Gaussian Definition and 706 Threads

Homework Statement In a youtube video() it is explained how gaussian beams propagate through an optical lens. Using the complex parameter q \frac{1}{q} = \frac{1}{R} - \frac{j\lambda}{\pi n w^2} (with R the radius of curvature), one can use the ABCD matrix to calculate the effect of an optical...

Hi there, I have following problem at hand: (any help would be appreaciated, I have a test tommorrow) 1. Homework Statement Homework Equations The Attempt at a Solution Above is obviously the teachers answer to the problem, what I don't get though; is, where does the 3/2 come from, at r...

Homework Statement Consider a system made up of joining together ##N## beads and ##N-1## springs. The positions of the beads is indicated by ##N## real numbers ##\left\{x_i\right\}_{i=1,...N}.## The Hamiltonian which characterises it is $$\mathcal H =...

Homework Statement Consider the Gaussian Distribution ## p(x) = Ae^{-\lambda(x-a)^{2}} ##, where ## A ##, ##a##, and ##\lambda## are constants. (Look up any integrals you need.) (a) Determine ##A## (I only need help with this (a)) Homework Equations ##\int_{-\infty}^{\infty} p(x)dx = 1##...

Hi all, I study binding of analytes in a platform where I have 10.000 sensors. Theres is one binding event per sensor and I identify it as a sudden positive change in the signal. I do first a control experiment without analytes. I measure the maximum change in the signal for each sensor and I...

Hi all, The Gaussian wave packet is widely discussed in the text. I got the following expression for wave packet in momentum space ##\psi(p, 0) = A \exp\left[-(p-p_0)^2/ (2\sigma_p^2)\right]## with ##A=\sqrt{2\sigma_p/\sqrt{2\pi}}## As my understanding, the corresponding wave packet in...

Homework Statement I'm encountering these integrals a lot lately, and I can solve them because I know the "trick" but I'd like to know actually how the cartesian to polar conversion works: ##\int_{-\infty}^{\infty}e^{-x^2}dx## Homework Equations ##\int_{-\infty}^{\infty} e^{-x^2} = I##...

Homework Statement a) 21.4-nC of charge is placed on a 4.8-m long steel tube with a d = 5.9-cm diameter. What is the magnitude of the electric field as a radial distance of r = d / 3? b) What is the magnitude of the electric field as a radial distance of r = 20 d? I was able to determine the...

Hi everyone, in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like: I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)} where a,b,c are...

Hi, I have a problem in my program and I cannot figure it out. In the last post I had a problem about some arrays, I perfectly resolved it thanks to you, but now my problem is a little bit subtle. I have a subroutine(here I'll post it has a program )that generates random numbers in order to...

This is a seemingly simple question, though I'm not exactly sure where I'm going wrong (if in fact I am going wrong). To start off: you have a 2D un-normalized Gaussian function centred at the origin and with a sigma of 4. If you integrate it over a circle of radius 4 also centred at the...

Hey everyone, I'm hoping I can get some input on this. As some of you may recall, I'm currently in an internship at Fermilab, and I've hit a snag in plotting my data. I'm measuring the intensity of the NuMI neutrino beam at multiple different points to look for discrepancies in the measured...

I have a 2D Gaussian: ## f(x,y) = e^{-[(x-x_o)^2 + (y-y_o)^2]/(2*{sigma}^2)}## which I converted into polar coordinates and got: ## g(r,θ) = e^{-[r^2 + r_o^2 - 2*r*r_o(cos(θ)cos(θ_o) + sin(θ)sin(θ_o))]/({2*{sigma}^2})} ## The proof for how this was done is in the attached file, and it would...

written as title, 1. If X(t) is gaussian process, then Can I say that X(2t) is gaussian process? of course, 2*X(t) is gaussian process 2. If X(t) is poisson process, then X(2t) is also poisson process?

I'm doing an analysis where I have a set of random variables with some known uncertainties (the uncertainties are different for each random variable). The random variable is roughly Gaussian distributed. I'd like to get a meaningful characteristic value and uncertainty for the whole set. I can...

I am attempting to find the determinant using gaussian elimination for the following matrix [4 3 2; 1 7 8; 3 9 3]. I have begun by attempting to form zeros below the diagonal. The answer should be -165, however i keep getting values up to -665. I have tried a variety of times and keep getting...

[Mentor's note: This post was moved from another thread as it raised a new question, off-topic in the originating thread] Albert Einstein in his book Relativity wrote " It is impossible to build up a system(reference body) from rigid bodies and clocks,which shall be of such a nature that...

The ##e^2## of Gaussian units becomes ##\frac{e^2}{4 \pi}## in Lorentz-Heaviside units. Is this the only difference between Gaussian units and Lorentz-Heaviside units?

Given the equation for a Gaussian as: ##z = f(x,y) = Ae^{[(x-x0)^2 + (y-y0)^2] /2pi*σ^2 }## , how would I go about converting this into cylindrical coordinates? The mean is non-zero, and this seems to be the biggest hurdle. I believe I read earlier that the answer is ~ ##z = f(r,θ) =...

I need to evaluate ∫xp e-x2 eix dx from -∞ to ∞ Can someone please give me some pointers on how to do this? I am completely lost. I just need some hints or something.

I attempting to find the determinant using gaussian elimination for the following matrix [1 2 3; 3 2 2; 0 9 8]. I have begun by attempting to form zeros below the diagonal. My first row operation was to make row 2 equal to 3(row 1) - row 2. This gives me [1 2 3; 0 4 7; 0 9 8] . I think i am...

Hallo math helpers , i am trying to understand how one could solve the following integrall: $$\int_a^b \mathcal{N}(f(x_1,...,x_n,t),g(x_1,...,x_n,t)) dt$$, where $$\mathcal{N}$$ is the normal distribution, and $$f(x_1,...,x_n,t): \mathbb{R}^{n+1} \rightarrow \mathbb{R}$$, $$g(x_1,...,x_n,t)...

Hi, so my question is along the lines of the following: If I'm looking at the decay channel H=>ZZ=>4 Muons, why would the resultant distribution for the mass of the parent Higgs be a Gaussian? Is it a case of the peak is the actual value of the Higgs at rest; those of greater mass are Higgs...

Homework Statement The electric field has been measured to be horizontal and to the right everywhere on the closed box shown in the figure. All over the left side of the box E1 = 80 V/m, and all over the right, slanting, side of the box E2 = 400 V/m. On the top the average field is E3 = 260...

I am a Post graduation student and I am doing a project on the conversion of a Gaussian beam into a flat top beam using optical components. I have found that the best component to do the same is a Powell lens. I would like to know more about the equations connecting the input beam diameter...

Homework Statement I'm trying to calculate the electric field through two oppositely charged plates with the same magnitude: http://imgur.com/uXCQqtW Homework Equations Flux = ∫E⋅dA = Qenclosed/ε0 The Attempt at a Solution I understand how the electric field is calculated using S1 and S4...

Homework Statement Evaluate the definite integral below numerically (between limits -1 and 1) using a couple of numerical methods, including Gauss-Legendre quadrature - and compare results. Homework Equations $$ \int{(1-x^2)^\frac{1}{2}} dx $$ "Gauss quadrature yields the exact integral if φ...

Hi all, so I've come across the following Gaussian integral in QFT...but it has a denominator and I am completely stuck! \int_{0}^{\infty} \frac{dx}{(x+i \epsilon)^{a}}e^{-B(x-A)^{2}} where a is a power I need to leave arbitrary for now, but hope to take between 0 and 1, and \epsilon is...

Homework Statement Recall that we have defined the Gaussian ##f_s## by ##f_s (t) = \sqrt{s}e^{-st^2}## and shown that ##\hat{f_s}(\lambda) = \frac{1}{\sqrt{2}}e^{\frac{-\lambda^2}{4s}}##. Show that ##f_3 \ast f_6 (t) = \sqrt{\pi}f_{1/2}(t) = \sqrt{\pi/2}e^{-t^{2}/2}## The Attempt at a...

1. We have two positive point charges (+q) at a distance from each other 2. Goal is to Find electric field at point A halfway between the point charges 3.By logic and summation of fields we should get ZERO 4. But how would I use a gaussian surface (sphere i assume ) to prove at point A...

Hi I am working on a programming assignment that requires me to implement the successive over-relaxation algorithm. We are given the wikipedia page for this: http://en.wikipedia.org/wiki/Successive_over-relaxation. I have read through the wikipedia page for this numerous times but am still...

Homework Statement I have to prove that I(a,b)=\int_{-\infty}^{+\infty} exp(-ax^2+bx)dx=\sqrt{\frac{\pi}{a}}exp(b^2/4a) where a,b\in\mathbb{C}. I have already shown that I(a,0)=\sqrt{\frac{\pi}{a}}. Now I am supposed to find a relation between I(a,0) and \int_{-\infty}^{+\infty}...

Why have the density perturbations of all lengths a relation to the constant radius c/H? I suppose this is the origin for scale invariance and gaussianity.

Hello everybody, nowadays I am working on shaping amlifiers and need guidiance about how to design one and do its simulations before pcb design and manufacturing. When googling about shaping amplifiers i found documents mostly about filters (active and passive), which i know works for rejecting...

So I've seen this type of integral solved. Specifically, if we have ∫e-i(Ax2 + Bx)dx then apparently you can perform this integral in the same way you would a gaussian integral, completing the square etc. I noticed on wikipedia it says doing this is valid when "A" has a positive imaginary part...

when considering the quantum harmonic oscillator, you get that the wave function takes the form psi=ae^{-\frac{m\omega}{2\hbar}x^2} I have been trying to integrate \psi ^2 to find the constant a so that the wave function is normalised, and I know the trick with converting to polar coordinates...

Homework Statement -h^2/2m (sqrt(2b/pi)) e^(-bx^2) d^2/dx^2 (e^(-bx^2)) dx from - to + infinity Homework Equations I tried differentiating e^(-bx^2) twice and it came up weird , I positioned the values and finally cam up with (-2b sqrt(pi/2b)...is there any other way to do it ? The...

I am looking for more information (e.g., reference, the CDF and descriptive stats) about a four-parameter skewed generalized Gaussian (SGG) distribution. I have come across the PDF for this distribution, but with no reference and not a lot of other information. Here is a snippet... On...

Homework Statement I have to determine the radius at which 50% of energy is in a Gaussian profile.Homework Equations The intensity is given by I=Ioe^(-r/2c)^2. This is just a gaussian function ofcourse. The Attempt at a Solution I know c is the standard deviation. I searched through charts...

Given a one-dimensional Gaussian distribution, distributed as following: f (x) = exp (-x ^ 2 / (2q)) / q / √ (2pi) proof which q is the standard deviation Thanks !The standard deviation is defined by: http://www.mathsisfun.com/data/standard-deviation-formulas.html

If I switch 2 rows, do I have to multiple by -1 each time? For example, I have If I switch row 2 and 3, will it become this: Or this? Each time I make a switch, do I have to also put a negative sign? Edit: Not really related to Gaussian elimination, but this is from a Gaussian...

Homework Statement I understand why, using Gauss's law, the net charge within a conductor should be zero at any point. However, when I try making a Gaussian surface that is so small so as to enclose a single proton, I cannot see why the enclosed charge should be zero for that situation as...

Homework Statement A particle of mass m in the harmonic oscillator potential V(x) = (mω2x2)/2 is described at time t = 0 by the wavefunction χ(x, t = 0) = 1/[(2πσ2)1/4] exp[-x2/(4σ2)] What is <E> at time t? Homework EquationsThe Attempt at a Solution <T>+<V>= <E> I've found the expectation...

In John Stillwell's book: Elements of Number Theory, Chapter 6 concerns the Gaussian integers, $$\mathbb{Z} = \{ a + bi \ | \ a, b \in \mathbb{Z} \}$$. Exercise 6.1.1 reads as follows: ------------------------------------------------ "Show that the units of $$\mathbb{Z} $$ are $$ \ \pm 1, \...

Homework Statement make a gaussian elimination in C the one we used on matrix to find x1 x2 and x3 if the matrix is 3x3 Homework EquationsThe Attempt at a Solution #include <stdio.h> #include <stdlib.h> #include <math.h> int main() { int i,j,k,n; float...

https://www.physicsforums.com/attachments/73840 This is the problem I don´t understand, I know I hace to determine the distance z1 which correspond to the widht W1, but I try to solve equation of W1 and W2 for z1 and use the fact z2 = z1 + 0.1m, but I, my question is how to determine the waist...

Homework Statement Let a,b be real with a > 0. Compute the integral I = \int_{-\infty}^{\infty} e^{-ax^2 + ibx}\,dx. Homework Equations Equation (1): \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi} Equation (2): -ax^2 + ibx = -a\Big(x - \frac{ib}{2a}\Big)^2 - \frac{b^2}{4a}The Attempt...

Homework Statement I need to evaluate the following integral: \sqrt{\frac{2}{\pi}}\frac{\sigma}{\hbar}\int\limits_{-\infty}^{\infty}p^2 e^{-32\sigma^2(p-p_0)^2/\hbar^2}\,dp Homework Equations Integrals of the form...

Does anyone know if there is a limitation of the number of basis functions for freq calculation in Gaussian 09? I have 2 freq calculations, one has 879 basis functions and the other one has 1047 basis functions. Gaussian 09 couldn't finish the 1047 basis function calculation no matter how...

Homework Statement Wave function of an electron: ##\frac{1}{N}∫_a^b[e^{ik_0x}(1+\frac{1}{2}e^{i\frac{0.1}{2}x}+\frac{1}{2}e^{-i\frac{0.1}{2}x})]dx## The integrand becomes zero both to the left and right of x = 0 . Let a be the first time it hits zero to the left and b the first time it...