Dirac Definition and 859 Threads

I'm reading Daniel T. Gillespie's A QM Primer: An Elementary Introduction to the Formal Theory of QM. In the section on continuous eigenvalues, he admits to playing "fast and loose" with the laws of calculus, with respect to the Dirac delta function. I'd like to understand it better, or, if such...

Homework Statement Solve the given symbolic initial value problem: y''-2y'-3y=2\delta (t-1)-\delta (t-3) ;y(0)=2,y'(0)=2 The attempt at a solution Let Y(s):= L{y(t)}(s) Taking laplace transform of both sides: [s^{2}Y(s)-2s-2]-2[sY(s)-2]-3Y(s)=2e^{-s}-e^{-3s}...

The problem is very easy, maybe just something about eigenvectors that I'm missing. Go to the first two pages of the 5th chapter of ''Principles of Quantum Mechanics'', by Shankar, 2nd edition. Homework Statement Shankar wants to find the solution for a free particle in Quantum Mechanics...

Homework Statement An ideal particle of energy E is incident upon a rectangular barrier of width 2a and height V_{0}. Imagine adjusting the barrier width and height so that it approaches V(x)=\alpha \delta(x). What is the relationship between V0, alpha and a? Homework Equations The...

In most of the physical systems, if we have a Lagrangian L(q,\dot{q}), we can define conjugate momentum p=\frac{\partial L}{\partial{\dot{q}}}, then we can obtain the Hamiltonian via Legendre transform H(p,q)=p\dot{q}-L. A important point is to write \dot{q} as a function of p. However, for the...

I am confused about two minor things right now. The following illustrates both which I pulled from my QM book: <x|p_{op}|0>=\int_{-\infty}^{\infty}dp<x|p_{op}|p><p|0>=\int_{-\infty}^{\infty}dp~p<x|p><p|0>...

Homework Statement y[n] - (2/3)y[n-1] = x[n] what is y[n] if x[n] = diracdelta[n] The Attempt at a Solution for some reason, i argued that y[n-1] = diracdelta[n-1] so y[n] = diracdelta[n] + (2/3)diracdelta[n-1] Im pretty sure this is wrong, anybody can help?

I believe Dirac spinors are not in any Hilbert space since it has no positive definite norm. However one QM axiom I learned told me any quantum state is represented by a state vector in Hilbert space, so what is happening to Dirac spinor?Or is it just that the axiom is not for relativistic QM?

As is well known, a Dirac Lagrangian can be written in a symmetric form: L = i/2 (\bar\psi \gamma \partial (\psi) - \partial (\bar\psi) \gamma \psi ) - m \bar\psi \psi Let \psi and \psi^\dagger be independent fields. The corresponding canonical momenta are p = i/2 \psi^\dagger...

We all know that the free Lagrangian for a spin-1/2 Dirac field is \mathcal{L}=\bar\psi(i\gamma_\mu\partial^\mu-m)\psi. But, if I were to invent a Lagrangian, I would have tried \mathcal{L}=\partial_\mu\bar\psi\partial^\mu\psi-m^2\bar\psi\psi. What's wrong with this second Lagrangian? Why...

Homework Statement Prove that \displaystyle \int_{-\infty}^{\infty} \delta (at - t_0) \ dt = \frac{1}{ | a |} \int_{-\infty}^{\infty} \delta (t - \frac{t_0}{a}) \ dt For some constant a. The Attempt at a Solution Edit: Looking at this again, I really don't understand where this is coming...

Hydrogen normalized position wavefunctions in spherical coordinates: \Psi_{n \ell m}\left(r,\theta,\phi\right) = \sqrt{{\left( \frac{2}{n r_1} \right)}^3 \frac{\left(n - \ell - 1\right)!}{2n\left[\left(n + \ell\right)!\right]}} e^{-\frac{r}{n r_1}} \left({2r \over {n r_1}}\right)^{\ell} L_{n -...

Homework Statement Prove the statement http://www.mathhelpforum.com/math-help/vlatex/pics/60_32c8daf48ffa5f233ecc2ac3660e517e.png The Attempt at a Solution I am clueless as to how I would go about doing this, I know the basic properties. I think it has to do with using epsilon...

In book Quantum Electrodynamics, Feynman wrote that the Dirac equation is a relativistic form of the Pauli equation, not a correct form of Klein-Gordon equation. But, I think that the electron spin is only assumed in Pauli equation, but Dirac equation derives it? I went through derivation in...

I'm really confused! The Dirac equation describes spin -1/2 particles - i.e. particles of definite spin. And yet the spin operator does not commute with the Dirac Hamiltonian! The reason I'm confused is because I thought if you were going to describe particles of a given kind - that is...

Homework Statement From Mandl and Shaw (exercise 4.5): Deduce the equations of motion for the fields: \psi_L(x)\equiv{1 \over 2} (1-\gamma_5)\psi(x) \psi_R(x)\equiv{1 \over 2} (1+\gamma_5)\psi(x) for non-vanishing mass, and show that they decouple in the limit m=0. Hence show that the...

Hello all, I'm still plugging away at the meaning of spin, and spin orbital coupling. I am at the stage where I am testing out various formulations of corrections to Schrodinger's equation and beginning to test my ideas against data. Right now I am looking at Hydrogen spectra because being a...

Anyone know where I can find a discourse on the dirac delta function in spherical or polar coordinates, in particular why it is the form it is with correction coefficients? Thank you.

Hi all, I have a question about the actual value associated with the probability p(r) where p(r) is infinite for r=0. I realize that this p(r) can only be a distribution and only exist under an integral, and can't represent a pdf. My p(r) is a radially symmetric laplace distribution in 2d...

Homework Statement Dear all, I have a problem when I using MATLAB to get the Fourier transform of dirac delta function. below is my code.Homework Equations clear all; clc; close all; % t=0:0.002:2; t=0:0.002:4; dt=t(2)-t(1); u=zeros(size(t)); pos0=find(t>=1,1); u(pos0)=1/dt...

I think I'm missing something real simple on trace theorems and Dirac matrices, but am just not seeing it. In the Peskin and Schroeder QFT text on page 135 we have: gamma^(mu)*gamma^(nu)*gamma_(mu) = -2*gamma^(nu) But, why can't we anti-commute and obtain the following...

Homework Statement Suppose we have a spin 1/2 Particle in a prepared state: \left|\Psi\right\rangle = \alpha \left|\uparrow\right\rangle + \beta\left|\downarrow\right\rangle where \left|\uparrow\right\rangle \left|\downarrow\right\rangle are orthonormal staes representing spin up and...

Homework Statement (Introduction to Elementary Particles, David Griffiths. Ch 7 Problem 7.8 (c)) Find the commutator of H with the spin angular momentum, S= \frac{\hbar}{2}\vec{\Sigma}. In other words find [H,S] Homework Equations For the Dirac equation, the Hamiltonian...

Please teach me this: We know that 0-spin particles obey Klein-Gordon equation and 1/2spin particles obey Dirac equation.But I do not know whether higher integer spin particles obey Klein-Gordon equation or not.Similarly,do higher half integer spin particles obey Dirac equation?Because if we...

i don't really understand the dirac delta function in 3D. is it right that integral of f(r)d3(r-a)dt = f(a) where a = constant ,r is like variable x in 1D dirac delta function? so why when i have f(r')d3(r-r') , it picks out f(r)? where r is now a constant and r' is a...

Hello all, I joined this amazing forum just today.I hope that my question will get answered soon. So here it is.I am unable to understand a some steps in calculation. Please help me understand. Here is a linear homogeneous first order differential equation whose solution a research...

Although I am an aspiring physicist, I cannot cope with the physicist's love for vagueness when it comes to yielding math. Exactness is simply not a luxury that can be ignored, certainly not in theoretical physics. But okay, I realize the dirac delta function can be made exact by the use of...

Hi togehter. I encountered the following problem: The timeordering for fermionic fields (here Dirac field) is defined to be (Peskin; Maggiore, ...): T \Psi(x)\bar{\Psi}(y)= \Psi(x)\bar{\Psi}(y) \ldots x^0>y^0 = -\bar{\Psi}(y)\Psi(x) \ldots y^0>x^0 where \Psi(x) is a Dirac...

So I've been told that the Dirac delta functional is a distribution, but I don't see why that's the case. I had an introduction to distributions in my calculus IV course, but as I remember it, a distribution involves and integral containing a the product of a function from the Schwartz space and...

i need help trying to find the laplace transform of te-t\delta(t) i know the laplace transform of te-t is 1/(s+1)2 but i don't know how to find the laplace transform of a product with the Dirac delta

The component solutions of the Dirac equation are also solutions of the Klein-Gordon equation. But these solutions are not scalars since the coefficients contain quantities like energy and momentum[the phase part is of course an invariant] These are neither zero spin nor half spin...

Hi this is my first post here so I'm sorry if my question seems trivial. I haven't worked a lot with the dirac delta function before, so i always thought that the shifting property would only work as: \int\delta(x-h)\;f(x)\;dx=f(h) Now I've been reading some articles and I came across...

Hey guys, i'm stuck (yet again! :) ) I am somewhat confused by Dirac spinors u,\bar{u}. Take the product (where Einstein summation convention is assumed): u^r u^s\bar{u}^s Is this the same as u^s\bar{u}^s u^r? Probably not because u^r is a vector while the other thing is a matrix...

I understand that the Dirac delta function can be taken as a distribution. And that one can calculate the Shannon entropy or information content of any distribution. So what is the information content of the Dirac delta function? I think it is probably identically zero, but I'd like to see the...

Homework Statement The Dirac function (unit impulse) is defined as \delta(t) = 0 where t \neq 0 the integration of d(t) between -ve inf and +ve inf is 1. Now I picture this as a rectangle with no width and infinite height. In fact I think of the width (along the x axis) as (1/inf =...

I am currently reading Dirac Equation from Peskin-Schroeder. In a particular para it says, "Now let us find Dirac Matrices \gamma^\mu for four-dimensional Minkowski Space. It turns out that these matrices must be at least 4X4." What is the proof of the above statement? I think (not sure)...

Homework Statement Suppose a relativistic particle with spin 1/2 at rest. Show that if we apply an electrical field at t=0 there's a probability fot t>0 of finding the particle in a negative energy state if such negative energy states are assumed to be originally empty. Homework Equations...

To quote Weinberg Vol1, Pg 14 : And immediately he said: So to speak, Dirac equation alone cannot determine g-factor uniquely, but quantum field theory can? How?

I was wondering if anybody knows of any code available to perform tensor analysis in Python or in other language; I was wondering if there is any computational method for finding constraints in a lagrangian via the Dirac Algorithm.

Not a Homework problem, but I think it belongs here. Homework Statement Consider four dirac matrices that obey M_i M_j + M_j M_i = 2 \delta_{ij} I knowing the property that Tr ABC = Tr CAB = Tr BCA show that the matrices are traceless. Homework Equations Tr MN = Tr NM The Attempt...

sorry if this looks ugly but I couldn't find out how to write out bras and kets on the Latex thing. I have these inner products <f|g> = i<x|(AB - A<B> - <A>B + <A><B>)|x> and <g|f> = -i<x|(BA - B<A> - <B>A + <A><B>)|x> where |x> is some arbitrary ket and A and B do not commute. I'm trying to...

I am reading about the electron flow in graphene and the article said this "This behavior is not described by the traditional mathematics (Schrodinger equation) but by the mass-less Dirac equation" What does this mean and what is the massless Dirac equation... the whole paragraph is...

Amusingly, a search on these three words here in PF does not show a lot of postings, so I am creating this thread so you can ask all your doubts about N-dimensional Majorana, Weyl and Dirac particles, their representations, their Lagragians, masses, and whatever you have always wanted to know...

Homework Statement a.) Given \delta_n=\frac{ne^{-{n^2}{x^2}}}{\pi} Show: x{\frac{d}{dt}\delta_n}=-\delta_n b.) For the finite interval (\pi,-\pi) expand the dirac delta function \delta(x-t) in sines and cosines, sinnx, cosnx, n=1,2,3... They are not orthogonal, they are normalized to...

I have a very simple question about the Dirac equation that I just cannot see the answer to. In these notes: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf In equation 4.115, I keep getting u( \vec{p} ) = \begin{pmatrix} \sqrt{p \cdot \sigma} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ \sqrt{...

Does anybody know a good thread, homepage or book that takes up different interpretations of Pauli and Dirac matrices with the connection to for example quaternions or bivectors? Maybe someone could comment on this?

Homework Statement Let's say we have a wire of finite length L with total charge Q evenly spread along the wire so that lambda=Q/L, linear charge density, is constant. The wire is shaped in x-y plane in some well behaved curve y = f(x). Find the surface charge density sigma(x,y). Homework...

Homework Statement Show that a Lorentz transformation preserves the sign of the energy of a solution to the Dirac equation. The Attempt at a Solution I'm not sure how to approach this. So I apply the Lorentz transform to the Dirac equation, and work through it to obtain the energy...

Homework Statement Please see attached Homework Equations The Attempt at a Solution Ok so basically a bit confused about notation.. does |psi> = sum over all r of ar |ur> ? any help would be great..thanks

In hamiltonian formulation of GR there appears some constraints (it may be found e.g. in "Modern canonical quantum GR" by Theimann, ch. 1.2). I would like to find a Dirac algebra of the constraints (i.e. compute Poisson bracket between constraints), but my results are not consistent with...