Gaussian Definition and 706 Threads

Homework Statement $$ \begin{pmatrix} -1&3&0\\ 2&0&-1\\ 0&-6&1 \end{pmatrix} $$ Finding the ImT basis of this The Attempt at a Solution I got it down to $$ \begin{pmatrix} 1&0&-1/2\\ 0&1&1/6\\ 0&0&1 \end{pmatrix} $$ I know that by the principle of having pivots as the only non-zero...

Homework Statement Find the normalization constant N for the Gaussian wave packet \psi (x) = N e^{-(x-x_{0})^{2}/2 K^{2}} Homework Equations 1 = \int |\psi (x)|^{2} dx The Attempt at a Solution 1 = \int |\psi (x)|^{2} dx = N^{2} \int e^{-(x-x_{0})^{2}/K^{2}} dx Substitute y=(x-x_{0})...

Homework Statement Apply Gaussian elimination to the following matrix 2 -3 0 3 4 -5 1 7 2 -1 -3 5 I understand how to get the answer. The Attempt at a Solution My question is why doesn't the following method work: Switch Z over into the first column, move row 1 to row 3, then exchange rows...

Homework Statement I'm having difficulty solving the following integral. \int_{-\infty}^{\infty} x^{4}e^{-2\alpha x^{2}} \text{d}x Homework Equations \int_{-\infty}^{\infty} e^{-\alpha x^{2}} \text{d}x = \sqrt{\frac{\pi}{\alpha}} \int_{-\infty}^{\infty} x^{2}e^{-\alpha x^{2}}...

Homework Statement Two bars, each 30cm long, and each having a resistance of 2-ohms, are connected to a 1,500 volt battery. The bars are attached to each other with 3 insulating springs, each having a spring constant of 9N/m. The two bars are initially at rest, 4cm apart. The switch is...

Homework Statement A cloud of electrons are drifting from a negative plate to a positive plate after being liberated by a laser pulse, (separated by a distance z = 10cm with an original potential difference of 15V) at an instant in time the centre of the cloud has traveled 25mm from the...

Homework Statement I will not post a specific problem, but rather, I would like to ask a general question. Say I am given a Gaussian wavepacket (function psi(x,t) ) and asked to find the expectation value for x-squared and momentum squared. Now, x-squared is rather straightforward...

Homework Statement Check the energy-time uncertainty principle for: \Psi(x,0)=Ae^{-ax^2}e^{ilx} using the observable x. Homework Equations \Delta{E}\Delta{t}\geq\hbar/2\\ \Delta{E}=\sigma_H=\sqrt{<H^2>-<H>^2}\\ \Delta{t}=\frac{\sigma_x}{\frac{d<x>}{dt}} The Attempt at a...

Hi everyone, I always have trouble on decomposing number into irreducible factors inside Gaussian integer field. I keep trying to express number as product as (a+bi)(c+di), and trying to solve a,b,c,d inside of integers (Z), then see if they are irreducibles, which of course end of very messy...

Homework Statement I am reading about electrodynamics, and in different formula, there is a SI form and a Gaussian form. Homework Equations What is the relationship in the different unit system for c or B or E and other relative variables? The Attempt at a Solution sometimes...

So we were taking measurements for an experiment in our radio astronomy lab. For the first part of the experiment, we recorded the intensity of a far away point source ( the signal from a TV satellite was used for the point source ) detected by commercial satellite dish and receiver. When we...

I need to factorise 70 into primes, how do I go about this? So far I have 2,5,7 as primes in Z. So I suppose I need to factorise these in Z[i]? 2 = (1+i)(1-i) How do I go around doing the other two, is it possible that they're primes in Z[i]? Edit: I have a corollary where if p is a prime...

Hey ^^, new here but I already have a question haha Does anyone here know how the coefficient (x-μ)^2 was derived in the following equation: σ^3=(1/√2∏)∫(1/σ)*(x-μ)^2*exp((x-μ)^2)/(2σ^2)) I know the general equation for density probability is (1/σ)*exp((x-μ)^2)/(2σ^2)) but in this case...

Homework Statement Consider two long coaxial metal cylindrical tubes, with radii a and b and length L. (You may assume a,b<<L. Also a<b.) Suppose the inner cylinder is given a charge +Q and the outer cylinder a charge -Q. Using Gauss' Law, compute the electric field for all r between a and...

Homework Statement Hi, the question is from a piece of coursework and before hand we were asked to find the Fourier transform G(K) of the function g(x)= e^(-∏(x^2)) (where g(x)= ∫ G(K)e^2∏ikx dx (integral from -∞ to ∞)). We were told to find G(K) by forming a differential equation in H(K)...

Hi folks, I'm trying to get from the established relation: $$ \int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} = a^{-2}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$ to the similarly derived: $$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = 3a^{-4} \int_{-\infty}^{\infty}...

Hello Everyone, I am a PhD student working in the field of mineral and material processing. One part of my project involves use of DFT calculations to study adsorption of small to mid sized molecules on mineral crystals. I am using Gaussian 09 simulation package and currently I am in the...

Homework Statement This is problem 2.22 from D.J. Griffiths Introduction to Quantum Mechanics A free particle has the initial wave function: \Psi(x,0)=Ae^{-ax^{2}} Find \Psi(x,t). Hint Integrals of the form: \int_{-\infty}^{\infty}e^{-(ax^{2}+bx)}dx can be handled by completing the square...

This has to do with number theory along with group and set theory, but the main focus of the proof is number theory, so forgive me if I'm in the wrong place. I've been struggling to understand a piece of a proof put forth in my book. I know what the Gaussian integers are exactly, and what a...

Show that N(a+bi) = even => a+bi divisible by 1+i So, N(a+bi) = a2+b2 = even so 2 divides a2+b2 Write 2 = (1+i)(1-i) so we have 1+i divides a2+b2 so 1+i divides either (a+bi) or (a-bi) if 1+i divides a+bi we are done what if 1+i divides a-bi though? Thats where I'm stuck!

This is inspired by Kardar's Statistical Physics of Particles, page 45, and uses similar notation. Homework Statement Find the characteristic function, \widetilde{p}(\overrightarrow{k}) for the joint gaussian distribution: p(\overrightarrow{x})=\frac{1}{\sqrt{(2\pi)^{N}det...

X = (Y(t))^2 where Y(t) is zero mean Gaussian process and correlation function R_YY = exp(-λ|τ|) i want to check if X is weakly stationary.So i guess for the first part, i checked if mean is constant σ^2=R_YY = exp(-λ|0|) = 1 E(X^2) = μ^2+ σ^2 = 1 since μ is zero and σ = 1 I wanted to check if...

Homework Statement In this video; the uploader states at 0:47 that there are three rules. 1. Can swap rows 2. Can multiply a row by a constant 3. Can add a multiple of one row to another Do these rules apply only to rows? Can they be applied to columns? The uploader has not stated...

Let's say I have a Gaussian sphere 1 light year across with synchronized clocks and sensors all over its surface. All clocks are co-moving, not accelerating, and the spatial curvature is negligible. If I have only one charge inside the Gaussian sphere, 1 centimeter from its surface for an entire...

If I have an oscillating charge inside of a sphere, will the integral of E(t), where t=proper time of the sphere, over the sphere's surface area result in a value of electric flux equal to the value of the charge?

Question: A solid sphere of radius R has a non-uniform charge distribution of p=Ar^2, where A is constant. Find total charge Q within the volume of the sphere. p=roe p=Q/dV EdA=qenclosed/Enaught Can you use Gauss' Law for this problem when sphere is solid? If so, how? Since p...

Homework Statement Consider a one-dimensional wave packet with \phi(p) = A \Theta ((h/2\pi) / d -|p-p0|) Determine the constant A and find the spatial wave function ψ(x). Ignore temporal evolution. Homework Equations The Attempt at a Solution Honestly this is a little embarrassing, but the...

Hi I have a stationary gaussian process { X_t } and I have the correlation function p(t) = Corr(X_s, X_(s+t)) = 1 - t/m, where X_t is in {1,2,..,m}, and I want to simulate this process. The main idea is to write X_t as sum from 1 to N of V_i, X_t+1 as sum from p+1 to N+p V_i and so on, where...

So I have come across this integral identity in Krall's Principles of Plasma Physics (right after equ. 6.4.4 in the 1st edition) and I have not been able to show the identity is true. The reason that I would like to understand the integral is that I am trying to solve a similar problem to the...

If I have a spread of electrical charges contained inside a Gaussian surface, and if I cause those electrical charges to move at relativistic speeds, the electric fields of those charges should be subject to relativistic contraction. What happens then to electric flux that cuts through that...

Dear all, I have a problem in understanding how to bound a Gaussian distribution. LEt me describe the problem at hand: Let's say that we have a Gaussian distribution in the x-coordinate and a Gaussian distribution in the y-coordinate. Further, assume that the independent random variables x...

Dear all: When I was just reading about how Gaussian wave packets allow us to derive(because it has) the lowest limit on uncertainty. So what would be the (mathmetical?) derivation to just obtain that limit before proving/identifying that it's true for Gaussian packet?

Hi, are there two non isomorphic surfaces with the same gaussian curvature? thank's Hedi

how can the gaussian wavepacket presents a physical picture of the origin of position-momentum uncertainty?

Homework Statement Find the Gaussian integral: I = \int_{-\infty}^{\infty} e^{-x^2-4x-1}dx (That's all the information the task gives me, minus the I=, I just put it there to more easily show what I have tried to do) 2. The attempt at a solution I tried to square I and get a double...

Homework Statement What is the Gaussian elimination matrix for the vector for the vector v=(2,-10,16,2)? The Attempt at a Solution I don't understand the question. What is a "Gaussian elimination matrix for a vector?" I know that Gaussian elimination is a method for solving a system of...

I am not sure how to do the following homework question: Suppose X1, X2, ... are independent Gaussian variables with mean zero and variance 1. Consider the event that X1 + X2 + ... + X2n ≥ 2na wth a > 0 Compare the chance of observing this event in the following two ways: (i) by...

Going over my Lecture Notes my Lecturer as Started with Show that a Gaussian Distribution Corresponds to a CTS random variable. Then she has i) Taken the f(x) = [p.d.f] and shown a) f(x) >= 0 for all x member of real numbers. b) Integral over all real numbers = 1 ii) Found the M.G.F then...

Can anyone tell me the basic difference between Maxwell and Gaussian distribution? Thanks in Advance

Homework Statement The spectroscopic observation of an interstellar gas cloud shows emission lines H (λ=6563 A), Hβ λ=4862 A) e OI (λ=1304 A), each one with a gaussian profile of the kind: g(\nu)=A e^[-\frac{(\nu-\nu_0)^2}{2\sigma^2}] Per Hα si osserva σ_{Hα} = 1.2 \times 10^{10} Hz...

In "Classical Theory of Fields", by Landau & Lifgarbagez, they give, for example, the force on a charged particle by a magnetic field as: F = \frac{ev}{c} x H where H is the magnetic field intensity. Now, normally written in SI units, this expression would use B and no factor of 1/c...

Hi all! Let's start from the begin to see where I get lost. Extrinsic curvature defines the way an object relates to the radius of curvature of circles that touch the object (a couple of further nicer definitions come from physics, for the moments I am not mentioning them), and intrinsic...

Hi all! I'm curious as to how one would go about actually computing this integral \int_0^\pi \exp(\frac{-x^2}{2c^2})\sin(\frac{m\pi x}{2}) dx I start off by using integration by parts but I am unsure how to solve this integral v = \int_0^\pi \exp(\frac{-x^2}{2c^2}) dx Any help...

Homework Statement If you have a simple Gaussian diagram of spherical shape, and arrows pointing out from the centre indicating [E] how do you know if [E] is constant or if it varies? Is it because the arrows are of slightly different lengths? (Though all eminating from the centre and...

Homework Statement Use the variational method with a gaussian trial wavefunction ψ(x) = Ae^{\frac{-a^{2}x^{2}}{2}} to prove that in 1 dimension an attractive potential of the form shown, no matter how shallow, always has at least 1 bound state. *Figure is of a potential V(x) that has a minimum...

Let x(u,v) be a coordinate patch. Define a new patch by y(u, v) = c x (u, v) where c is a constant. Show that K_y = \frac {1}{c^2}K_x where K_x is the gaussian curvature calculated using x(u, v) and K_y is the gaussian curvature calculated using y(u, v). my book expects us to use the...

I need to solve three coupled differential equations. The equations are as follows: dE1/dz = f(E2,E3) dE2/dz = f(E1,E3) dE3/dz = f(E1,E2) Where E1,E2,E3 represents field amplitudes. In case of plane waves these amplitudes will be constant in the transverse direction therefore i can...

How to show that the variance of the gaussian distribution using the probability function? I don't know how to solve for ∫r^2 Exp(-2r^2/2c^2) dr .

Homework Statement I'm re-hashing a problem from my notes; given the wave function \psi(x)=Ne^{-(x-x_0)/2k^2} Find the expectation value <x>. Homework Equations The normalization constant N for this is in my notes as N^2=1/\sqrt{2 \pi k^2} N=1/(2\pi k^2)^{(1/4)} It should be...

I am studying the Gaussian distribution and am doing one of the problems for practice. The problem states that the standard deviation is equal to 15 and the actual value recorded in the experiment is 385.0. It then asks what is the probability that a single measurement lies in the range of the...