Dirac equation Definition and 166 Threads

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

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...

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 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 +...

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?

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 .

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...

Say we a have a sum of spin up plane wave solutions to the Dirac equation which represent the wave-function of a localized spin-up electron which is 90% likely to be found within a distance R of the origin of a spherical coordinate system. Four complex numbers at each spacetime point are needed...

Hello, I'm looking at the Dirac Equation, in the form given on Wikipedia, and (foolishly) trying to understand it. \left( c \boldsymbol{\alpha}\cdot \mathbf{\hat{p}}+\beta mc^2 \right ) \psi = i\hbar\frac{\partial \psi}{\partial t}\,\! So I picture a wavefunction in an eigenstate of the...

Hello, I'm reading Griffiths' introduction to elementary particles and he seems to claim that the Schrödinger equation can be seen as a non-relativistic limit of the Dirac equation. I was wondering how one could deduce this, e.g. how do we go from \mathcal L = \bar{\psi} \left( i \gamma^\mu...

So it is said that a basis for the plane wave solutions to the Dirac equation are of the form (p denotes the four-momentum vector) e^{-i p \cdot x} u^{(s)} (for particles) and e^{i p \cdot x} v^{(s)} (for antiparticles), with s = 1 or 2 (and u and v having predetermined structure). I'm...

Hi, Everybody! Currently, I am reading the book "Lectures on Quantum Field Theory" (by Ashok Das) But I am a bit confusing. Why does Dirac Equation describe spin 1/2 particles? I have already known that Dirac Equation bears some angular momentum structure, but why it just describe spin...

What does "couples as the 4th component of a vector" mean in the Dirac equation? I'm doing an exercise regarding the spin-orbit operator and the Dirac equation/particles, and I'm having trouble understanding the link between terminology and mathematics. The particular phrase I'm having trouble...

Hi! Homework Statement 1. Substituting an ansatz \Psi(x)= u(p) e^{(-i/h) xp} into the Dirac equation and using \{\gamma^i,\gamma^j\} = 2 g^{ij}, show that the Dirac equation has both positive-energy and negative-energy solutions. Which are the allowed values of energy? 2. Starting...

I am reading about Dirac's equation for relativistic electron in Feynman's book "Quantum Electrodynamics". Factor \gamma =(1-v^2)^{-1/2} (units c=1) is almost always presented in non quantum calculations of Special relativity. But in his book I also find it on page 44 in lecture "Relativistic...

Hi Everyone, I'm a math grad student working on numerical procedures for the Dirac equation, and I'd like to be able to incorporate the neutral current interaction neutrino + fermion -> Z bozon -> neutrino + fermion <- poorly impersonated Feynman diagram into the Dirac equation as a...

This isn't really a homework problem, just a form of writing I don't quite understand. The Dirac equation is: ("natural units") (i\gamma^{\mu}\partial_{mu}-m)\Psi = 0 When I try to build the conjugated equation, where \bar{\Psi} := \Psi^{+}\gamma^{0}, I get...

Demonstrations of Dirac equation covariance state: The Dirac equation is (i γ^{μ} ∂_{μ} - m)ψ(x) = 0. \ \ \ \ \ \ \ \ \ \ [1] If coordinates change in a way that x \rightarrow x' = Lx, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2] where L is a Lorentz transformation, [1] should...

Hello, I have a question concerning the current in the Dirac equation and its corresponding operator. One can construct a current density that is \textbf{j}^{i} = \psi^{\dagger}\gamma^{i}\psi If I want to have the current, I will have to integrate: I = \oint \textbf{j} \cdot \textbf{n} \, dA...

Is this the correct form for a Dirac electron in a Higgs field with scalar potential \phi and an electromagnetic field with vector potential A_\mu i \gamma^\mu \partial_\mu \psi = g \phi \psi + e \gamma_\mu A^\mu \psi where g is the coupling constant to the Higgs field and e is the...

If I take a modified Dirac Eq. of the form (i\gamma^\mu \partial_\mu -M)\psi=0 where M=m+im_5 \gamma_5, and whish to square it to get a Klein-Gordon like equation would I multiply on the left with (i\gamma^\nu \partial_\nu +m+im_5\gamma_5) or (i\gamma^\nu \partial_\nu +m-im_5\gamma_5)? I was...

I need to show that u^{+}_{r}(p)u_{s}(p)=\frac{\omega_{p}}{m}\delta_{rs} where \omega_{p}=\sqrt{\vec{p}^2+m^{2}} [itex]u_{r}(p)=\frac{\gamma^{\mu}p_{\mu}+m}{\sqrt{2m(m+\omega_{p})}}u_{r}(m{,}\vec{0})[\itex] is the plane-wave spinor for the positive-energy solution of the Dirac equation...

The Schrodinger wavefunction for the hydrogen atom scales as r^l for small r, where l is the orbital angular momentum. Is this changed in any dramatic way for the Dirac equation wavefuction? Does the small component of the Dirac spinor have the same small-r asymptotic behaviour as the large...

Hi could someone please explain the story (if there is one) about the Dirac equation with an anomalous magnetic moment term, I have seen this in several old papers but it never seems to be mentioned in textbooks. Was this an old confusion in formulating QFT. In this context I believe the Dirac...

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 -...

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...

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...

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 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...

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{...

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...

Would someone tell me some website where I can find the relativistic treatment of the hydrogen atom using Dirac's Equation? I am not trying the find the method which uses Schrodinger's equation and adds as perturbations fine and hyperfine structures? Thank you. So far i have not find anything...

i am now studying dirac equation and klein paradox if we confine to one dimension, we only need one alpha matrix, not three so in lower dimensions, maybe the dirac spinor is not of four components but fewer? i am curious about this question because it seems that as for the Klein...

Is it possible to incorporate into Dirac equation for proton the possibility of its transformation into neutron (isospin freedom)?

I wonder how Dirac equation transform under change of coordinates (in flat spacetime). Should I simply express partial derivaties of one coordinates in another or it is necessary to transform Dirac matrices as well?

Hi! I was taught that the dirac matrices are AT LEAST 4x4 matrices, so that means that I can find also matrices of higher dimensions. The question is: what do these higher-dimension-matrices represent? Are they just mathematical stuff or have they got a physical meaning? I ask that because in...

Can Dirac equation be used for many particles (fermions) system (i.e. a nucleus with many electrons)? And in this case how do you incorporate the anti-symmetry nature of the wavefunctions? Obviously Slater determined will complicate the equation to a point where it’s almost impossible to solve...

The dirac equation for massless particles can be decoupled into separate equations for left and right handed parts. i \tilde{\sigma}^\mu\partial_\mu \psi_R= 0 and i \sigma^\mu\partial_\mu \psi_L= 0. Now we can have four solutions for each of the above equations. For the equation i...

This is probably a stupid question, but when I apply the Euler-Lagrange equation to the Lagrangian density of the Dirac field I get for the conjugate field \bar{\psi} (-i \partial_\mu \gamma^{\mu} -m) = 0 (derivative acts to the left). But when I take a hermitian conjugate of the Dirac...

Homework Statement Show that \mathbf\alpha\equiv\left[\begin{array}{cc} 0&\mathbf\sigma\\ \mathbf\sigma&0\end{array}\right] is hermitian. The Attempt at a Solution My first instinct was to say that \mathbf\sigma must be equal to its complex conjugate (as it would if it was a scalar...

Consider the Dirac equation in the ordinary form in terms of a and \beta matrices i\frac{{\partial \psi }} {{\partial t}} = - i\vec a \cdot \vec \nabla \psi + m\beta \psi The matrices are hermitian, \vec a^\dag = \vec a,\beta ^\dag = \beta . Daggers denote hermitian...

Hi all, As a blind follower of QFT from the sidelines (the joys of the woefully inadequate teaching of theory to exp. particle physics students...), I have just realized that I've never actually gone further than deriving the Dirac equation, and then just used the Dirac Lagrangian density as...

I've seen the derivation of Dirac Equation using Inhomogeneous Lorentz Group in L H Ryder's QFT book.Can anybody give some comprehensible descriptions of this method?

I was hoping someone could help me with a seeming paradox involving the Dirac equation. I have taken a non-relativistic QM course, but am new to relativistic theory. The Dirac equation is (following Shankar) i\frac{\partial}{\partial t}\psi = H\psi where H = \vec{\alpha}\cdot...

ψ(x) is the four-component wave function of the Dirac equation，that is ψ(x) can be expressed by a column vector (ψ1(x) ψ2(x) ψ3(x) ψ4(x)） ,under a lorentz transformation,it will become ψ'(x').I am confused that how ψ'(x') can be expressd in the form which is stated by textbooks: ψ'(x')=S(a)ψ(x)...

Hi, I have learned the Dirac equation recently and I managed to solve it for a free particle (following Greiner book “relativistic quantum mechanics” and Paul Strange book “Relativistic Quantum Mechanics”). I was asked to solve the Dirac equation in the stationary frame for a free particle (no...

Some confusions from some recent lectures; I asked the prof, but I still don't fully understand what is going on. We began with the action (tau is some worldline parameter, dots indicate tau derivatives; they are hard to see): S = \int d\tau \; \left\{ \dot x^{\mu} p_{\mu} - \frac12 e(\tau)...

The http://en.wikipedia.org/wiki/Dirac_equation" , which itself is based upon the relativistic energy-momentum relation E^2 = p^2 + m^2 (natural units). And here comes my question then: Why do we throw away the negative energy solutions in relativity but do we keep them when we combine it...

During my research a while ago, I have unexpectedly derived a "modified Dirac equation" with a \gamma_{5} mass term. (\gamma^{\mu}\partial_{\mu}+\gamma^{5}m)\psi(x)=0 I was quite surprised, and went about asking a few people. The answer I got is this equation is not new and has been...