Gaussian Definition and 706 Threads

Okay, I have solved the schrödinger equation numerically by making it dimensionless (though I am still confused about this proces). And then approximating it on a finite interval and solving the resulting eigenvalue equation. This allows me to solve for the wave function of different potentials...

1. A Nd:YAG laser operates at a wavelength of 1064 nm and is used for frequency doubling in the non-linear material LiNbO3. The laser outputs 1 W of power in a TEM00 mode with a beam waist ω0 = 1.0 mm on the output mirror. A lens of focal length of 50 mm is placed immediately outside the...

I'm in a high school pre-calculus class and a statistics class. For the latter, we are given z-tables for some of our tests. I don't like these z-tables. Thus, I decided that a more direct approach (fundamental theorem of calculus) would be more accurate and, more importantly, more fun. My...

I have the integral \int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2} where \xi is a constant. I would like to transform by some change of variables in the form \int_{-\infty}^{\infty}dx F(x) \int_{-\infty}^{\infty}dy G(y) the problem is...

does anyone know how dirac diffusion looks like, i.e. The solution of dirac equation with no potential and initial condition for 1 spinor component psi(x,t0) being delta(x-x0) ? Are the solution gaussian like the schroedinger case ? Thanks.

Homework Statement Consider the Gaussian Distribution ρ(x) = A e^{-λ(x-a)^{2}} where A, a, and λ are constants. Determine the normalization constant A. Homework Equations \int^{∞}_{-∞}ρ(x) dx = 1 The Attempt at a Solution The problem recommends you look up all necessary integrals, so I...

Hello I was wondering If anyone could give intuitive explanations for the multivariate Gaussian distribution function and mahalanobis distance? My professor didn't explain these in probability class, they were only defined... Where did the formula come from? Why is the Gaussian function the...

Homework Statement See image, p(y|θ) is the Likelihood function which has to be rearranged in the form of equation (3). θ is a vector variable. Homework Equations None? The Attempt at a Solution I first expanded the exponent in the original function, equation (2)...

Please help me find some references where public bus headways are described(modeled) as gaussian processes. Best Regards

Hi PF, I have a device (with a microcontroller) which generates random numbers. when I analyzed those numbers using Matlab software I found that it is following a uniform distribution. How can I mathematically (any algorithm?) convert this random output to a gaussian one. Also I would...

We all know about the famous equation: \int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}. How about \int_{-\infty}^\infty e^{-x^4} dx? Or, in general, can we calculate any integral in the form \int_{-\infty}^\infty e^{-x^n} dx, where n is an even counting number?

Hi, having a bit of trouble with this question "In an argon ion laser ( λ = 514nm) the minimum beam waist is 1.0mm and is close to the plane mirror. Calculate the radius of curvature of the beam at the output mirror. 1.15m away" Attempt at a solution: θ = 2λ/pi W02 R = Z + ZR2/Z ZR =...

I have an integral of the form (from -inf to inf): ∫exp((a+ib)x2+icx)dx How do I solve that? I have tried setting y = √(a+ib)(x+ic/√(a+ib)) And you then get an integral of exp(-y2) times a constant. But I don't even know if this is legal since part of the exponentials are complex...

Homework Statement I am looking at finding the Fourier transform of: f(t)=\exp \left[ \frac{-(t-m)^2}{2 \sigma^2}\right] Homework Equations \hat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt The Attempt at a Solution I did it a little differently that my...

Hi all, If I have a Gaussian with the equation: G(x,y) = h*exp(-(x-x0)^2/a -(y-y0)^2/b) where x0, y0, a, b and h are the parameters which may vary, what's the equation for the elliptical contour line at the half-max of G? I'm getting myself confused! Thanks for help

I'm doing my senior project about Compressed sensing, I have some problems with some algorithms. 1)Can anybody help me about Gaussian random matrix, how we can explain it briefly. 2)Does the randn(m,n) building function in Matlab working bases on Probability density function? How does randn...

hey can anyone give me some literature on how work with those programs (Orca, gaussian, gamess) ? I need to do optimization on large quantum system with those programs but at my university i can not find any literature.

I have a book on Gaussian but it mainly just explains how to use the program, it only gives very brief explanations of the theory behind how Gaussian does what it does. I want to gain an understanding of how Gaussian performs the various types of calculations that it can do. I understand the...

I know that \int_{- \infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}. What if I tried evaluating this over the complex plane? If I created a contour that ran along the real axis, then circled around from positive to negative infinity this should not change things as along the curve at...

Hi everybody Any ideas how to simulate a Gaussian stationary process in R language using predefined variance and mean? I have a uni-variate normal distribution for my real life process Thank you

Hi all, I got an idea why noise is having a gaussian pdf but didnot understood why it should be having zero mean. Why can't noise contain a dc content. -Devanand T

I read that B3LYP is a hybrid functional which uses some HF method and some DFT method for its calculations. According to this page: my professor told me that you can set the proportions of each method that the B3LYP uses yourself, so for example you can make it so it uses 70% HF and 30% DFT or...

Homework Statement The random variables X and Y have a joint probability distribution of f_{XY}(x,y)= \frac{1}{2\pi \sigma _X \sigma _Y \sqrt{1-\rho ^2}} \exp \{ \left [ -\frac{1}{2(1-\rho ^2)} \right ] \left [ \left ( \frac{x-\mu _X}{\sigma _X} \right )^2 + \left ( \frac{y-\mu...

I was assigned a multistep task by my professor so I first made the molecule with ChemDraw, then I ran an optimisation using B3LYP/6-31G. I then ran a frequency scan and checked for any imaginary frequencies. There were no negative frequencies and my understanding is that this means that...

I posted this in two forums because the question might be too chemistry related for the people in the calculus forum. I'm a visual thinker so I struggle a bit to get my head around calculus concepts. So as an example, here's a potential energy surface: lets say this represents the structure of...

I'm learning about how Gaussian works and I'm reading about potential energy surfaces which Gaussian uses to calculate very properties of a molecule. Right now I'm reading about optimisations, I understand the concept that to optimise the structure of a molecule, it needs to find the global...

Homework Statement I must find the characteristic function of the Gaussian distribution f_X(x)=\frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2} \left [ \frac{(x- \langle x \rangle )^2}{\sigma ^2} \right ]}. If you cannot see well the latex, it's the function P(x) in...

Homework Statement I have to prove: ∫(-infinity:infinity) cos(pi*v/2L)*e^-((L-L_av)^2/sqrt(2pi)*sigma^2) dL proportional to cos(pi*v/2L_av)*e^-(t/tau)^2 tau is some constant, and sigma << L_av. The Attempt at a Solution i can change the integral to 0:infinity, since sigma <<...

Homework Statement For some Gaussian distribution, let's say e^{-x^2} times a constant, I want to find the expectation value of x^2. In other words, I want to evaluate: \int^{\infty}_{-\infty} e^{-x^2}x^2dx Homework Equations Integration by parts: \int udv = uv - \int vdu The...

Here is a link to a course which i am studying, http://quantummechanics.ucsd.edu/ph130a/130_notes/node89.html#derive:timegauss My problem comes from the k' term attached to Vsub(g) (group velocity). I used the substitution k' = k - k(0), factored out all exponentials with no k'...

Hey, Whenever I read Relativity I get stuck around Gaussian coordinates, i can't seem to find much out about them, do they have another name? does anyone know any good resources explaining them? or am I just in over my head? thanks guys,

Homework Statement let \varphi:\mathbb{Z}[i]\rightarrow \mathbb{Z}_{2} be the map for which \varphi(a+bi)=[a+b]_{2} a)verify that \varphi is a ring homomorphism and determine its kernel b) find a Gaussian integer z=a+bi s.t ker\varphi=(a+bi) c)show that ker\varphi is maximal ideal in...

I think everyone knows that Q(x)= P(X>x) where X is a Gaussian Random variable. Now I was reading about it and it says that Q(x) is bounded as follows Q(x)≤ (1/2)(e-x2/2) for x≥0 and Q(x)< [1/(√(2∏)x)](e-x2/2) for x≥0 and the lower bound is Q(x)> [1/(√(2∏)x)](1-1/x2) e-x2/2 for x≥0 Can...

Homework Statement Homework Equations The Attempt at a Solution Looking at equations 17 and 18, I don't see how that follows. If you substitute infinity for x you're going to get infinity divided by some real number which is infinity

We know that momentum is proportional to k so by adding more waves to localise our particle we are adding more waves with independent momentum values Upon measurement, the gaussian wavepacket must collapse into one eigenstate of momentum, but if we have a very localised packet there will be...

Hi, I'm studying about Chebyshev Quadrature and i found so little and confused information about this. I don't know if Gauss-Chebyshev Quadrature is the same of Chebyshev Quadrature. The only good information that i found was from Wolfram...

Homework Statement for some reason I am having real trouble trying to solve this system. Homework Equations The Attempt at a Solution a=(10,11,4) v1=(2,1,4) v2=(-1,-2,1) v3=(3,3,-1) the qstn does a belong to the span (v1,v2,v3)? so I do gauss elim but I keep running...

Homework Statement Find the convolution of g(x) = e^{-πx^{2}} with itself from -∞ to ∞ using the definition of convolution, not the Fourier Transform. The Attempt at a Solution See my attachment. My professor said that you have to use integration by parts, but I keep getting stuck...

Hello, I have a set of data on an x-y plot in Matlab and I'm trying to calculate the Gaussian distribution of best fit, I only want the right hand side of the Gaussian. I tried applying the least squares method but it gets messy. can you help me?

\int_0^\infty e^{-x^2}dx \int_0^\infty e^{-y^2}dy = \int_0^\infty \int_0^\infty e^{-(x^2+y^2)} dxdy Under what conditions we could do the same for other functions? I don't get how Poisson (or Euler, or Gauss, whoever that did this for the first time) realized that this is true. It looks...

Which Gaussian surfaces have an electric flux of +q/εo through them? I think is b since in c, +q and -q cancel each other out, d is only -q is that the right approach? thanks!

Hello Colleagues, I am curious about a problem in Quantum Mechanics that incorporates the evolution of a Gaussian Wave Packet under the Gravitational Potential. What I am interested in is equation (3) in the following paper: "On the quantum analogue of Galileo's leaning tower...

Is there any circuit for generating Gaussian Noise with a mean and variance. we should have control to change mean as well as variance accordingly. thanks.

I would normally use Gaussian ELimination to solve a linear system. If we have more unknowns than equations we end up with an infinite number of solutions. Are there any real life applications of these infinite solutions? I can think of solving puzzles like Sudoku but are there others?

Is there a closed form expression for finding all the roots of the derivative of a k-component gaussian mixture model?

What is meant by phase of a pulse? (Gaussian or sech or any other pulse) How can we mathemaitcally express it and what is the physical meaning of it? I know about the phase of a wave but not sure about that of a pulse since it is a combination of waveforms of different frequencies. Thanks

Hi I have the following code: lorentz[A_, Ox_, Oy_, FWHM_, x_] := A (1/3.14) FWHM/((x - Ox)^2 + FWHM^2) + Oy; gauss[A_, Ox_, Oy_, x_, C_] := A Exp[-(x - Ox)^2 C] + Oy; Convolve[lorentz[1, 0, 0, 1, x], gauss[1, 0, 0, x, 1], x, y] It takes extremely long time for this to finish -- is it...

I'm trying my very best to understand it, but really, I just couldn't get it. I read four books now, and some 6 pdf files and they don't give me a clear cut answer :( Alright, so this integral; ∫e-x2dx from -∞ to ∞, when converted to polar integral, limits become from 0 to 2∏ for the outer...

λHomework Statement A symmetric confocal resonator with mirror spacing d =16 cm, mirror reflectances 0.995, and n = 1 is used in a laser operating at λ[o] = 1 μm. (a) Find the radii of curvature of the mirrors. (b) Find the waist of the (0,0) (Gaussian) mode. (c) Sketch the intensity...

I put this under statistics because of your knowledge of the Gaussian. I have run into an elementary problem. I was considering what the average value, <x>, is for a Gaussian with an x offset, and got results which don't make sense to me. First, it is obvious that for P1(x)=e^(-(x^2)) the...