Gaussian Definition and 772 Threads

Gaussian is a general purpose computational chemistry software package initially released in 1970 by John Pople and his research group at Carnegie Mellon University as Gaussian 70. It has been continuously updated since then. The name originates from Pople's use of Gaussian orbitals to speed up molecular electronic structure calculations as opposed to using Slater-type orbitals, a choice made to improve performance on the limited computing capacities of then-current computer hardware for Hartree–Fock calculations. The current version of the program is Gaussian 16. Originally available through the Quantum Chemistry Program Exchange, it was later licensed out of Carnegie Mellon University, and since 1987 has been developed and licensed by Gaussian, Inc.

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  1. Einj

    Gaussian functional integral with constant operator

    Hello everyone. What it the result for a Gaussian functional integral when the "matrix" is nothing but a number? Mathematically speaking is the following true? $$ \int \mathcal{D}\phi e^{-\int d^3k f(k) |\phi(k)|^2}\propto \left(f(k)\right)^{-1/2} $$ Here ##f(k)## is just a function of k, not...
  2. pretty_fly

    Designing and Simulating Shaping Amplifiers for Detector Applications

    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...
  3. T

    Gaussian integral w/ imaginary coeff. in the exponential

    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...
  4. 2

    How to Normalize the Quantum Harmonic Oscillator Wave Function?

    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...
  5. T

    Integration seems gaussian but the answer does not match

    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...
  6. G

    Skewed Generalized Gaussian Distribution

    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...
  7. P

    How Do You Compute Gaussian Functional Integrals in Quantum Field Theory?

    I am new to path integral and struggling with the computation involving Gaussian functional integrals. Could anyone show me the steps of computing the following integral? \int D \phi e^{-S}, where S = \int dx~d \tau [(\frac{\partial \phi}{\partial x})^2+2 i \frac{\partial \theta}{\partial...
  8. U

    Cubic phase modulation of a gaussian beam

    Hi all, I'm reading a paper on light sheet microscopy, i won't go into detail here but they are generating the light sheet by use of an Airy beam. They are using a term i don't understand when describing how the airy beam is generated. Quote "An Airy beam can be generated through the Fourier...
  9. N

    Determining radius at which 50% of energy is in the profile.

    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...
  10. C

    Calculate the standard deviation of Gaussian distribution ,thanks

    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
  11. P

    Finding determinant through Gaussian elimination

    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...
  12. R

    Gaussian surface surrounding only a proton inside a conductor

    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...
  13. P

    A gaussian wavefunction of the harmonic oscillator

    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...
  14. Math Amateur

    MHB Units of the Gaussian Integers, Z[i]

    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, \ \pm i \ ."...
  15. D

    Simple C programming for gaussian

    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...
  16. D

    How Do You Calculate the Waist Radius and Location of a Gaussian Beam?

    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...
  17. homer

    How Do You Compute the Complex Gaussian Integral I = ∫ e^(-ax^2 + ibx) dx?

    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...
  18. A

    Efficiently Solve Gaussian Integrals with our Homework Help Guide

    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...
  19. A

    1000 Basis functions limitation - Gaussian 09?

    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...
  20. B

    Gaussian Wave Packet: Solve Homework Equations

    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...
  21. J

    How to Solve the Tough Gaussian Integral with a Constant in the Exponential?

    Homework Statement I'm trying to solve the Gaussian integral: \int_{-∞}^{∞}xe^{-λ(x-a)^2}dx and \int_{-∞}^{∞}x^2e^{-λ(x-a)^2}dxHomework Equations I can't find anything online that gives the Gaussian integral of x times the exponential of -λ(x+(some constant))squared. I was hoping someone here...
  22. M

    Is there a table of Abel transforms for Gaussian distributions available?

    Hello. I'm wondering if anyone has a table of transforms showing the result of an Abel transform on a Gaussian distribution. I have been unable to find the solution to this. Many thanks for any help. I'm reconstructing an an image from a picture that fits a Gaussian very well, hence I'm hoping...
  23. T

    Electric Field Zero on Gaussian Surface with No Charges

    why is the electric field zero , if the closed gaussian surface doesn't enclose any charges. but,, if the charges are present at a distance outside the conductor?? and why isn't the electric field zero inside the gaussian surface, if there are charges distributed on the surface of the...
  24. D

    Euclidean Algorithm Gaussian Integers

    Hi, Just wondering when using the Euclidean Algorithm to find gcd of 4+7i and 1+3i. Where does 2 and 2+i come from in the follwoing? 4+7i = 2*(1+3i)+(2+i) 1+3i=(1+i)*(2+i) +0? I know you didvide them to get (5-i)/2 and then take closest Gaussian integer then not sure where to go.
  25. A

    Integral of Gaussian function, for squared x

    Homework Statement I am trying to compute an integral, as part of the expected value formula (using a Gaussian PDF) \int_{-∞}^{∞} (x)^2 p(x) dx Where p(x) is the Gaussian probability density function: \frac{1}{\sigma \sqrt(2 \pi)} \exp(\frac{-x^2}{2 \sigma^2}) My aim after this is...
  26. J

    What is the solution to the Gaussian integral?

    Homework Statement I am asked to evaluate ##\displaystyle\int_{-\infty}^{\infty} 3e^{-8x^2}dx## Homework Equations I know ##\displaystyle\int_{-\infty}^{\infty} e^{-x^2}dx = \sqrt{\pi}## The Attempt at a Solution based on an example in the book it seems a change of variables...
  27. Greg Bernhardt

    Understanding the Thin & Gaussian Lens Equations

    Definition/Summary A lens is most often used to focus light rays from an object to form an image. It can also be used to collimate light rays from an object, or to bring a collimated beam of light rays to a focus. Equations See "Extended explanation" for definitions of terms. The well-known...
  28. Greg Bernhardt

    Understanding Gaussian Beams: Definition, Equations, and Parameters

    Definition/Summary A Gaussian beam is an electromagnetic wave, usually a laser beam, with a Gaussian cross-sectional irradiance pattern. The Gaussian irradiance profile results in minimal spreading due to diffraction effects. The spot size w represents the radius or half-width at which the...
  29. M

    Use Gaussian elimination with back substitution?

    Use Gaussian elimination with back substitution to solve the following system: x1+x2+x3=1, x1+2x2+2x3=1, x1+2x2+3x3=1. The answer is (1, 0, 0) and I know how to solve the problem but I just don't know if I should use bracket or the big parentheses for this type of problem when I solve...
  30. G

    MHB Joint Probability Density from two Gaussian Distributions

    I've been reading the following paper entitled "An Improved Algorithm to Generate a Wi-Fi Fingerprint Database for Indoor Positioning": An Improved Algorithm to Generate a Wi-Fi Fingerprint Database for Indoor Positioning In Part 3.3 (Step 6), it states: Use the fingerprint database to...
  31. V

    Gauss' Law and a Gaussian Sphere

    Homework Statement A Gaussian Sphere with a radius of 1m surrounds an unknown charge at the center. At this surface a uniform outward directed electric field is 1 N/C. Use Gauss' Law to calculate the amount of charge enclosed by the sphere. Homework Equations E = q/4∏εor^2 The Attempt at a...
  32. A

    Simulation of a Gaussian beam in an optical waveguide

    I am trying to simulate a Gaussian beam through an optical waveguide having a circular cross-section in matlab. I am familiar with the theory of modes in an optical fiber and can analytically calculate the evolution of the beam by breaking down the beam into a sum of infinite modes. However, I...
  33. S

    Does Gaussian function give bound states for a particle?

    Hello everyone. I was yesterday asked in an interview to draw a gaussian curve. I drew. And then they asked in what region would this give rise to bound states? I am really confused how to conclude if a function gives bound state or not. Please help. Thanks.
  34. M

    Electric Field of a charged sphere with cylindrical gaussian surface

    So the problem statement is: A conducting solid sphere (R = 0.167 m, q = 6.63·10–6 C) is shown in the figure. Using Gauss’s Law and two different Gaussian surfaces, determine the electric field (magnitude and direction) at point A, which is 0.00000100 m outside the conducting sphere. (Hint: One...
  35. U

    Solving Equations with Gaussian Elimination: Troubleshooting Mistakes

    Homework Statement When solving 2 equations using gaussian elimination, can we divide one equation by the other? Can you help me find where I went wrong? Homework Equations 4x + 2y = 14 2x-y=1 The Attempt at a Solution
  36. facenian

    Problem with Gaussian Integral

    I'm reading a book on Path Integral and found this formula \int_{-\infty}^{\infty }e^{-ax^2+bx}dx=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}} I Know this formula to be correct for a and b real numbers, however, the author applies this formula for a and b pure imaginary and I do not understand why...
  37. D

    MHB Linear Quadratic Gaussian (LQG) regulators

    How do LQG regulators work? I have read the Matlab page about them, Wikipedia, and a few schools notes on them but it isn't either clear to me or they are not adequately explaining how they work. All see is that we want to control something giving a quadratic cost. Is there are more robust...
  38. D

    MHB Kalman filter and simulating Gaussian noise

    Water level assumed constant. Static state model \(P_0 = 1000\), system noise \(Q = 0.0001\), measurement noise \(R = 0.1\). I want to use Kalman filtering to solve this problem. I know how to do this but I need to generate \(\mathbf{y} = y\) using Gaussian random noise. \(P_0\) is also call...
  39. V

    Quantum Physics: Gaussian Wave Packets

    Homework Statement This is the problem sheet that I am solving at the moment: 2. The attempt at a solution I have already solved 17. Here is my solution to 17: Now I am working on 18. I am trying to show that the probability density function is conserved. I.e the integral of the...
  40. skate_nerd

    Is there a formula for this gaussian integral

    Is there a formula for this gaussian integral: $$int_{-\infty}^{\infty}{x^4}{e^{-a(x-b)^2}}dx$$ I've tried wikipedia and they only have formulas for the integrand with only x*e^... not x^4e^... Wolframalpha won't do it either, because I actually have an integral that looks just like that...
  41. M

    Gaussian signal, extract uniform distribution of values

    Hello, From an offset zener diode breakdown circuit, I have collected a set of bytes from an ADC. The values distribute normally as integers between 0 and 1024 with a mean of 512. I would like to use the data to create a set of random integers that distribute uniformly. So far, I have...
  42. skate_nerd

    MHB General formula for this weird Gaussian integral?

    Is there a formula for Gaussian integrals of the form $$\int_{-\infty}^{\infty}{x^n}{e^{-a(x-b)^2}}dx$$ I've looked all over, and all I could find were formulas saying $$\int_{-\infty}^{\infty}{e^{-a(x-b)^2}}dx=\sqrt{\frac{\pi}{a}}$$ and...
  43. K

    Computation about Gaussian and Dirac Delta Function

    I have a Gaussian distribution about t, say, N(t; μ, σ), and a a Dirac Delta Function δ(t). Then how can I compute: N(t; μ, σ) * δ(t > 0) Any clues? Or recommender some materials for me to read? Thanks!
  44. B

    What is Known About Zero-th Gaussian Periods of Type (N,r)?

    Hi, I'm new to this subject and wondering if anything is known specifically on the zero-th Gaussian periods of type (N,r), where N is a product of distinct primes and r = p^s is a power of a prime. I know there are some very general results out there, but I haven't seen this so far. Thanks...
  45. R

    Comparing curves using gaussian process regression

    Hi guys, I have run multiple simulations on networks that are all slightly perturbed from each other. They produce slightly different curve outputs onto an x-y graph which I need to now analyse (it has been about 5 years since I did statistical analysis hence why I am here). A couple of the...
  46. A

    Understanding the Relationship Between Gaussian Math and Graph Width

    For a gaussian of the form: exp(-ax2) How is a related to the width of the graph? I can see that if I compare with a normal distribution a = 1/2σ2. Is that correct?
  47. C

    Gaussian - Resuming a calculation

    I heard that if Gaussian crashes, you can just resume the calculation by loading the incomplete log file. How do I do that, I tried opening the incomplete .log file with GaussView and Gaussian and neither work (get error messages). Do I need to create a checkpoint file or something? This time I...
  48. G

    Definition of a Gaussian wave packet for a Initial State

    Hi :) I'm reading a didactic paper and the author defined the initial state ket as |\Phi_{in}> = {\int}dq\phi_{in}(q)|q> where q is a coordinate and \phi_{in}(q) = <q|\Phi_{in}> = exp\left[\frac{-q^{2}}{4\Delta^{2}}\right] I don't know if I'm missing something but isn't this definition a...
  49. Choisai

    Difference between Gaussian and Airy

    What is the difference between a Gaussian beam profile and an Airy disk? Both seem to be pretty much the same to me. An Airy disk looks like this: And a Gaussian beam profile looks like this: So they seem more or less the same. What is the difference between the two?
  50. craigi

    Many Worlds, Indistinguishability, Pascal's Triangle and the Gaussian

    Suppose we have a quantity which can take discrete equally spaced values. Iteratively, we can increase or decrease this quantity by one quantum, splitting into two new worlds each time. After multiple iterations we have some indistinguishable worlds, as described by Pascal's triangle. As the...
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