Dirac Definition and 859 Threads

Homework Statement I'm specifically having trouble with taking the Fourier transform of f(t) in order to sketch F(w) and also to move on with the rest of the problem. Homework Equations f(t) = (5+rect(t/4))cos(60pi*t) mixed_signal = cos(60pi*t) The Attempt at a Solution I attempted to...

Hi, I am having trouble following the Peskin and Schroeder and their derivations to show that a Dirac particle is a spin 1/2 particle (page 60 and 61). I understand how he gets the first (unnumbered) equation on page 61. However, I don't understand how he gets to the second equation...

Homework Statement This is just an example, not a specific problem. So if I have ∫σ(sinx), for example, and my limits of integration are, for example, 1 to 10, what I need to do to solve that is to find a value of x that would make the argument of the delta function 0. So for sinx, 0 makes...

Hi everyone, my problem is this Using Dirac notation show that \frac{d}{dt}<\varphi|\hat{A}|\varphi> = \frac{i}{\hbar}<\varphi|[\hat{H},\hat{A}]|\varphi> where A does not explicitly depend on t I am given as a hint that the hamiltonian operator in Dirac notation is...

Homework Statement The Potential V(r) is given: A*e^(-lambda*r)/r, A and lambda are constants From this potential, I have to calculate: E(r), Rho(r) -- charge density, and Q -- total charge. Homework Equations The Attempt at a Solution I know that E(r) is simply minus...

What is the sum of an infinite Dirac series and why? 1 or infinity? \sum_{n=-\infty}^{\infty}\delta (n) I can see it being 1 because it's like a series version of the integral: \int_{-\infty}^{\infty}\delta (t)dt = 1 But for the series where n=0, \delta (0) = \infty :confused:

1.) an inner product of a state vector represent by <\psi|\psi>. sometimes the notation is like <\phi|\psi> is mean transfer from state |\psi> to <\phi|.it mean the former 1 do not transfer the state? what is the difference between both? 2.) what is mean by <x|\psi>? is it mean x(position)...

Homework Statement Prove this theorem regarding a property of the Dirac Delta Function: $$\int_{-\infty}^{\infty}f(x)\delta'(x-a)dx=-f'(a)$$ (by using integration by parts) Homework Equations We know that δ(x) can be defined as...

Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?

http://en.wikipedia.org/wiki/Dirac_comb Please have a look at the Fourier Series section, and its last equation. Let T = 1. After expanding the Equation x(t) = 1 + 2cos(2∏t) + 2cos(4∏t) + 2cos(6∏t) ... Now this does not give the original Dirac Comb. Eg: at t = 1/2 x(1/2) = 0 But RHS =...

Hi All, I found (Wikipedia page on Helmotz's decomposition theorem) the follwoing equality, which puzzles me: $$\delta(x-y) = - (4 \pi)^{-1} \nabla^{2} \frac{1}{\vert x - y \vert}$$ I am not sure I understand, the r.h.s seems to me a proper function. The page mentions this a sa position...

What does the Dirac Delta Function do? ##\delta^3(\vec{r})## How do you evaluate it? What are its values from -inf to +inf?

Why does the \psi of the Dirac equation return four complex numbers instead of one, as in the Schrodinger equation? I know it has something to do with spin, but I'm not finding a clear answer to this question in my sources. What do these four complex numbers represent?

It's been quite some time now since I decided to stop self-studying physics and to pay more attention to the math behind. I'm working towards gaining an understanding of 100% rigorous mathematics for now. One thing that has always bothered me is the Dirac delta function. What I want to know...

Classically as well as quantum-mechanically, the source of the Maxwell field is the electron/four-current (Dirac field), so the use of the Green Function propagator for the Maxwell field makes perfect sense: the Maxwell field is inhomogenous in the presence of matter. But what about the source...

Homework Statement Good day. May I know, for Dirac Delta Function, Is δ(x+y)=δ(x-y)? The Attempt at a Solution Since δ(x)=δ(-x), I would say δ(x+y)=δ(x-y). Am I correct?

From what I can tell, it seems that 1/x + δ(x) = 1/x because if we think of both 1/x and the dirac delta function as the following peicewise functions: 1/x = 1/x for x < 0 1/x = undefined for x = 0 1/x = 1/x for x > 0 δ(x) = 0 for x < 0 δ(x) = undefined for x = 0 δ(x) = 0 for x > 0...

I got 2 questions to ask! I have finished one but not sure if it's correct so I need to double check with someone :) http://imageshack.us/a/img708/1324/83u8.png Here is my worked solution, I took this picture with my S4 and I wrote is very neatly as I could! The reason I didn't type it all...

Homework Statement Consider the double Dirac delta function V(x) = -α(δ(x+a) + δ(x-a)). Using this potential, find the (normalized) wave functions, sketch them, and determine the # of bound states. Homework Equations Time-Independent Schrodinger's Equation: Eψ = (-h^2)/2m (∂^2/∂x^2)ψ +...

In Griffith's Introduction to Quantum Mechanics, on page 56, he says that for scattering states (E > 0), the general solution for the Dirac delta potential function V(x) = -aδ(x) (once plugged into the Schrodinger Equation), is the following: ψ(x) = Ae^(ikx) + Be^(-ikx), where k = (√2mE)/h...

In page 555, Appendix B of Intro to electrodynamics by D Griffiths: \nabla\cdot \vec F=-\nabla^2U=-\frac{1}{4\pi}\int D\nabla^2\left(\frac{1}{\vec{\vartheta}}\right)d\tau'=\int D(\vec r')\delta^3(\vec r-\vec r')d\tau'=D(\vec r) where ##\;\vec{\vartheta}=\vec r-\vec r'##. Is it supposed to be...

I want to proof [SIZE="4"](1)##\delta(x)=\delta(-x)## and [SIZE="4"](2) ## \delta(kx)=\frac{1}{|k|}\delta(x)## [SIZE="4"](1) let ##u=-x\Rightarrow\;du=-dx## \int_{-\infty}^{\infty}f(x)\delta(x)dx=(0) but \int_{-\infty}^{\infty}f(x)\delta(-x)dx=-\int_{-\infty}^{\infty}f(-u)\delta(u)du=-f(0) I...

My question is in Griffiths Introduction to Electrodynamics 3rd edition p48. It said Two expressions involving delta function ( say ##D_1(x)\; and \;D_2(x)##) are considered equal if: \int_{-\infty}^{\infty}f(x)D_1(x)dx=\int_{-\infty}^{\infty}f(x)D_2(x)dx\;6 for all( ordinary) functions f(x)...

Why does the Dirac equation not have a potential energy term? The Schrödinger equation does, and the Dirac equation is supposed to be the special relativity version of the Schrödinger equation, no?

I know this probably belongs in one of the math sections, but I did not quite know where to put it, so I put it in here since I am studying Electrodynamics from Griffiths, and in the first chapter he talks about Dirac Delta function. From what I've gathered, Dirac Delta function is 0 for...

I know the trace tr[\gamma_5 a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/] in 4-dimensional space-time, how is the result of it in D dimension? Is it the same as in 4 dimension?

It seems that notions of quantum field and wave function are utterly different from each other.Then is Dirac equation being equation for field or for relativistic wave function or for the both?

I am interested in learning about how the Dirac Equation was derived, how it allowed special relativity and QM to be unified, and how it predicted the existence of animatter. The explanations I have found so far are too advanced for me mathematically, and I was wondering if anybody could...

An electron field is a superposition of two four-component Dirac spinors, one of them multiplied with a creation operator and an exponential with negative energy, the other multiplied with an annihilation operator and an exponential with positive energy. So I assume one Dirac spinor creates a...

Do not solve the problem just look at the picture. http://i208.photobucket.com/albums/bb33/DanusMax/giro2_zps11d2056b.jpgWell its the end of the semester and I found out that I had only one of the required books for my undergraduate course. Anyways back to the question. As you can see in the...

Hi All, so I'm trying to tackle this DEQ: f''[x] = f[x] DiracDelta[x - a] - b, with robin boundary conditions f'[0] == f[0], f'[c] == f[c] where a,b, and c are constants. If you're curious, I'm getting this because I'm trying to treat steady state in a 1D diffusion system where...

Homework Statement 1. Given that |ψ> = eiπ/5|a> + eiπ/4|b>, express <ψ| as a linear combination of <a| and <b|. 2. What properties characterise the bra <a| that is associated with the ket |a>? Homework Equations The Attempt at a Solution 1. <ψ| = e-iπ/5<a| + e-iπ/4<b| 2. a. The bra <a|...

In almost all the books on field theory I've seen, the authors list out the different types of quantities you can construct from the Dirac spinors and the gamma matrices, but I'm confused by how these work. For instance, if $$\overline\psi\gamma^5\psi$$ is a pseudoscalar, how can...

Hello gang, I wanted to get these lectures to you earlier but I was "temporarily indisposed," or should I say, "temporarily disposed" from the site. I guess there's only room for one bad boy in the physics community... http://www.scientificamerican.com/article.cfm?id=bad-boy-of-physics...

Hello, I'm in an introductory course about quantum computing. My math experience is fairly solid, but not very familiar with Dirac (bra-ket) notation. Just would like to clarify one thing: In a single cubit space, we have |0 \rangle , and | 1 \rangle . I understand that these form an...

(1,a^2,a^2,a^2)) from the action; \mathcal{S}_{D}[\phi,\psi,e^{\alpha}_{\mu}] = \int d^4 x \det(e^{\alpha}_{\mu}) \left[ \mathcal{L}_{KG} + i\bar{\psi}\bar{\gamma}^{\mu}D_{\mu}\psi - (m_{\psi} + g\phi)\bar{\psi}\psi \right] I can show that, i\bar{\gamma}^{\mu}D_{\mu}\psi -...

So I definitely believe that the continuity of the Dirac equation holds, there is one thing that annoys me, which is that c \alpha . (-i \hbar \nabla \psi ) = c (i \hbar \nabla \psi^\dagger ) . \alpha from the first part of the Dirac Hamiltonian because the momentum operator should be...

Just to clarify in the dirac equation (i\gamma^{\mu}\partial_{\mu} -m)\psi=0 Is it equal to (-i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i} -m)\psi=0 in (-,++++) notation?

\nabla \cdot \frac{\mathbf{r}}{|r|^3}=4 \pi \delta ^3(\mathbf{r}) What's the proof for this, and what's wrong with the following analysis? The vector field \frac{\mathbf{r}}{|r|^3}=\frac{1}{r^2}\hat{r} can also be written \mathbf{F}=\frac{x}{\sqrt{x^2+y^2+z^2}^3}\hat{x}+...

Hi can anyone explain how to derive an expression for the Dirac Hamiltonian, I thought the procedure was to use \mathcal{H}= i\psi^{\dagger}\Pi -\mathcal{L}, but in these papers the have derived two different forms of the Dirac equation H=\int d^{3}x...

Professor Susskind describes the Dirac Sea. He says remove a negative energy electron, and replace it with a positive energy electron and a hole. In other words an electron-positron pair. I'm having trouble equating holes with positrons because positrons have mass but holes don't.

Prove that. \int_a^b f(x)g' (x)\, dx = -f(0) This is supposed to be a delta Dirac function property. But i can not prove it. I thought using integration by parts. \int_a^b f(x)g' (x)\, dx = f(x)g(x) - \int_a^b f(x)'g (x)\, dx But what now? Some properties: \delta...

In texts about dirac delta,you often can find sentences like "The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin". If we take into account the important property of dirac delta: \int_\mathbb{R} \delta(x) dx=1 and the fact that it is zero...

What potential would one use when evaluating the Dirac equation of the hydrogen atom? Would it simply be in the form used when examining the hydrogen atom-Schrodinger equation or does it need modification?

Homework Statement Given the gauge invariant Dirac equation (i\hbar \gamma^\mu D_{\mu} - mc)\psi(x, A) = 0 Show that the following holds: \psi(x, A - \frac{\hbar}{e} \partial\alpha) = e^{i\alpha}\psi(x, A) Homework Equations The covariant derivative is D_\mu = \partial_{\mu} +...

Homework Statement Show that (\Delta A)^{2} = \langle \psi |A^{2}| \psi \rangle - \langle \psi |A| \psi \rangle ^{2}\\ \phantom{(\Delta A)^{2} }=\langle \psi | (A - \langle A \rangle )^{2} | \psi \rangle , where \Delta A is the uncertainty of an operator A and \langle A \rangle is the...

What does the square of a Dirac delta function look like? Is the approximate graph the same as that of the delta function?

Hi when trying to derive this equation, i am stuck on: [\Gamma_{\mu}(x),\gamma^{\nu}(x)]=\frac{\partial \gamma^{\nu}(x)}{\partial x^{\mu}} + \Gamma^{\nu}_{\mu p}\gamma^{p} . This [\Gamma_{\mu}(x) term is the spin connection, if this is an ordinary commutator: a) is it a fermionic so +...

While studying the Dirac Equation, we come across the gamma matrices. Can we consider these matrices as the components of a 4-vector ?

I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...