Dirac Definition and 859 Threads

In Dirac's "General Theory of Relativity", he develops the "comprehensive action principle" in chapter 30. Simply put, he writes a combined action for the gravitational field and all other matter-energy fields ##I=I_g+I'##. Varying this: $$\delta(I_g+I')=\int ( p^{\mu\nu}\delta g_{\mu\nu} +...

Question concerns a remark by Dirac in his "General Theory of Relativity", chap. 33, p. 65-66. Dirac is working with plane waves, so the metric $$g_{\mu\nu}(x) = g_{\mu\nu}(l_\sigma x^\sigma)$$ where ##l_\sigma## is the wave vector. Writing ##\xi = l_\sigma x^\sigma##, Dirac defines...

Why do the enery levels calculated with the Dirac/Pauli euqations always lie lower than the results calculated with the Schrödinger equation? I assume it has to do something with relativistic effects and the changing masses because of this.

In Dirac's "General Theory of Relativity", Chap. 34 on the polarization of gravitational waves, he introduces a rotation operator ##R##, which appears to be a simple ##\pi/2## rotation, since $$R \begin{pmatrix} A_0 \\ A_1 \\ A_2 \\ A_3 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 &...

The four components of Dirac equations obey the Klein-Gordon equation for a particle of mass m. This is always explained when introducing Dirac equation, but it is never exploited further. I am wondering: Can we then write a bosonic lagrangian for these four "particles"? Is this related to the...

Let's say we can solve the Dirac equation numerically with a powerful computer. What experiments do you recommend to take a look at to compare the result of the simulations with the real data. Maybe chemical reactions?

Hello shipmates, Instead of imagining a Dirac Delta as the limit of a normal, like this: $$ \delta\left ( x \right ) = \lim_{a \to 0}\frac{1}{|a|\sqrt{2\pi}}\exp\left [ -\left ( x/a \right )^2 \right ] $$ Could we say the same thing except starting with a lognormal, like this? $$ \delta_{LN}...

In Dirac ("GTR") p. 39 he says, "For a covariant vector ##A_\mu##, we have $$A_{\mu;\nu}-A_{\nu;\mu} = A_{\mu,\nu} - \Gamma^\rho_{\mu\nu} A_\rho - \left( A_{\nu,\mu} - \Gamma^\rho_{\nu\mu}A_\rho \right) = A_{\mu,\nu} - A_{\nu,\mu}.$$ This result may be stated: covariant curl equals ordinary...

Dirac in his "GTR" (Chap 19, page 34-35) finds a coordinate system ##(\tau, \rho)## which has no coordinate singularity at ##r=2m##. Explicitly, the transformation looks like (after some algebra): $$\tau=t + 4m\sqrt{\frac{r}{2m}} + 2m\log{\frac{\sqrt{r/2m}-1}{\sqrt{r/2m}+1}}$$ $$\rho=t +...

Dirac in "General Theory of Relativity" (top of p. 20) says "Even if one is working with flat space ... and one is using curvilinear coordinates, one must write one's equations in terms of covariant derivatives if one wants them to hold in all systems of coordinates." This comment follows his...

The sigma tensor composed of the commutator of gamma matrices is said to be able to represent any anti-symmetric tensor. \sigma_{\mu\nu} = i/2 [\gamma_\mu,\gamma_\nu] However, it is not clear how one can arrive at something like the electromagnetic tensor. F_{\mu\nu} = a \bar{\psi}...

What is ##<x|P|x'>##? (for particle in 1d, and ##\hbar = 1##)?\begin{align*} <x|P|x'> &= \int dp' <x|P|p'><p'|x'> \\ &= \int dp' \ p' <x|p'> <p'|x'> \\ &= \int dp' \ p' \frac{1}{\sqrt{2\pi}} e^{ip'x} \frac{1}{\sqrt{2\pi}} e^{-ip'x'} \\ &= \frac{1}{2\pi} \int dp' \ p' e^{ip'(x-x')} \end{align*}

Hi If A is a linear operator but not Hermitian then the expectation value of A2 is written as < ψ | A2| ψ >. Now if i write A2 as AA then i have seen the expectation value written as < ψ | A+A| ψ > but if i only apply the operators to the ket , then could i not write it as < ψ | AA | ψ > ? In...

I have found a new formula in Dirac calculus. The formula is elementary, so probably I'm not the first who found it. Yet, I have never seen it before. As many other formulas in Dirac calculus, it is not rigorous in the sense of functional analysis. Rather, it is a formal equality, which is only...

I'm going to be a bit sketchy here, at least to start with. If you want me to show you exactly where I am I might post a pdf, if that's okay. (Only because it will simplify coding several pages of LaTeX.) Briefly, what I'm trying to do is take this system of equations: ##F^{ \prime } +...

Under the entry "Quantum electrodynamics" in Wikipedia, the Dirac equation for an electron is given by $$ i\gamma^{\mu}\partial_{\mu}\psi - e\gamma^{\mu}\left( A_{\mu} + B_{\mu} \right) \psi - m\psi = 0 ,\tag 1 $$ or $$ i\gamma^{\mu}\partial_{\mu}\psi - m\psi = e\gamma^{\mu}\left( A_{\mu} +...

Could anyone help with some of the later parts of the derivation for Dirac spinors, please? I understand that an arbitrary vector ##\vec v## $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} $$ can be defined as an equivalent matrix V with the components $$ \begin{bmatrix} z & x - iy \\ x + iy...

I'm having trouble following a proof of what happens when the Dirac Lagrangian is put into the Euler-Lagrange equation. This is the youtube video: and you can skip to 2:56 and pause to see all the math laid out. I understand the bird's eye results of the Dirac Lagrangian having an equation of...

I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in confidently interpreting his use of Dirac Notation in Section 1.9 ... in Section 1.9 we read the following: I need some help to confidently interpret and proceed with Neuenschwander's notation...

My goal is to develop the equation 21. You should asume that \delta(r_2-r_1)^2 =0. These is named renormalization. Then my question is , do my computes are correct with previous condition ?

I am trying to convert the attached picture into dirac notation. I find the LHS simple, as it is just <ψ,aφ>=<ψIaIφ> The RHS gives me trouble as I am interpreting it as <a†ψ,φ>=<ψIa†Iφ> but if I conjugate that I get <φIaIψ>* which is not equiv to the LHS. *Was going to type in LaTex but I...

I am learning Dirac notations in intro to quantum mechanics. I don’t understand why the up arrow changes to down arrow inside the equation in c). My own calculation looks like this:

Hi, I have to verify the sifting property of ##\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} e^{-sa}e^{st} ds## which is the inverse Mellin transformation of the Dirac delta function ##f(t) = \delta(t-a) ##. let ##s = iw## and ##ds = idw## ##\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iwa}e^{iwt}...

Hi, I found Laplace transform of this Dirac delta function which is ##F(s) = e^{-st}## since ##\int_{\infty}^{-\infty} f(t) \delta (t-a) dt = f(a)## and that ##\delta(x) = 0## if ##x \neq 0## Then the Mellin transform ##f(t) = \frac{1}{2 \pi i} \int_{\gamma - i \omega}^{\gamma +i \omega}...

Why is the Laplacian of ##1/r## in spherical coordinates proportional to Dirac's Delta, namely: ##\left(\frac{\partial^2 }{\partial r^2}+\frac{2}{r}\frac{\partial }{\partial r}\right)\left(\frac{1}{r}\right)=-\frac{\delta(r)}{r^2}## I get that the result is zero.

I feel that this problem can be directly answered from the E>0 case of the attractive Dirac delta potential -a##\delta##(x), with the same reflection and transmission coefficients. Can someone confirm this hunch of mine?

I am working on my physics paper and just realized that after explaining everything I did not add the equasion. Now I am wondering where I can find a good source on them, so that I can add them.

Forgive me if you've heard this song before, but I don't understand how to interpret the \psi_3 and \psi_4 components of the Dirac equation. For instance, at 8:27 of this video we see that while an electron at rest can be in a state like [1,0,0,0], the same electron as viewed from a...

It's said that every ket has a unique bra. For any vector ##|v> ∈ V## there is a unique bra ##<v| ∈ V*##. I'm not sure what that means. What is unique? Can anyone please help me understand. Thank you

I'm new to relativistic quantum mechanics and quantum field theory and was trying to learn about the Dirac equation. Unfortunately, I got a little stumped by the interaction between matter and antimatter. It seems like the time derivative of matter is dependent on the spatial derivative of...

I'm looking for complete and detailed references on constrained Hamiltonian systems and Dirac brackets. While my main interest is electrodynamics, I would prefer a complete exposition of the theory from the ground up. So far, my knowledge about the topic comes from books in QFT, like Weinberg...

I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept...

Hello! I went over a calculation of the hydrogen wavefunction using Dirac equation (this one) and I am a bit confused by the angular part. The final result for the wavefunction based on that derivation is this: $$ \begin{pmatrix} if(r) Y_{j l_A}^{m_j} \\ -g(r)...

Summary:: Operator acts on a ket and a bra using Dirac Notation Please see the attached equations and help, I Think I am confused about this

Hello, I was reviewing a part related to electromagnetism in which the charge and current densities are defined by the Dirac delta: ##\rho(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x} - \underline{x}_n(t))## ##\underline{J}(\underline{x}, t)=\sum_n e_n \delta^3(\underline{x} -...

We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##. $$ \delta(x - \mu) =...

In the 4-dimensional representation of ##\beta##, ## \beta=\begin{pmatrix}\mathbf I & \mathbf 0 \\ \mathbf0 & -\mathbf I\end{pmatrix} ,## and we can suppose ## \alpha_i=\begin{pmatrix}\mathbf A_i & \mathbf B_i \\ \mathbf C_i & \mathbf D_i\end{pmatrix} ##. From the anti-commutation relation...

Is it correct to say that $$\int e^{-i(k+k’)x}\,\mathrm{d}x$$ is proportional to ##\delta(k+k’)##?

Hey guys! Sorry if this is a stupid question but I'm having some trouble to express this charge distribution as dirac delta functions. I know that the charge distribution of a circular disc in the ##x-y##-plane with radius ##a## and charge ##q## is given by $$\rho(r,\theta)=qC_a...

I'm trying to the following exercise: I've proven the first part and now I'm trying to do the same thing for fermions. The formulas for the mode expansions are: What I did was the following: $$\begin{align*} \sum_s \int d\tilde{q} \left(a_s(q) u(q,s) e^{-iq \cdot x}+ b_s^\dagger(q) v(q,s)...

hi, there. I am doing some frequency analysis. Suppose I have a function defined in frequency space $$N(k)=\frac {-1} {|k|} e^{-c|k|}$$ where ##c## is some very large positive number, and another function in frequency space ##P(k)##. Now I need integrate them as $$ \int \frac {dk}{2 \pi} N(k)...

Shankar Prin. of QM 2nd Ed (and others) introduce the inner product: <i|V> = vi ...(Shankar 1.3.4) They expand the ket |V> as: |V> = Σ vi|i> |V> = Σ |i><i|V> ...(Shankar 1.3.5) Why do they reverse the order of the component vi and the ket |i> when they...

Hi, I have a quick question about something which I have read regarding the use of dirac delta functions to represent conditional pdfs. I have heard the word 'mask' thrown around, but I am not sure whether that is related or not. The source I am reading from states: p(x) = \lim_{\sigma \to...

Given \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y) \end{split} \end{equation} where ##\epsilon > 0## Is the following also true as ##\epsilon \rightarrow 0## \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon}...

Hello, everyone. Need I understand Dirac equation if I plan to learn QED?

Indeed, if we take a vector field which dual to the covector field formed by the gradient from a quadratic interval of an 8-dimensional space with a Euclidean metric, then the Lie algebra of linear vector fields orthogonal (in neutral metric) to this vector field is isomorphic to the...

Hi! I am studying Dirac's equation and I already have understood the derivation. Following Griffiths, from factoring Einstein's energy relation with the gamma matrices: ## (\gamma^\mu p_\mu + m)(\gamma^\mu p_\mu - m) = 0 ## You take any of the two factors, apply quantization and you arrive to...

As a part of a bigger problem, I was trying to evaluate the D'Alambertian of ##\epsilon(t)\delta(t^2-x^2-y^2-z^2)##, where ##\epsilon(t)## is a sign function. This term appears in covariant commutator function, so I was checking whether I can prove it solves Klein-Gordon equation. Since there's...

A beam of length L with fixed ends, has a concentrated force P applied in the center exactly in L / 2. In the differential equation: \[ \frac{d^4y(x)}{dx^4}=\frac{1}{\text{EI}}q(x) \] In which \[ q(x)= P \delta(x-\frac{L}{2}) \] P represents an infinitely concentrated charge distribution...

hi i am recently following the nptel course in quantum mechanics (The Course ) and it seems like a really good course , but i can't find the book that it based on . my question is : had anyone saw that course before to suggest a QM book related to it ? - she began by an introduction to vector...