Delta Definition and 1000 Threads

Dear all, I was revising on a bit of tensor calculus, when I stumbled upon this: $$\delta^i_j = \frac{\partial y^i}{\partial x^\alpha} \frac{\partial x^\alpha}{\partial y^j}$$ And the next statement reads, "this expression yields: $$|\frac{\partial y^i}{\partial x^j}|...

Is there a time domain function whose Fourier transform is the Dirac delta with no harmonics? I.e. a single frequency impulse

Something i ran into while doing hw Homework Statement starting with \int{dx} e^{-ikx}\delta(x) = 1 we conclude by Fourier theory that \int{dk} e^{+ikx} = \delta(x) Now, i try to compute \int{dk} e^{-ikx} (I've dropped the normalization factors of 2\pi. I believe no harm is done by...

So part of the idea presented in my book is that: div(r/r3)=0 everywhere, but looking at this vector field it should not be expected. We would expect some divergence at the origin and zero divergence everywhere else. However I don't understand why we would expect it to be zero everywhere but...

How can I find the derivative of a step function?

It is pretty straight forward to prove that the Kronecker delta \delta_{ij} is an isotropic tensor, i.e. rotationally invariant. But how can I show that it is indeed the only isotropic second order tensor? I.e., such that for any isotropic second order tensor T_{ij} we can write T_{ij} =...

Hi every one, I am examining a prototype device that is designed to analyse current from an electrochemical O2 sensor (current source), The sensor will output 1.124 uV per PPM (cross 47 ohms @0.023 uA), and has acuracy of +- 2 PPM, with max 1000 PPM. it ueses 16 bit Sigma Delta ADC with...

I'm trying to explicitly show that \varepsilon^{0 i j k} \varepsilon_{0 i j l} = - 2 \delta^k_l I sort of went off the deep end and tried to express everything instead of using snazzy tricks and ended up with \begin{eqnarray*} \delta^{\mu \rho}_{\nu \sigma} & = & \delta^{\mu}_{\nu}...

Integrate[f[qs] DiracDelta'[qs (1 - 1/x)], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] Integrate[f[qs] DiracDelta'[qs - qs/x], {qs, -\[Infinity], \[Infinity]}, Assumptions -> 0 < x < 1] This is on Mathematica 8 for windows. The results differ by a sign. They are effective...

Homework Statement Background: The problem is to find the uncertainty relationship for the wave equation for a delta function potential barrier where ##V(x)=\alpha\delta(x)##. Check the uncertainty principle for the wave function in Equation 2.129 Hint: Calculating ##\left< p^2 \right> ##...

As part of a physics calculation, I have the following integral $$\int d \bar x a^{\sigma} \left[-\partial_{\mu}\left(\frac{\delta x^{\nu}}{\delta a^{\sigma}}\right) (\partial_{\nu}\Phi )\frac{\partial L}{\partial (\partial_{\mu}\Phi)} + \partial_{\mu}\left(\frac{\delta x^{\mu}}{\delta...

Homework Statement Delta functions said to live under the integral signs, and two expressions (##D_1(x)## and ##D_2(x)##) involving delta functions are said to be equal if: ##\int _{ -\infty }^{ \infty }{ f(x)D_{ 1 }(x)dx } =\int _{ -\infty }^{ \infty }{ f(x)D_{ 2 }(x)dx }## (a)...

Is it possible to solve a differential equation of the following form? $$\partial_x^2y + \delta(x) \partial_x y + y= 0$$ where ##\delta(x)## is the dirac delta function. I need the solution for periodic boundary conditions from ##-\pi## to ##\pi##. I've realized that I can solve this for some...

How to calculate ##\int^{\infty}_{-\infty}\frac{\delta(x-x')}{x-x'}dx'## What is a value of this integral? In some youtube video I find that it is equall to zero. Odd function in symmetric boundaries.

I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also like to go ahead and apologize for not knowing how to format correctly. I was given that a...

I am self studying the 17th Chapter of "Mathematical Methods for Physics and Engineering", Riley, Hobson, Bence, 3rd Edition. It is about eigenfunction methods for the solution of linear ODEs. Homework Statement On page 563, it states: "As noted earlier, the eigenfunctions of a...

Homework Statement If f(x) = 3x+1 en assume δ > 0. Assume ε>0. Give a δ > 0 with the following property : |x-1|< δ => |f(x) - f(1)| < ε Homework Equations The Attempt at a Solution |f(x) - f(1)| < ε <=> |3x+1-(3*1+1)| < ε <=> |3x-3| < ε <=>...

The question In my work \mu is the mass per unit length, therefore I believe I can say m=\mu\Delta xbecause Michael Fowler from the University of Virginia does the same at http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/AnalyzingWaves.htm (the 2nd line bellow the graph) I start...

I have a PDE of the following form: f_t(t,x,y) = k f + g(x,y) f_x(t,x,y) + h(x,y) f_y(t,x,y) + c f_{yy}(t,x,y) \\ \lim_{t\to s^+} f(t,x,y) = \delta (x-y) Here k and c are real numbers and g, h are (infinitely) smooth real-valued functions. I have been trying to learn how to do this...

I am working on an experiment where I need to measure the Delta T across a heat ex-changer. I have two independent T Type thermo couple, one at the inlet and the other at the outlet. Both tied to an Agilent data logger. I normally put the data into an excel sheet and generate Delta T = T1 - T2...

Homework Statement ##\frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}(E-\alpha\delta(x))\psi(x)=0## Show that ##\psi(x)## is continuous and that first derivative has discontinuity ##\frac{2m\alpha}{\hbar^2}\psi(0)##.Homework Equations The Attempt at a Solution I'm not sure how to show that function...

Homework Statement Prove that ## lim_{x\implies 1} \frac{2}{x-3} = -1 ## Use delta-epsilon. The Attempt at a Solution Proof strategy: ## | { \frac{ 2}{x-3} +1 } | < \epsilon #### \frac{x-1}{x-3} < \epsilon ## , since delta have to be a function of epsilon alone and not include x. I...

Homework Statement How to find out the equivalent capacitance using delta star conversion? Homework Equations Delta star conversion formula of capacitors The Attempt at a Solution Using the formula of resistors but not coming.What is the formula of delta star in capacitors?

Hello Forum! I have this review package for my final full of weird mistakes. Problem is that it is hard for me to know if the solutions are right or not: Could you please look at this problem I attached? They bizarrely switch from HA to HB. Is that just a typo? Then, the sign of DG°...

I am doing a project in Chemistry and I need to use Hess' Law to cancel two equations and if in one equation the NH4NO3 is solid and in the second one the NH4NO3 is aqueous. The equations are: 1: NH4NO3 (s) + HCl (aq) --> HNO3 (aq) + NH4Cl (aq) 2: NH4OH (s) + HNO3 (aq) --> H2O (l) + NH4NO3...

these proofs are always confusing but here's my take on it.. since $x\rightarrow +\infty$ we don't need absolute values and since $ \displaystyle \frac{1}{10^2}=0.01 $ then we could use $N=10$ letting $L=0$ since it is a horz asymptote then we have $ \displaystyle...

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!

Homework Statement I'm looking for the bound energy of a triple delta potential: V(x) = -w \left [ \delta(x-a) + \delta(x) + \delta(x+a) \right ] What is the correct transcendental equation for kappa? Homework Equations My wave function is \psi_1(x) = A e^{\kappa x} for x < -a, \psi_2(x) =...

Sorry if the question seems naive but if we have the Dirac delta function delta(x-y) is it the same as delta(y-x)?? Or there are opposite in sign? And why ? Thank you for your time

let E = episolon and D = delta; the problem is as follows: let f(x) = (2x^2 - 3x + 3). prove that lim as x approaches 3 f(x) = 21, we write |f(x) - 21| = |x^2 + 2x - 15| = |x + 5||x - 3| to make this small, we need a bound on the size of |x + 5| when x is close to 3. For example...

Homework Statement I am studying my lecturer's notes and in this part he uses a delta potential to illustrate a simple example of Fermi's golden rule, that the rate of excitation is ##\propto t##. Homework Equations The Attempt at a Solution I've managed to get the bound states, by solving...

Simplification -- complicated summation involving delta functions Homework Statement \frac{1}{\sqrt{(2^3)}}\sum[δ(k+1)+δ(k-1)]|k> for k=0 to 7 Homework Equations The Attempt at a Solution I am trying to simplify the above expression. I get \frac{1}{∏*\sqrt{(2^3)}} |1>, which is...

Prove: tδ(t) = 0 The answer our TA has given isn't doing it for me: \int dt \delta(t)f(t) = (0)f(0) = 0 I'm wanting to write: t \frac{d}{dt}\int \delta(t) dt = t \frac{d}{dt}(1) = t * 0 = 0 Am I right here? This doesn't make use of a test function. I'm very sloppy with proofs! Thanks for...

Homework Statement Prove the following sequence {an} converges to L=1/2 an = n2/(2n2+n-1) The Attempt at a Solution Given ε>0 we can determine an N∈N so that |an - L|<ε for n≥N. We have: |an-L|=|(n2/(2n2+n-1)-(1/2)| = |(-n+1)/(2(2n-1)(n+1))| I'm not sure what to do once I get to this...

I have an evil TA (who makes the assignments) who likes to give us torturously difficult assignments on stuff we haven't been taught (and in many cases don't even understand conceptually). Homework Statement The input signal, x(t) is a real-valued bandlimited signal with bandwidth W. Find...

I am not understanding something from my textbook. This is related to Fermi's Golden rule. It's about what happens when the matrix element of the perturbation H' ends up being a Dirac delta for chosen normalization. Here is Fermi's Golden rule. \Gamma_{ba} = 2\pi \left|\langle b \mid H'\mid a...

If you ask me define Dirac delta function, i can easily define it and prove its properties using laplacian or complex analysis method. But still i don't understand what is the use of DIRAC DELTA FUNCTION in quantum mechanics. As i have done some reading Quantum mechanics from Introduction to...

Consider the following potential function: V=αδ(x) for x=0 and V=∞ for x>a and x<-a , solve the shroedinger equation for the odd and even solutions. solving the shroedinger equation I get ψ(x)=Asin(kx) +Bcos(kx) for -a<x<0 and ψ(x)=Asin(kx) +Bcos(kx) for 0<x<a is it...

Every time I try to read Peskin & Schroeder I run into a brick wall on page 15 (section 2.2) when they quickly derive the Euler-Lagrange Equations in classical field theory. The relevant step is this: \frac{∂L}{∂(∂_{μ}\phi)} δ(∂_{μ}\phi) = -∂_{μ}( \frac{∂L}{∂(∂_{μ}\phi)}) δ(\phi) + ∂_{μ}...

Every time I try to read Peskin & Schroeder I run into a brick wall on page 15 (section 2.2) when they quickly derive the Euler-Lagrange Equations in classical field theory. The relevant step is this: \frac{∂L}{∂(∂_{μ}\phi)} δ(∂_{μ}\phi) = -∂_{μ}( \frac{∂L}{∂(∂_{μ}\phi)}) δ(\phi) + ∂_{μ}...

Hello! By manipulating Maxwell's equation, with the potential vector \mathbf{A} and the Lorentz' gauge, one can obtain the following vector wave equation: ∇^2 \mathbf{A}(\mathbf{r}) + k^2 \mathbf{A}(\mathbf{r}) = -\mu \mathbf{J}(\mathbf{r}) The first step for the solution is to consider a...

Homework Statement Consider the TISE for a particle of mass m moving along the x-axis and interacting an attractive delta function potential at origin: Part(a): What is the difference between a bound state particle and a free particle? Part(b): Show ##\psi _{(x)} = exp (-|k|x)## is a...

Prove Lim x^2=9. With the epsilon/delta definition of a limit. x->3 My work so far. For every ε>0 there is a δ>0 such that if 0<|x-3|<δ , Then |x^2-9|<ε so, |(x-3)(x+3)|<ε |x-3|* |x+3|∠ε what do I do from here? My book is not very clear (Stewart Calculus 7ed)...

Homework Statement I need to prove for arbitrary functions φ(x) that: \lim_{\lambda \to 0} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx = \varphi(0), which, in the sense of distributions is basically the delta...

Homework Statement Given the delta function -α[δ(x+a) + δ(x-a)] where α and a are real positive constants. How many bound states does it possess? Find allowed energies for \frac{hbar2}{ma} and \frac{hbar2}{4ma} and sketch the wave functions. Homework Equations I know there are three parts of...

If the velocity of the object of 5 kg changed from 5 m/s to 4 m/s. Then is the change in kinetic energy -22.5 J or +22.5J. doesn't change mean final velocity minus initial velocity?

For two body decay, in CM frame, we know that the magnitude of the final particle momentum is a constant, which can be described by a delta function, ##\delta(|\vec{p^*}|-|\vec{p_0^*}|)##, ##|\vec{p_0^*}|## is a constant. When we go to lab frame (boost in z direction), what's the Lorentz...

Find the limit L. Then use the epsilon-delta definition to prove that the limit is L. $\sqrt(x)$ as x approaches 9 I figure out the first part of the question. the Answer is three. Yet I have some difficulty to answer the second part of the question.Thank you Cbarker11

I have been reviewing some details on the Dirac Delta function and I've hit a little bit of a road block with trying to wrap my head around how the Translation/Sifting property of the function is justified. Now according to my text the overall definition is the generalized function with the...