Dirac Definition and 859 Threads

Hi, I try to prove, that function f_n = \frac{\sin{nx}}{\pi x} converges to dirac delta distribution (in the meaning of distributions sure). On our course we postulated lemma, that guarantee us this if f_n satisfy some conditions. So I need to show, that \lim_{n\rightarrow...

Hey, I'm trying to determine the probability density and current of the Dirac equation by comparison to the general continuity equation. The form of the Dirac equation I have is i\frac{\partial \psi}{\partial t}=(-i\underline{\alpha}\cdot\underline{\nabla}+\beta m)\psi According to my...

Hello, Is this correct: \int [f_j(x)\delta (x-x_i) f_k(x)\delta (x-x_i)]dx = f_j(x_i)f_k(x_i) If it is not, what must the left hand side look like in order to obtain the right handside, where the right hand side multiplies two constants? Thanks!

Hi there, I'm trying to comprehend Dirac Delta functions. Here's something to help me understand them; let's say I want to formulate Newton's second law F=MA (for point masses) in DDF form. Is this correct: F_i = \int [m_i\delta (x-x_i) a_i\delta (x-x_i)]dx Or is it this: F_i = [\int...

Hey, My question is on the Dirac equation, I am having a little confusion with the steps that have been taken to get from this form of the Dirac equation: i\frac{\partial \psi}{\partial t}=(-i\underline{\alpha}\cdot \underline{\nabla}+\beta m)\psi to -\frac{\partial^2 \psi}{\partial...

Hi i am trying to derive the Dirac equation of the form: [i\gamma^0 \partial_0 + i\frac{1}{a(t)}\gamma.\nabla +i\frac{3}{2}(\frac{\dot{a}}{a})\gamma^0 - (m+h\phi)]\psi where a is the scale factor for expnasion of the universe. I understand that the matter action is S=\int d^{4}x e...

Homework Statement I am trying to integrate the function \int _{-\infty }^{\infty }(t-1)\delta\left[\frac{2}{3}t-\frac{3}{2}\right]dt Homework Equations The Attempt at a Solution I think the answer should be \frac{5}{4} because \frac{2}{3}t-\frac{3}{2}=0 when t=9/4. then (9/4-1) = 5/4...

Homework Statement For a function ρ(x,y,z) = cδ(x-a), give the meaning of the situation and describe each variable. Homework Equations As far as units go, I know that: ρ(x,y,z) = charge density = C/ m^3 δ(x-a) = 1/m and if those two are correct, then b must have units of (C/m^2)...

Does the Dirac delta fuction have a residue? Given the close parallels between the sifting property and Cauchy's integral formula + residue theory, I feel like it should. Unfortunately, I have no idea how they tie together (if they do at all).

Homework Statement compute the integral: \int_ {-\infty}^\infty \mathrm{e}^{ikx}\delta(k^2x^2-1)\,\mathrm{d}xHomework Equations none that I have The Attempt at a Solution I don't actually have any work by hand done for this because this is more complex than any dirac delta integral I have...

Hi everyone, I'm trying to follow how Dirac went about deriving his wave equation. I know there's a couple of different ways to "derive" this, but I'm trying to follow the original method that Dirac used. There's a lot of good stuff online for this, but there's a step that I get stuck at and...

Hi all, I'm familiar with the fact that the dirac delta function (when defined within an integral is even) Meaning delta(x)= delta(-x) on the interval -a to b when integral signs are present I want to prove this this relationship but I don't know how to do it other than with a limit...

hello, Please attached snapshot. Does the integral 7.9 equal to 0.5 \inthμσ dzμ/dτ dzσ/dτdτ ? I'm confused as to the fourth power of Dirac's Delta. Then where does the derivative on x go ? For more on this, see MTW pg180 thanks

If this seems like a homework question please forgive me, but it is merely for understanding and confidence building. I've been reading into the Bohr Model and the Wave Mechanics Model and I read through de Broglie and have proceeded to Schrodinger, Heisenberg and Dirac. At this stage, my mind...

Homework Statement To whom it may concern, I am trying to understand the central force problem of the Dirac equation. In particular, I am following Sakurai's Advanced Quantum Mechanics book. There (section 3.8, p.122), it is shown that there is an operator K = \beta(\Sigma . L +...

Hello! I have a small question, and I am not sure if I am missing something: Today I glanced at the wikipedia page for Pions, and saw this: Statistics: Bosonic Can anyone explain to me why a quark paired with a anti-quark obey Bose-Einstein Statistics? If quarks obey Fermi-Dirac statistics...

Hi All, I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution. My question are: 1. How can I find the time derivative...

I have been wondering exactly how one would express the Dirac delta in arbitrary spaces with curvature. And that leads me to ask if the Dirac delta function has a coordinate independent expression. Is there an intrinsic definition of a Dirac delta function free of coordinates and metrics? Or as...

does anyone know how dirac diffusion looks like, i.e. The solution of dirac equation with no potential and initial condition for 1 spinor component psi(x,t0) being delta(x-x0) ? Are the solution gaussian like the schroedinger case ? Thanks.

Homework Statement I have to integrate: \int_0^x \delta(x-y)f(y)dy Homework Equations The Attempt at a Solution I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity: \delta(0)=\infty I have to express the integral in terms of function...

Hi, It is well known that \int f(x) \delta(x-a) dx = f(a) \quad\mathrm{and}\\ \int f(x) \delta^{(n)}(x-a) dx = (-1)^n f^{(n)}(a) Similarly, Is there a way to express a distribution integral with multiplicative Diracs in a compact form (e.g., a sum)? \int f(x) \delta(t-x-l_1)...

Homework Statement L[t^{2} - t^{2}δ(t-1)] Homework Equations L[ t^{n}f(t)] = (-1^{n}) \frac{d^{n}}{ds^{n}} L[f(t)] L[δ-t] = e^-ts The Attempt at a Solution My teacher wrote \frac{2}{s^{3}} -e^{s} as the answer. I got \frac{2}{s^{3}} + \frac{e^-s}{s} + 2 \frac{e^-s}{s^2} + \frac{2e^-s}{s^3}

Is the Dirac Equation generally covariant and if not, what is the accepted version that is? For general coordinate changes beyond just Lorentz, how do spinous transform?

Hello, I've read that Dirac introduced the idea of the creation and annihilation operators in the solution to the quantum harmonic oscillator problem, but can anyone tell me where he did this? In a paper, or maybe in a book? I've had a little search online, but I've yet to discover...

Homework Statement Homework Equations I really wish they existed in my notes! *cry*. All I can think of is that integrating or in other words summing the dirac delta functions for all t, would be infinite? None the less the laplace transform exist since its asked for in the question and i...

Hi everyone Homework Statement I have a question: Am I allowed to do the following, where the a's are operators and the alphas are quantum states. \langle \alpha \mid a^\dagger a^\dagger a a \mid \alpha \rangle = \langle \alpha \mid a^\dagger a^\dagger \mid \alpha \rangle \langle...

Alright... So I'm in an 'introductory' Q.M class in college right now, it's the only one that this two-year college has, so I don't have an upper division Q.M Profs to talk to about this, and since my prof is equally confused, I turn to the internet. Okay, so everyone knows that <ψ|Aψ> = <a>...

I need to know the mathematical argument that how the relation is true $(C^{-1})^T\gamma ^ \mu C^T = - \gamma ^{\mu T} $ . Where $C$ is defined by $U=C \gamma^0$ ; $U$= non singular matrix and $T$= transposition. I need to know the significance of these equation in charge conjuration .

Homework Statement Imagine you have two vectors |a> and |b> such that: |c> = |a> + |b> Now imagine you want the dot product: <c|a> Is that the same as: <c|a> = [ <a|*+<b|* ] |a> = <a*|a> + <b*|a> where * represents the complex conjugate of the vector? Homework Equations...

Homework Statement The question was way too long so i took a snap shot of it http://sphotos-h.ak.fbcdn.net/hphotos-ak-snc7/397320_358155177605479_1440801198_n.jpg Homework Equations The equations are all included in the snapshotThe Attempt at a Solution So for question A I've done what the...

How would you simplify this expression: <a|<b|a>|a> where ψ = |a>|b> and I'm finding ψ*ψ.

Homework Statement This is an issue I'm having with understanding a section of maths rather than a coursework question. I have a stage of the density function on the full phase space ρ(p,x); ρ(p,x) = \frac {1}{\Omega(E)} \delta (\epsilon(p,x) - E) where \epsilon(p,x) is the...

Hi I would like to know what is the dirac delta function of x^2, I read somewhere it is equal to delta of x itself but why?

If i have an operator (for example Sz) and then add it on two kets in a row i e. Sz|1>|0> How could you calculate this? should i first add it on the |1> and then the |0> ? help please

Hello! I should prove: \delta'(\lambda x) = \dfrac{1}{\lambda \vert \lambda \vert} \delta(x), where lambda is just a constant. If we make use of the scaling property and the definition of the distributional derivative, we find: \left( \delta'(\lambda x), f \right) =...

The Dirac delta function is defined as: \int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1 Or more generally the integral is, \int_{ - \infty }^{ + \infty } {\delta (\int_{{x_0}}^x {dx'} )dx} But if the metric varies with x, then the integral becomes, \int_{ - \infty }^{ + \infty }...

Homework Statement I need help to expand some matrices Homework Equations \pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0 The Attempt at a Solution How do I expand i\hbar \gamma^0 the matrix in this term, I am a bit lost. All the help would be...

So in 1927 Davisson and Germer showed that electrons shot at a crystal do indeed have a DeBroglie wavelength inversely proportional to their momentum (h/p)? That would mean that their wavelength is a function of their velocity, the voltage used to accelerate them, etc. But I seem to remember...

Homework Statement Consider the matrix elements of \hat{x} in momentum space. That is, evaluate \langle p | \hat{x} | \psi(t) \rangle in terms of the momentum space wave equation \langle p | \psi(t) \rangle . Homework Equations \langle x | p \rangle = \frac{1}{\sqrt{2 \pi \hbar}}...

Hi. I'm currently reading about (negative frequency) solutions to the Dirac equations which can be written on the form \Psi = ( \sqrt{p \cdot \sigma} \chi, \sqrt{p \cdot \bar{\sigma}} \chi)^T e^{-i p \cdot x}For any two component spinor Chi. But the dot product with the four vector p and the...

Prove that x \frac{d}{dx} [\delta (x)] = -\delta (x) this is problem 1.45 out of griffiths book by the way. Homework Equations I attempted to use integration by parts as suggest by griffiths using f = x , g' = \frac{d}{dx} This yields x [\delta (x)] - \int \delta (x)dx next I tried...

Homework Statement I am using the time differentiation property to find the Fourier transform of the following function: Homework Equations f(t)=2r(t)-2r(t-1)-2u(t-2) The Attempt at a Solution f'(t)=2u(t)-2u(t-1)-2δ(t-2) f''(t)=2δ(t)-2δ(t-1)-?? Can somebody explain what the...

Can anyone point me to how to interpret Dirac notation expressions as wave functions and integrals beyond the basics of     <α| = a*(q)     |β> = b(q)     <α|β> = ∫ a* b dq For example in the abstract Dirac notation the expression     |ɣ> (<α|β>) can be evaluated as     (|ɣ><α|) |β>  ...

Homework Statement Consider a particle moving in one dimension and bound to an attractive Dirac δ-function potential located at the origin. Work in units such that m=\hbar=1. The Hamiltonian is given, in real (x) space, by: H=-\frac{1}{2}\frac{d^2}{dx^2}-\delta (x) The (non normalized)...

The Dirac delta "function" is often given as : δ(x) = ∞ | x = 0 δ(x) = 0 | x \neq 0 and ∫δ(x)f(x)dx = f(0). What about δ(cx)? By u=cx substitution into above integral is, ∫δ(cx)f(x)dx = ∫δ(u)f(u/c)du = 1/c f(0). But intuitively, the graph of δ(cx) is the same as the graph of...

I originally asked this in the Calculus & Analysis forum. But perhaps this is better suited as a question in Abstract algebra. For the set of all Dirac delta functions that have a difference for an argument, we have the property that: \int_{ - \infty }^\infty {{\rm{\delta (x -...

Dirac Matrix Property? Possible Book mistake? Derive KG from Dirac I got a copy of QFT demystified and on pg. 89 he says he can write \gamma_{\nu} \gamma^{\mu} = g_{\nu \sigma} \gamma^{\sigma} \gamma^{\mu} = g_{\nu \sigma} \frac{1}{2} (\gamma^{\sigma} \gamma^{\mu} + \gamma^{\mu}...

Square integrable functions -- Hilbert space and light on Dirac Notation I started off with Zettilis Quantum Mechanics ... after being half way through D.Griffiths ... Now Zettilis precisely defines what a Hibert space is and it includes the Cauchy sequence and convergence of the same ... is...

I'm having trouble seeing how an operator can be written in matrix representation. In Sakurai, for an operator X, we have: X = \sum \sum |a''> <a''| X |a'> <a'| since of course \sum |a> <a| is equal to one. Somehow, this all gets multiplied out and you get a square matrix with the...

In Sakurai's Modern Quantum Mechanics, he develops the Dirac notation of bras and kets. In one part, he states (page 17): <B|X|A> = (<A|X^|B>)* = <A|X^|B>* where X^ denotes the Hermitian adjoint (the conjugate transpose) of the operator X. My question is, since a bra is the conjugate...