Read about dirac equation | 40 Discussions | Page 1

  1. peguerosdc

    A Getting particle/antiparticle solutions from the Dirac Equation

    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...
  2. M

    I Condtion on transformation to solve the Dirac equation

    The problem is given in the summary. My attempt: Assume that ##\psi^\prime (x^\prime)## is a solution of the Dirac equation in the primed frame, given the transformation ##x\mapsto x^\prime :=\Lambda^{-1}x## and ##\psi^\prime (x^\prime)=S\psi(x)##, we have $$ \begin{align*} 0&=(\gamma^\mu...
  3. J

    I Changing the effective mass of an electron using electric potentials?

    The Dirac equation for an electron in the presence of an electromagnetic 4-potential ##A_\mu##, where ##\hbar=c=1##, is given by $$\gamma^\mu\big(i\partial_\mu-eA_\mu\big)\psi-m_e\psi=0.\tag{1}$$ I assume the Weyl basis so that $$\psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}\hbox{ and...
  4. redtree

    I Relativistic quantum mechanics

    Given that the Minkowski metric implies the Lorentz transformations and special relativity, why do the equations of relativistic quantum mechanics, i.e., the Dirac and Klein-Gordon equations, require a mass term to unite quantum mechanics and special relativity? Shouldn't their formulation in...
  5. T

    A Quantum Field Theory - Evaluate matrix spin dependent term in quadratic Dirac equation

    I derive the quadratic form of Dirac equation as follows $$\lbrace[i\not \partial-e\not A]^2-m^2\rbrace\psi=\lbrace\left( i\partial-e A\right)^2 + \frac{1}{2i} \sigma^{\mu\nu}F_{\mu \nu}-m^2\rbrace\psi=0$$ And I need to find the form of the spin dependent term to get the final expression $$g...
  6. sakh1012

    A Dirac Field quantization and anti-commutator relation

    Can anyone explain while calculating $$\left \{ \Psi, \Psi^\dagger \right \} $$, set of equation 5.4 in david tong notes lead us to $$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{-iqy} b_p^s u^s(p)e^{ipx}].$$ My question is how the above mentioned terms can be written as...
  7. Y

    Primary calculation involving the Dirac gama matrices

    When working on the exercise 3.2 of Peskin's QFT, I find one of the calculating steps confused for me. I read the solution, which is showed in the picture. I just don't understand the boxed part. I know it involved the Dirac equation, and the solution seems to treat the momentum as a operator...
  8. A

    I Chirality projection operator

    Hello everybody! I have a doubt in using the chiral projection operators. In principle, it should be ##P_L \psi = \psi_L##. $$ P_L = \frac{1-\gamma^5}{2} = \frac{1}{2} \begin{pmatrix} \mathbb{I} & -\mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} $$ If I consider ##\psi = \begin{pmatrix}...
  9. M

    Proving Even Parity for this Expression

    My idea was straight forward calculation: $$\begin{align*}\bar { \psi }' ( x' ) \gamma ^ { \mu } \partial _ { \mu }' \psi ( x' ) &= \psi^\dagger\gamma^{0\dagger}\gamma^0\gamma^\mu \partial_\mu'\gamma^0\psi = \bar\psi\underbrace{\gamma^0\gamma^\mu\gamma^0}_{=\gamma^{\mu\dagger}=-\gamma^\mu}...
  10. M

    Understanding solutions of Dirac equation

    some notes: There was actually no proof given why ##u^s(p)## or ##v^s(p)## should solve the Dirac equation, only a statement that one could prove it using the identity $$(\sigma\cdot p)(\bar\sigma\cdot p)=p^2=m^2.$$ We were using the Wely-representation of the ##\gamma##-matrices, if this should...
  11. M

    Calculating field transformation

    Homework Statement Let ##\psi(x)=u(p)e^{-ipx}##, where $$ u((m,0)) = \sqrt{m}\begin{pmatrix} \xi\\\xi \end{pmatrix}\quad\text{where}\quad \xi = \sum_{s\in \{+,-\}}c_s\xi^s\quad \text{and}\quad \xi^+\equiv\begin{pmatrix} 1\\ 0 \end{pmatrix}\quad \xi^-\equiv\begin{pmatrix} 0\\ 1 \end{pmatrix}, $$...
  12. N

    A Perturbation solution and the Dirac equation

    I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$ The solution up to first order is $$ \psi(x) =...
  13. P

    A Dirac's solution to the Klein-Gordon equation

    Dirac wanted to fix the problems with the Klein-Gordon equation by seeking a new solution to it. He wanted a relativistic solution so it makes sense that the solution needed to satisfy Einstein's energy-momentum relation. But why did it need to be of first order in time- and...
  14. C

    I Spin conservation in the Dirac equation

    Here I am considering the one particle free Dirac equation. As is known the spin operator does not commute with the Hamiltonian. However, the solutions to the Dirac equation have a constant spinor term and only an overall phase factor which depends on time. So as the solution evolves in time...
  15. It's me

    Dirac hydrogen atom vs spin symmetry

    Homework Statement Exact spin symmetry in the Dirac equation occurs when there is both a scalar and a vector potential, and they are equal to each other. What physical effect is absent in this case, that does exist in the Dirac solution for the hydrogen atom (vector potential = Coulomb and...
  16. referframe

    I Dirac's Equation vs. QFT

    I’m attempting to learn QFT on my own and would like to get an idea of just how much I still do not know. Consider a system consisting only of electrons and for the purpose of this question, pretend that particle creation and annihilation never occur. QUESTION: Would Dirac’s famous...
  17. Gene Naden

    Positive and negative plane wave solutions of Dirac equation

    I continue to be occupied with the first chapter of Lessons on Particle Physics by Luis Anchordoqui and Francis Halzen. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf. I am on page 24, where they derive equations 1.5.67, which are: ##(\gamma^\mu p_\mu-m)u(p)=0## and...
  18. Gene Naden

    A Transformation of Dirac spinors

    So I am working through Lessons in Particle Physics by Luis Anchordoqui and Francis Halzen, the link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf I am in the discussion of the Dirac equation, on page 21, trying to go from equation 1.5.49 to 1.5.51. And I get stuck. Equation...
  19. W

    I Why this system has a rotational symmetry in Dirac equation?

    why system has a rotational symmetry in dirac equation? is that a general property of all systems?
  20. B

    I The Relation Between Wavefunctions in Dirac Equation

    Can the wave function in four dimensions be expressed as e^i(kx+ky+kz-wt)?
  21. B

    I Dirac Matrices and the Pythagorean Theorem

    I understand that momentum, rest mass and energy can be put on the sides of a right triangle such that the Pythagorean Theorem suggests E^2=p^2+m^2. I understand that the Dirac equation says E=aypy+axpx+azpz+Bm and that when we square both sides the momentum and mass terms square while the cross...
  22. Turbotanten

    A What does it mean for the Hamiltonian to not be bounded?

    If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian, see Peskin and Schroeder $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 \Big( a^{s\dagger}_\textbf{p}a^s_\textbf{p}...
  23. bleist88

    A The Lagrangian Density and Equations of Motion

    Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...
  24. M

    Breit Interactions

    I am finding references and good books for the conceptual understanding of Breit Interactions. Are there any books which specifically include the topic? The related topics explaining the formulation Breit Hamiltonian and its involvement to the correction to the atomic structure calculation are...
  25. L

    A How the g factor comes from QFT?

    I'm reading the book Quantum Field Theory and the Standard Model by Matthew Schwartz and currently I'm studying the chapter 17 titled "The anomalous magnetic moment" which is devoted to computing the corrections due to QFT to the g factor. My main issue is in the begining of the chapter, where...
  26. Sophrosyne

    B The Dirac equation and the spectrum of the hydrogen atom

    I was reading that one of the successes of the Dirac equation was that it was able to account for the fine structure of some of the differences in the spectrum of the hydrogen atom. But the Dirac equation is about subatomic particles moving at relativistic velocities. But an electron around the...
  27. P

    Dirac Lagrangian invariance under chiral transformation

    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...
  28. T

    I How is Graphene's Hamiltonian rotationally invariant?

    Graphene's Hamiltonian contains first order derivatives (from the momentum operators) which aren't invariant under simple spatial rotations. So it initially appears to me that it isn't invariant under rotation. From reading around I see that we also have to perform a rotation on the Pauli...
  29. T

    A Help to rewrite Dirac equation

    $$i\frac{\partial \phi}{\partial t} = \frac{1}{2m} (\sigma .P)(\sigma .P)\phi + eφ\phi$$ 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...
  30. T

    I Need help with Dirac Equation

    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 \\...
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