qunatum mechanics. I wanted to ask if there is any analogous density function in classical mechanics. Obviously if we solve Hamilton equations we get fully deterministic trajectory. But it should be possible to find function which shows...

Summary: In the past, physicists talked of the phenomenon of "wave function collapse" very freely, whereas now there seems to be some reservation about it. Why? In reading older popular physics literature, physicists used to talk about "wave function collapse" freely and more often...

I am experimenting with using a radiometer as an approximate indicator of pressure in my homemade high vacuum system, running a small turbo pump. I am interested in the relationship between pressure and vane rotation speed, with light intensity being constant. I have only been able to find...

There are meaningful ways to assign values to things like 1 - 1 + 1 + ... or 1 - 2 + 3 - 4 + ... In a similar spirit, is it possible to assign a value to the integral of a function like this: ##f(x)=x*sin(x)## or this one: ##g(x)=Re(x^{1+5i})## (Integrals from some value, say zero, up...

## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ## ## u_y = -6xy## and ## v_x = -6xy## To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x## Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0 However...

I have been playing around with Taylor expansion to see if I can get anything out but nothing is jumping out at me. So any hints, suggestions and preferably explanations would be greatly appreciated as I’ve spent so so long messing around with it and I need to move on... But as always, thank you

Hello everyone. I am currently using the pca function from MATLAB on a gaussian process. Matlab's pca offers three results. Coeff, Score and Latent. Latent are the eigenvalues of the covariance matrix, Coeff are the eigenvectors of said matrix and Score are the representation of the original...

Hello, From wikipedia, this is the partition function for a "classical continuous system": This is the pillar of classical statistical physics, but it can be seen as a mere kind of "mathematical transform" . It can be used even without thinking to statistics or temperature. If we focus only...

I am not able to understand what the question asks of me in Q75, part a)

I am reading Zettili’s “Quantum Mechanics: Concepts and Applications” and I am in the section on functions of operators. It starts with how ##F(\hat A)## can be Taylor expanded and gives the particular and familiar example: $$e^{a \hat A} = \sum_{n=0}^\infty \frac{a^n}{n!} \hat A^n...

In Peskin book they calculate the QED Vertex Function named Gamma which depend on two scalar functions called form factors F1(q^2), F2(q^2). they are calculating F1(q^2) and they do not really finish calculating for what I understand. They are just showing the Infrared divergent is canceled by...

Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following $$ 0< |x-a| < \delta $$ From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...

First, let me introduce the notation; given a Hamiltonian ##H## and a momentum operator ##\vec{P}##, and writing ##P=(H,\vec{P})##. Let ##|\Omega\rangle## be the ground state of ##H##, ##|\lambda_\vec{0}\rangle## an eigenstate of ##H## with momentum 0, i.e. ##\vec{P}|\lambda_\vec{0}\rangle=0##...

There are no restrictions for ##a,b,f_1,f_2##. One solution is the first order Taylor series expansion of course with ##f_1(a)=a,f_2(b)=b##, but are there any other solutions? I tried the Bhaskara formula but I couldn't express it in this form.

I mean the first question has derivative form and the second is linear form so what the difference here in steps of converting both to transfer function... please need some ellaboration to make sure i am solving correctly or not... is it correct to apply the same rule on both: Transfer function=...

My questions are now... Do the steps of converting this space to transfer function include any laplace ? or just we do get [SI-A]-1 and then transfer function is = C* [SI-A]-1 * B As [1 0] * [s-1/det -0.5/det ; 0.5/det s-0.5/det] * [0; 1] = -0.5/s^2+s+0.5 I mean do we need any laplace after that...

I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me? ##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\ I(R_j) =...

We always think in terms of isolated particles. It's better to analyze it with solids. If wave functions were just calculational tools. Molecules like the following still interact by wave functions, right? So how can it be calculational tool? And if it is, then what model do you use to...

I tried to apply the chain rule $$X_{ik} = \frac {\partial U}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}} \frac {\partial x_i}{\partial \xi_{ik}} = \frac {\partial U}{\partial x_{i}} $$ and I got the force x-component of the force acting on ##P_i## I guess. but I do not know what...

I came across an example of a solution to finding the potential of a charged layer using the Green function (here, pdf). The standard algorithm for finding the Green function by boundary conditions for many problems is understandable: \begin{align*} G_\mathrm{Left} = Ax+ B \\ G_\mathrm{Right} =...

In interpretations where the wave function represents something real, like Many worlds, Copenhagen with objective wave function and spontaneous objective collapses. I'd like to understand which of them has true non-locality. First. Is Many Worlds not having true non-locality due to the...

I am studying a beginner's book on QFT. In a chapter on electromagnetic form factors, the authors have written, using normalized states, $$\begin{eqnarray} \langle \vec{p'}, s'| j_\mu (x) |\vec{p}, s \rangle \ = \ \exp(-i \ q \cdot x) \langle \vec{p'}, s'| j_\mu (0) |\vec{p}, s \rangle...

We have a periodic function ##f: \mathbb{R} \rightarrow \mathbb{R}## with period ##T, f(x+T)=f(x)## The statement is the following: $$\frac{1}{T}\int_0^T f(x)dx =0 \implies \frac{1}{T}\int_0^T\frac{d}{dx} f(x)dx =0$$ Can you give me a hint on how to prove/disprove it? The examples I tried all...

Hello! I am facing a difficulty into developing a multivariable function of a dependent variable "x". Let's assume that "x" is a function of 6 independent variables a,b,c,d,e,f,g. From experimental data i have developed 6 functions, each representing how "x" changes by each of the paremeters...

So I'm having trouble understanding the physics behind evaporative cooling. This is what I know: I want to cool some water so I nebulize it and I let an air flow (coming from the outside) pass through this mist of water. Now some water has to evaporate. Here I am stuck because I don't understand...

I first took out the variation of conductivity along the radius of cylinder.Also we know that J=sigmaE.Therefore i have to find variation of E also.But how will i find that as potential is also not given.Help.

If for a field say ##f=xyz## ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any ##(x,y,z)## of choice??

Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right), \end{equation} with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...

I have calculated the normalization constant, but I'm struggling with the discontinuities in the derivatives of the wave function. Due to the symmetry, it should suffice to consider the first two cases. The results should be (according to WolframAlpha): \left( \frac{\partial^{2}}{\partial...

Hi. If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?

If I'm trying to solve the problem of a particle in free space (H = P2/2m ). the eigenfunctions of the Hamiltonian cannot be normalized. now assume I have a legitimate wave function expressed in terms of the eigenfunction of H and I want to measure its energy. what will happen to the...

I'm trying to determine if a certain bicycle tire size will fit my bike, and that determination is based on the inflated diameter (or width) of the tire. As such, I'm trying to come up with a formula that will give me the diameter of a bicycle tire as a function of the tire's carcass width and...

Here is a picture of these plots from a paper: When I try to reproduce the 3rd graph above (yellow line below), I get sharp discontinuities: Those jump discontinuities should not occur, and the function should never rise to the high value of the two other plots. So, what could be the cause...

I understand how to reach $$\int_0^\phi \frac{d\theta}{\sqrt{1-k^{2}sin^{2}\theta}}=\sqrt \frac g l t$$ from physics but from there I don't get how to turn that into this new (for me) sn(u) form.

We know from first law of thermodynamics for a closed system that ##dE##=##\delta Q## -##\delta W## , my question is that for a closed adiabatic system net heat transfer =0 this mean net change in energy = work done , does that mean for an adiabatic system work done is a point function as...

What exactly is this equation telling me? How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE? I'm extremely sorry if this question comes off ignorant.

I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun...

Problem Statement: Determine whether f is continuous at c. (see image for piecewise function f) EDIT: Sorry if it is a little blurry that is x^3 in the numerator of the rational function and x^2 in the denominator Relevant Equations: Basic understanding of limits My work: Since the...

Because the Taylor series centered at 0, it is same as Maclaurin series. My attempts: 1st attempt \begin{align} \frac{1}{1-x} = \sum_{n=0}^\infty x^n\\ \\ \frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n\\ \\ \frac{1}{x^2} = \sum_{n=0}^\infty (1-x^2)^n\\ \\ \frac{1}{(2-x)^2} =...

Hi, I'm reading "Wave Physics" by S. Nettel and in chapter 3 he introduces the Green's function for the 1-dimensional wave equation. Using the separation of variables method he restricts his attention to the spatial component only. Let ##u(x)## be the spatial solution to the wave equation and...

I want to compute the gradient of some smooth function at many points by taking the value of the function at point x(i) subtracted from the value of the function at point x(i+1) and then divide the result by ( x(i+1)-x(i) ). My function has a struct as an argument and within that struct I have...

Red line being the function and blue an approximation of the derivative. Does it look right?

##r,\theta,\phi## are the usual spherical polar coordinate system. ##\int_v\nabla•(\frac{\hat r}{r})dv## over a spherical volume of radius ##R## reduces to ##\int_s(\frac{\hat r}{r})•\vec ds=4\pi R## Now ##r## runs from 0 to ##R,\theta## from 0 to ##\pi## and ##\phi## from 0 to ##2\pi##. In...

The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...