Dirac Definition and 859 Threads

This may seem like a stupid question, but i can't get my head around this so please bear with me. I just looked at the derivation of Dirac equation and my question is: do the solutions for a free particle obey special relativity? because if yes why? I mean I thought using E2=(mc2)2+(pc)2 would...

Consider the Dirac Lagrangian, L =\overline{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi, where \overline{\psi}=\psi^{\dagger}\gamma^{0} , and show that, for \alpha\in\mathbb{R} and in the limit m\rightarrow0 , it is invariant under the chiral transformation...

In some sources QM is explained using bracket notation. I quite understand algebra of bracket notation, but I do not understand how is this notation related with physically meaningful things? How is bracket notation related to wavefunction notation? Could you tell me whether following is true...

Is the square of a Dirac delta function, ##(\delta(x))^2##, still a Dirac delta function, ##\delta(x)##? A Dirac delta function peaks at one value of ##x##, say 0. If it is squared, it still peaks at the same value, so it seems like the squared Dirac delta function is still a Dirac delta...

Homework Statement Calculate ##\int_{r=0}^\inf δ_r (r -r_0)\,dr## Homework Equations ##\int_V \delta^3(\vec{r} - \vec{r}') d\tau = 1## The Attempt at a Solution $$\int_V \delta^3(\vec{r} - \vec{r}') d\tau = \int_V \frac {1}{r^2 sinθ}\delta_r(r-r_0) \delta_θ (θ-θ_0) \delta_Φ (Φ-Φ_0) r^2...

Hello friends, I'm trying to construct transformation matrix S such that it transforms Dirac representations of gamma matrices into Chiral ones. I know that this S should be hermitian and unitary and from this I arrived an equation with 2 matrices on the LHS (a known matrix multiplied by S from...

I just want to make sure that I am understanding the Dirac Delta function properly. Is the following correct?: For two variables ##x## and ##y##: \begin{equation} \begin{split} \delta(x-y) f(x) &= f(y) \end{split} \end{equation} And: \begin{equation} \begin{split} \delta(x-x) f(x) &=...

Homework Statement I have the following question (see below) Homework Equations The eigenvalue equation is Au = pu where u denotes the eigenstate and p denotes the eigenvalue The Attempt at a Solution I think that the eigenvalues are +1 and - 1, and the states are (phi + Bphi) and (phi-Bphi)...

I'm an undergraduate student, so I understand that it may be difficult to provide an answer that I can understand, but I have experience using both the Dirac delta function and residue calculus in a classroom setting, so I'm at least familiar with how they're applied. Whether you're integrating...

So, I have to show that in the non-relativistic limit the lower two components of the positive energy solutions to the Dirac equation are smaller than the upper two components by a factor of ##\beta##. I started with the spinor $$\psi = \begin{pmatrix} \phi \\ \frac {\vec \sigma \cdot \vec p}...

Homework Statement I have a vector V with components v1, v2in some basis and I want to switch to a new (orthonormal) basis a,b whose components in the old basis are given. I want to find the representation of vector V in the new orthonormal basis i.e. find the components va,vb such that |v⟩ =...

I am familiar with the derivation of the resolution of the identity proof in Dirac notation. Where ## | \psi \rangle ## can be represented as a linear combination of basis vectors ## | n \rangle ## such that: ## | \psi \rangle = \sum_{n} c_n | n \rangle = \sum_{n} | n \rangle c_n ## Assuming an...

Hi everyone! I'm having a problem with calculating the fermionic propagator for the quantized Dirac field as in the attached pdf. The step that puzzles me is the one performed at 5.27 to get 5.28. Why can I take outside (iγ⋅∂+m) if the second term in 5.27 has (iγ⋅∂-m)? And why there's a...

Currently I'm set to pursue solid state physics in a EE department, working on more practical theory. However I'm seeing a lot of papers studying mathematically obfuscatory topics such as topological materials, Berry's phase, quantum phase transitions, and other abstruse (albeit important and...

I promise that anytime I have question about Dirac notation I will ask it in this thread. I do not know how to parse the following Dirac notation. |\Psi'\rangle = |u\rangle |U\rangle Can someone please convert the Dirac notation to matrix notation?

I know that the Dirac equation is ##i\gamma^{\mu}\partial_{\mu}\psi=m\psi##. How do I use this to show that ##(\partial_{\mu}\bar{\psi})\gamma^{\mu}=im\bar{\psi}##?

Hello! I have a question regarding the construction of solutions to the Diracequation for generell \vec{p} . In my lecturenotes (and also in Itzykson/Zuber) it is stated that it is easier than boosting the restframe-solutions, to construct them by using...

When Dirac tried to combine Quantum Mechanics and Special Relativity. Wasn't he initially worried that one was undeterministic (QM) and the second was continuous (SR). They are supposed to be incompatible. yet he combined them. Did Dirac do it by just considering the time dilation and other...

Hello, I am having a real problem trying to figure out what a Majorana mass is. The only thing I gather so far is that dirac mass is the mass that is the result of the Higgs Mechanism. What exactly is the Majorana mass, and for which particles does it exist.Thank you

Homework Statement \begin{equation} \int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau \end{equation} where: \begin{equation} \vec{r}' = 3\hat{x} + 2\hat{y} + \hat{z} \end{equation} Where d $\tau$ is the volume element, and V is a solid sphere with radius 4, centered...

Hey guys, Hope all is well. I am trying to understand the process that takes us from the Pauli equation to the Dirac equation. Whilst I understand the motivation is to have a lorrentz covariant equation I don't really understand A.) how this was done B.) what the physical result...

This is a companion question to https://www.physicsforums.com/threads/why-su-3-xsu-2-xu-1.884004/ Of course the Higgs mechanism over the standard model produces this low-energy group, SU(3)xU(1), which acts on Dirac fermions (this is, no Left-Right asymmetry anymore). Is there some reason...

So, we can break down the Dirac equation into 4 "component" equations for the wave function. I was going to post a question here a few days ago asking if a fermion (electron) could possesses a "spin" even if it were at rest, I.e., p=0. I did an internet scan, though, and found out that...

$$i\frac{\partial \phi}{\partial t} = \frac{1}{2m} (\sigma .P)(\sigma .P)\phi + eφ\phi$$ Rewriting the equation by using B = ∇ × A and e = −|e| (electron charge) leads to a Schr¨odinger like equation: $$i\frac{\partial \phi}{\partial t} =[ \frac{1}{2m} (-i∇ + |e|A)^2 + \frac{|e|}{2m} σ.B - |e|φ...

We know, $$\delta(x) = \begin{cases} \infty & \text{if } x = 0 \\ 0 & \text{if } x \neq 0 \end{cases}$$ And, also, $$\int_{-\infty}^{\infty}\delta(x)\,dx=1$$ Using Fourier Transformation, it can be shown that, $$\delta(x)=\lim_{\Omega \rightarrow \infty}\frac{\sin{(\Omega x)}}{\pi x}$$ Let's...

I just started learning this so I am a bit lost. This is where I am lost http://www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_7/node1.html . Why when E>0, we use $$\phi_p= \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} $$ or $$ \begin{pmatrix} 0 \\...

Homework Statement A particle of mass ##m## is in a spherically symmetric potential ##V = -\alpha\delta(|r|-a)##. Find the minimum value of ##\alpha## such that there is at least one bound state. Homework Equations ##u = \frac{R}{r}## ##-\frac{\hbar^2}{2m} \frac{d^2u}{dr^2} + \left[V +...

I am very much struggling with this problem: The set $\{\sin x, \cos x, x \sin x, x \cos x, x+2, x^2-1 \}$ on interval of $[0, \pi]$ is linearly independent and generates vector space $V$. Find the dimension of the kernel of the Dirac functionals in $V$. Here are what I know of the definitions...

In the representation theory of Lorentz transformations, the words Clifford algebra and Dirac algebra are used interchangeably. However, there is a distinction between the two. Indeed, the Dirac algebra is the particular Clifford algebra ##Cl_{4}({\bf{C}})\equiv Cl_{1,3}({\bf{C}})## with a basis...

Hi, I'm recently reading an introductory text about particle physics and there is a section about the Dirac equation. I think I can understand the solutions for rest particles, but the plane wave solutions appear to be a bit weird to me. For instance, when the upper states are (1 0), the lower...

Is the RHS of the conjugate relationship Ad(g)x = gxg-1 from the Lie algebra equivalent to: <g|λ|g> In the Dirac notation of quantum mechanics? I am looking at this in the context of gluons where g is a 3 x 1 basis matrix consisting of components r,g,b, g-1 is a 1 x 3 matrix consisting of...

Hi dear friends Please reffer to my work , I did part ( a ) Can someone help me to solve part( b ) Please

Hi dear friends Please reffer to my work , I did part ( a ) only Could you please help me to do part ( b ) I don't know how to do it.

Homework Statement hi i have to find the result of ##\int_{0}^{\pi}[\delta(cos\theta-1)+ \delta(cos\theta+1)]sin\theta d\theta## Homework Equations i know from Dirac Delta Function that ##\int \delta(x-a)dx=1## if the region includes x=a and zero otherwise The Attempt at a Solution first i...

I'm really interested in quantum theory and would like to learn all that I can about it. I'm looking books for learning quantum physics that contains derivation of Heisenberg uncertainty principle, dirac notation, pauli matrices, quantum operators, hawking radiation, etc. What are good books to...

The spin exchange operator would have the property $$\begin{align*}P\mid \chi_{\uparrow\downarrow} \rangle = \mid\chi_{\downarrow\uparrow} \rangle & &P\mid \chi_{\downarrow\uparrow} \rangle =\mid \chi_{\uparrow\downarrow} \rangle \end{align*}$$ This also implies ##P\mid \chi_{\text{sym.}}...

Dear all, I have a quick question, is the following statement true? $$\nabla_\textbf{x'} \delta(\textbf{x}-\textbf{x'}) = \nabla_\textbf{x} \delta(\textbf{x}-\textbf{x'})?$$ I thought I have seen this somewhere before, but I could not remember where and why. I know the identity ##d/dx...

The Dirac equation is the more generalized form of the Schrodinger equation and accounts for relativistic effects of particle motion (say an electron) by using a second order derivative for the energy operator. If you have an electron that is moving slowly relative to the speed of light, then...

In Dirac theory the electron velocity is equal to the speed of light. Why should that appear? Why should we try to solve this problem outside quantum mechanics hypothesis? Should we look for an alternative theory?

Hi, under what equation does the Dirac Equation fall under versus that of the Wave Function. Why is Antimatter from Dirac Equation really there but the wave function is not real? Because if Antimatter exist from an equation of complex numbers.. why can't the wave function be real too?

Is \frac{\partial}{\partial t} an operator on Hilbert space? I'm a little confused about the symmetry between spatial coordinates and time in relativistic QM. There is a form of the Dirac equation that treats these symmetrically: i \gamma^\mu \partial_\mu \Psi = m \Psi However, at least in...

The solutions to the Dirac equation are also solutions of the Klein-Gordon equation, which is the equation of motion for the real scalar field. I can see that the converse is not true, but why do spinors follow the equation for real-field particles? Is there any physical meaning to it?

If I understand it correctly Dirac developed his equation because he was looking for a relativistic first order wave equation for the electron, rather than a second-order one like the Klein-Gordon equation. Why did he wanted a first-order equation? Is it because the probability current is not...

Homework Statement δ(z*-z0*)δ(z+z0)=? δ(z*+z0*)δ(z-z0)=? where 'z' is a complex variable 'z0' is a complex number Formula is just enough, derivation is not needed.

Homework Statement What is the product of two Dirac delta functions δ(Real(z-c))δ(Img(z-c))=? 'z' and 'c' are complex numbers. This is not a problem, But I just need to use this formula in a derivation that I am currently doing. I just want the product of these two Dirac delta functions as a...

Homework Statement Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. Then solve the differential equation with...

Can somebody help me? I am studying Faddeev-Popov trick, following the Peskin and Schroeder's QFT book, but I can't understand one thing. After they inserted the Faddeev-Popov identity, $$I = \int {{\cal D}\alpha \left( x \right)\delta \left( {G\left( {{A^\alpha }} \right)} \right)\det \left(...

http://arxiv.org/abs/1001.2485 --- The above paper is about a possible two-time formulation of physics. It is by serious people. To understand it I'm trying to generalized the Dirac eqn. to 3+2 dimensions with signature (++---) I found the following (now closed post) useful...

How can I prove that the eigenvalues of the operator ## i\gamma^\mu \partial_\mu ## are non-negative? I've tried using the ansatz ## \psi=u(p) e^{ip_\nu x^\nu} ## but it didn't help. I've also tried playing with the equation using the properties of gamma matrices but that doesn't seem to lead...

In lectures, I have learned that F(k)= \int_{-\infty}^{\infty} e^{-ikx}f(x)dx where F(k) is the Fourier transform of f(x) and the inverse Fourier transform is f(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}f(k)dk . But on the same chapter in the lecture notes, there is an example solving...