Dirac Definition and 859 Threads

Homework Statement show x\frac{d}{dx}\delta(x)=-\delta)(x) using the gaussian delta sequence (\delta_n) and treating \delta(x) and its derivative as in eq. 1.151. Homework Equations the gaussian delta sequence given in the book is \delta_n=\frac{n}{\sqrt{\pi}}e^{-n^2x^2} and eq...

Homework Statement Distribution of matter is given in cylindrical coordinates: \rho(\vec{r})=\frac{1}{\rho}\delta(\rho^2-10\rho+9)\delta\left(\frac{z^2-a^2}{z^2+a^2}\right)\delta(\cot(\phi)) where a>0 is a constant. Find the complete mass of the object. Homework Equations The...

Hello, Dirac Delta Function is defined as the function that its amplitude is zero everywhere except at zero where its amplitude is infinitely large such that the area under the curve is unity. Sometimes it is used to describe a function consists of a sequence of samples such as...

Hi, I'm trying to work my way through Halzen and Martin's section 5.4. I'd appreciate if someone could answer the following question: How does j^{\mu}_{C} = -e\psi^{T}(\gamma^{\mu})^{T}\overline{\psi}^{T} become j^{\mu}_{C} = -(-)e\overline{\psi}\gamma^{\mu}\psi ? Is there some...

Using the defining property of the dirac delta function, \int{dx f(x) \delta(x-c)} show that \delta(ax)=\frac{1}{|a|}\delta(x) I think all I need to do is make the right u substitutions and it will come out right, but I can't think of how to make the substitutions...after a long time working...

Dirac delta function as the limit of a sequence Hi.. If I have a sequence which in some limit tends to infinity for x=0 and goes to zero for x\neq0, then can I call the limit as a dirac delta function? If not, what are the additional constraints to be satisfied?

Hi, I'm trying to get to Pauli's equation from Dirac's equation in the weak field regime. Specifically, if I substitute \psi = \left(\begin{array}{cc}\chi \\ \varphi \end{array}\right) into the Dirac equation, I get two coupled equations i\frac{\partial\chi}{\partial t} =...

Homework Statement Evaluate the following integrals: \int^{+\infty}_{-\infty}\delta[f(x)]dx and \int^{+\infty}_{-\infty}\delta[f(x)]g(x)dx Homework Equations \int^{+\infty}_{-\infty}\delta(x)dx=1 \int^{+\infty}_{-\infty}\delta(x)f(x)dx=f(0) \int^{+\infty}_{-\infty}\delta(x-a)f(x)dx=f(a)The...

Hi, I'm desperately searching for some literature which discusses the harmonic oscillator, preferably simple, in terms of the path integral formulation. I am unfamiliar with dirac notation and want something as simple as possible which gives general results of the harmonic oscillator in terms of...

does the Dirac measure still exist with complex variance? The Dirac delta function can be rigorously defined as a measure. See http://en.wikipedia.org/wiki/Dirac_delta_function#As_a_measure For the gaussian form of the Dirac delta function we have, \[ {\rm{\delta (x - x}}_0 ) =...

I'm trying to understand the multiple limit processes involved with the Dirac delta function. Does it matter which process you do first the integral or the delta parameter that approaches zero? The closest theorem I found that addresses the order of taking limits is the Dominate Convergence...

We all know that \frac{1}{2\pi}\int{e^{ik(x-x')}dk=\delta(x-x'). i am working a problem which appears to depend on the statement \int e^{z^*(z-w)}dz^*\propto\delta(z-w) Does anyone know if this is valid? \delta(z-w) is defined in the usual way so that...

Hi all, I'm working through Chandrasekhar's http://prola.aps.org/abstract/RMP/v15/i1/p1_1" and can not understand the steps to progress through Eq. (66) in Chapter 1. The integral is: \prod^{N}_{j=1} \frac{1}{l^{3}_{j}|\rho|}\int^{\infty}_{0} sin(|\rho|r_{j})r_{j}\delta...

This is probably a stupid question, but when I apply the Euler-Lagrange equation to the Lagrangian density of the Dirac field I get for the conjugate field \bar{\psi} (-i \partial_\mu \gamma^{\mu} -m) = 0 (derivative acts to the left). But when I take a hermitian conjugate of the Dirac...

I'm told that a product of distributions is undefined. See, http://en.wikipedia.org/wiki/Distribution_(mathematics)#Problem_of_multiplication where the Dirac delta function is considered a distribution. Now the Dirac delta function is defined such that, \[ \int_{ - \infty }^{ +...

Hi all, I want to calculate traces of Dirac matrices with a program like Mathematica. I found the package FeynCalc but it seems to be outdated. It is always producing results like this: 4 (-(DiracCanonical->False) (Factoring->False) (FeynCalcInternal->True) g^(mu nu)...

why in the problem of dirac delta potential, the integral \int^{\epsilon}_{-\epsilon}\phi''(x)dx is equal to \phi'(\epsilon)-\phi'(-\epsilon)? but \int^{\epsilon}_{-\epsilon}\phi(x)dx is equal to 0 if, for example\phi(x)=e^x then \phi(x)''=\phi(x) but, the firts integral is...

Consider the Dirac equation in the ordinary form in terms of a and \beta matrices i\frac{{\partial \psi }} {{\partial t}} = - i\vec a \cdot \vec \nabla \psi + m\beta \psi The matrices are hermitian, \vec a^\dag = \vec a,\beta ^\dag = \beta . Daggers denote hermitian...

I do not know how to execute the problem with the 2x in the problem. Evaluate the integral: \int_{-4}^{4} (x^2+2x+1) \delta(2x) dx

Expectation value of operator A is given by following formula in Dirac notation. <A> = <x|A|x> where A : Operator <A> : Expectation value of A |x> : State Somehow I am unable to convince myself that this formula is true. Would someone please explain it to me? Thanks

I had just reviewed back the properties of Delta Dirac Function, however I'm having a little confusing about the first property as stated : \delta\left(x-a)\right = 0 if x \neq a, \delta\left(x-a)\right = \infty if x = a;Here is my problem : when integrate over the entire region (ranging from...

I am trying to see why exactly the momentum eignenstates for a free particle are orthogonal. Simply enough, one gets: \int_{-\infty}^{\infty} e^{i (k-k_0) x} dx = \delta(k-k0) I can see why, if k=k0, this integral goes to zero. But if they differ, I don't see why it goes to zero. You have...

hello all, i am unaware of how to handle a delta function. from what i read online the integral will be 1 from one point to another since at zero the "function" is infinite. overall though i don't think i know the material well enough to trust my answer. and help on how to take the integral of...

I'm trying to work through the proof of the Lorentz invariance of the Dirac bilinears. As an example, the simplest: \bar{\psi}^\prime\psi^\prime = \psi^{\prime\dagger}\gamma_0\psi^\prime = \psi^{\dagger}S^\dagger\gamma_0 S\psi = \psi^{\dagger}\gamma_0\gamma_0S^\dagger\gamma_0 S\psi =...

Homework Statement I am really confused in my electrodynamics class. I have the following function. f(x) = \delta (x + \alpha ) + \delta(x -\alpha) How do i convert this into Fourier Tranform ? Those are dirac delta functions on either sides of the origin. Homework Equations...

hi, may someone help me to clarify my doubts... in my work, i encounter diracdelta square \delta(x-x_1)\delta(x-x_2) i am not sure what it means... it seems if i integrate it \int dx \;\delta(x-x_1)\delta(x-x_2) = \delta(x_1-x_2) is either zero of infinity. is this correct? thanks

Homework Statement I wonder how to deal with the square root of Dirac Delta function, \sqrt{\delta(x)}. Actually, this comes from a problem which asking readers to calculate the wave function of a free particle and of a harmonic oscillator at time t, provided that the wave function at time...

How would you write an infinite line charge with constant charge per unit length \lambda as a volume charge density using Dirac delta functions? Perhaps in cylindrical coordinates? I'm confused because if you integrate this charge distribution over all space, you should get an infinite amount...

I know that this question was posted before but I just couldn't get it using another way around. So your comments are highly appreciated. In the textbook, \nabla\bullet\left(\widehat{r}/r^{2}\right)=4\pi\delta^{3}\left(r\right). But when I want to calculate the divergence using Catesian...

Hi all, As a blind follower of QFT from the sidelines (the joys of the woefully inadequate teaching of theory to exp. particle physics students...), I have just realized that I've never actually gone further than deriving the Dirac equation, and then just used the Dirac Lagrangian density as...

I was reading some more quantum mathematics, and a question occurred to me. In the current treatment of the topic, the bra-ket notation is defined as a shorthand notation for more complex mathematical operations. But couldn't bra-ket notation be defined separately from quantum physics? In other...

I've seen the derivation of Dirac Equation using Inhomogeneous Lorentz Group in L H Ryder's QFT book.Can anybody give some comprehensible descriptions of this method?

since dirac delta function is not a literally a function but a limit of function,does it mean that dirac delta function is continuous and differentiable through out the infinity? is there any example of dirac delta function if yes then give meeeeeeee?

Hi everybody, I don't know if this is the right section to ask for such a question but I have been dealing with this problem for a while and there's something I still cannot grasp... Let us suppose that we have a dirac free particle with propagator (i'm sorry but I'm not able to obtain the...

The conserved current for a field \phi obeying the Klein-Gordon equation is (neglecting operator ordering) proportional to i\phi^{\dag}\partial_\mu \phi-i\phi\partial_\mu \phi^{\dag}. The conserved current for a four component field \psi obeying the Dirac equation is...

Homework Statement \[ \underset{\left|\underline{\xi}\right|=1}{\int}\delta_{0}\left(\underline{\xi}\cdot\underline{z}\right)dS_{\xi}=\intop_{0}^{2\pi}d\varphi\intop_{-r}^{+r}\delta_{0}\left(\varsigma\right)\frac{d\varsigma}{r}=\frac{2\pi}{r}\] The \delta_{0} is the dirac delta function.the...

Hi, I am not really sure whether its over the surface of the sphere or the Volume, the problem and the solution are given below, I want to know how it has been solved. The \delta_{0} is the dirac delta function. \[...

Homework Statement my apologies if this question should be posted in the math forum 3-d space spanned by orthonormal basis: (kets) |1>, |2>, |3>. Ket |a> = i|1> - 2|2> - i|3>. Ket |b> = i|1> + 2|3>. The question is to construct <a| and <b| in terms of the dual basis (kets 1,2,3)...

I have been reading papers for my research and I came across this equation twice: \lim_{\eta\to 0+}\frac{1}{x+i \eta} = P\left(\frac{1}{x}\right) - i \pi \delta(x) Where P is the pricipal part. It has been quite a while since I have had complex variables, but might it come from the...

I was hoping someone could help me with a seeming paradox involving the Dirac equation. I have taken a non-relativistic QM course, but am new to relativistic theory. The Dirac equation is (following Shankar) i\frac{\partial}{\partial t}\psi = H\psi where H = \vec{\alpha}\cdot...

Hi, What is the origin of the following commutation relation in Lorentz Algebra: [J^{\mu\nu}, J^{\alpha\beta}] = i(g^{\nu\alpha}J^{\mu\beta}-g^{\mu\alpha}J^{\nu\beta}-g^{\nu\beta}J^{\mu\alpha}+g^{\mu\beta}J^{\nu\alpha}) This looks a whole lot similar to the commutation algebra of...

Homework Statement \int_{-\infty}^t (cos \tau)\delta(\tau) d\tau Evaluate the integral. I'm supposed to evaluate this for all t I believe, so I'm concerned with t<0, t=0, t>0. Homework Equations \int_{-\infty}^{\infty} f(x)\delta(x) dx = f(0) The Attempt at a Solution...

Hi everybody, I am trying to get the partial derivative of the following with respect to Si[t] and Phi[t] separately: Integrate[<Phi[t]|H|Si[t]>] The operator H is the partial derivative with respect to t. I tried this in Mathematica, calling Needs["Quantum`Notation`"] but I...

Using perturbation theory, I'm trying to solve the following problem \frac{\partial P}{\partial \tau} = \frac{1}{2}\varepsilon^2 \alpha^2 \frac{\partial^2 P}{\partial f^2} + \rho \varepsilon^2 \nu \alpha^2 \frac{\partial^2 P}{\partial f \partial \alpha} + \frac{1}{2}\varepsilon^2 \nu^2...

A two-dimensional harmonic oscillator is associated with the group Su(2). What is that association? Solutions to the Dirac equation require a pair of spinors at each point? Can we think think of spacetime as having pairs of 2D harmonic oscillators at each point? Thanks for any help.

Is is right to say that the two dirac cones are described by two bi-spinors of different chirality?... Is it right to say that each of the dirac cones contains quasi-particles of different helicity (electrons of positive elicity and of holes negative elicity for one dirac cone and the...

Hello, I'm fuzzy on how Dirac notation works especially when operators are added in. Does anyone have a clear explanation (the simpler the better) that they can give to me, and or a website or book that does a good job of explaining it?

Hi people, I was asking myself... is it true that the elements of the base of solutions of the dirac equation usually used are eigenstates of elicity? Yesterday I tried the calculation following the notation of this site (it uses the dirac representation) and its set of solutions...

Hi guys. I play now a bit with EM fields and I have encountered some problems connected with Dirac delta. By coincidence I visited this forum and I thought I could find some help in here. The problem is that in order to get a potential in some point from a single charge you need to just...

Hi everyone I need the historical articles that bose and fermi integrals were defined for the first time. Can anyone help me?