What is Dirac equation: Definition and 169 Discussions

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1⁄2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.
The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.
Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1⁄2 particles.
The Dirac equation appears on the floor of Westminster Abbey on the plaque commemorating Paul Dirac's life, which was unveiled on 13 November 1995.

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  1. 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...
  2. R

    I Are special relativity rules encoded in the Dirac equation?

    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...
  3. 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...
  4. S

    A Manipulation with the Dirac equation

    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}##?
  5. K

    I Solution to the Dirac equation

    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...
  6. C

    B Dirac Equation: Combining Quantum Mechanics and Special Relativity

    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...
  7. 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...
  8. 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...
  9. 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 \\...
  10. J

    I Plane wave solution to Dirac equation

    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...
  11. DiracPool

    B Dirac Equation vs. Schrodinger Equation

    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...
  12. B

    B Dirac Equation vs Wave Function

    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?
  13. stevendaryl

    I Time/space symmetry in Dirac Equation

    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...
  14. PatrickUrania

    I Why are the gamma-matrices invariant?

    Hi, I've been studying Dirac's theory of fermions. A classic topic therein is the proof that the equation is covariant. Invariably authors state that the gamma-matrices have to be considered constants: they do not change under a Lorentz-transformation. I am looking for the reason behind this. It...
  15. H

    Gordon identity manipulation?

    The Gordon identity allows us to solve using $$2m \bar{u}_{s'}(\textbf{p}')\gamma^{\mu}u_{s}(\textbf{p}) = \bar{u}_{s'}(\textbf{p}')[(p'+p)^{\mu} -2iS^{\mu\nu} (p'-p)_{\nu}]u_{s}(\textbf{p}) $$ But how would we solve for $$2m \bar{u}_{s'}(\textbf{p}')\gamma^{\mu}v_{s}(\textbf{p}) $$ Would a...
  16. DiracPool

    Momentum-free spinors and the Dirac equation

    We can create a Dirac equation with no potential energy and zero momentum and still get spin? Is this correct? How do the Pauli spin matrices apply here? On the surface, the Dirac equation seems fairly straightforward, but when you dig even a little deeper, it's starts to become unwieldy...
  17. V

    Hyperfine structure in hydrogen

    Consider the Dirac equation for bounded electron in hydrogen atom. I am trying to get a clear physical explanation for all mathematical terms that appear in the Hamiltonian and energy spectrum. Kinetic and Coulombic potential and rest energies are the first terms and easy to identify. Then we...
  18. Spinnor

    Dirac equation in 1+1 spacetime, resources

    Trying to get a good understanding of the Dirac equation in 1 space dimension. Looking for resources and stumbled upon another source that should keep me busy over the weekend. Looks to be made as simple as possible while not leaving out the physics. Thanks to Hans for putting it online...
  19. S

    Dirac equation and structure of spacetime

    I've been reading about the Dirac equation, and most authors eventually make some statement to the effect that the fact of spin and antiparticles falling out of the equation reflects a deep connection to the structure of spacetime. Is the implication that the math requires four particle states...
  20. A

    Exploring the Dirac Equation: β,αk,pk, and Ψ(x,t)

    Hello there I have a problem about Dirac equation So I want to know what is matrices β,αk,pk value. And is it right that with Dirac equation we can calculate every particle spin and how we take dervitative of Ψ(x,t) and what is Ψ(x,t) value.
  21. LarryS

    Dirac Equation - Analytic Solution?

    The quantum harmonic oscillator is an analytic solution of the Schrodinger Equation. Does the original Dirac Equation for a free electron also have an analytic solution? Of course a "solution" of the Dirac Equation would consist of 4 functions. Thanks in advance.
  22. S

    Does charge conjugation affect parity?

    "Notice that these transformations do not alter the chirality of particles. A left-handed neutrino would be taken by charge conjugation into a left-handed antineutrino, which does not interact in the Standard Model." --https://en.wikipedia.org/wiki/C-symmetry The excerpt above seems to...
  23. R

    Satisfying Dirac equation, how to make equal to zero.

    Homework Statement Homework Equations Dirac equation: $$(\gamma ^{\mu}\rho_{\mu}-mc)\psi=0$$ The Attempt at a Solution If we multiply out the Dirac equation by inserting all it's components we get: which if I've multiplied it correctly gives $$ \begin{bmatrix} \frac{E}{c}-mc\\ 0 \\...
  24. LarryS

    Are there separate quantum fields for spin 1/2 particles in modern QFT?

    The spin observable for spin 1/2 particles is represented by Pauli Matrices acting upon a 2-dimensional Hilbert Space. In RQM, forgetting about the matter-antimatter duality for the moment, that TWO-state Hilbert Space is directly related, through the Lorentz Group, to the TWO separate...
  25. sk1105

    Make the Dirac Equation Consistent with Relativity

    Homework Statement The free Dirac equation is given by ##(i\gamma ^\mu \partial _\mu -m)\psi = 0## where ##m## is the particle's mass and ##\gamma ^\mu## are the Dirac gamma matrices. Show that for the equation to be consistent with Relativity, the gamma matrices must satisfy ##[\gamma ^\mu...
  26. C

    Dirac Equation and commutation relations

    Homework Statement Consider the Dirac Hamiltonian ##\hat H = c \alpha_i \hat p_i + \beta mc^2## . The operator ##\hat J## is defined as ##\hat J_i = \hat L_i + (\hbar/2) \Sigma_i##, where ##\hat L_i = (r \times p)_i## and ##\Sigma_i = \begin{pmatrix} \sigma_i & 0 \\0 & \sigma_i...
  27. blue_leaf77

    Relativity Introductory book on special relativity

    I'm looking for an introduction level books on special relativity. My goal is to get familiar with Dirac equation as I'm into atomic physics in this semester. My background on the subject is that I have taken a course in the past which was designed to be kind of introductory to modern physics in...
  28. HeavyMetal

    Dirac Equation and Pauli Matrices

    I have been reading through Mark Srednicki's QFT book because it seems to be well regarded here at Physics Forums. He discusses the Dirac Equation very early on, and then demonstrates that squaring the Hamiltonian will, in fact, return momentum eigenstates in the form of the momentum-energy...
  29. LarryS

    Generalized Dirac Equation for All Fermions?

    The original Dirac Equation was for the electron, a particle of spin 1/2. Is there a "Generalized Dirac Equation" that has been experimentally proven to work for all fermions, not just those of spin 1/2? Thanks in advance.
  30. T

    Dirac equation and clifford algebra

    Is it a must to know clifford algebra in order to derive the dirac equation? I recently watch drphysics video on deriving dirac equation and he use two waves moving in opposite directions to derive it, without touching clifford algebra. If this possible, what is the intuition behind it?
  31. T

    Solve 3D Dirac Equation: Simple Example

    Can anyone give me a really simple example on how to use the eqn above to solve it? The eqn is the modified schrodinger eqn that takes into account relativity.
  32. A

    Dirac Equation computational complexity

    How fast does the computational complexity of the Dirac equation, with regards to full* solution, grow with number of particles N? can we specify the order of time t(N) for this solution in terms of t(N=1)? (I assume that number of protons, neutrons and electrons combined is N - i.e. that...
  33. J

    Dirac Equation Charge Density

    I am confused about the coupling of the Dirac equation to electromagnetism. The 4-current that is the source for Maxwell's equation that arises from the Lagrangian \begin{equation} \mathcal{L}=i\overline{\psi}\gamma^\mu(\partial_\mu+ieA_\mu)\psi-m\overline{\psi}\psi \end{equation} is...
  34. D

    Derivative of a Noether current from Dirac Equation

    Homework Statement Hey guys, Consider the U(1) transformations \psi'=e^{i\alpha\gamma^{5}}\psi and \bar{\psi}'=\bar{\psi}e^{i\alpha\gamma^{5}} of the Lagrangian \mathcal{L}=\bar{\psi}(i\partial_{\mu}\gamma^{\mu}-m)\psi. I am meant to find the expression for \partial_{\mu}J^{\mu}. Homework...
  35. B

    How Should I Think About the Dirac Equation

    In Weinberg's QFT Vol. 1 he says the Dirac equation is not a true generalization of Schrodinger's equation, that it does not stand up to inspection when viewed in this light. He says it should be viewed as an approximation to a true relativistic quantum field theory of photons and electrons. a)...
  36. DrClaude

    Dirac equation in a central field (Schiff)

    Not really a homework problem, but I think it fits better in this section. Homework Statement I'm having a problem with eq. (53.12) in the book Quantum Mechanics by Schiff. In the context of the Dirac equation, we have $$ \hbar^2 k^2 = (\vec{\sigma}' \cdot \vec{L})^2 + 2\hbar (\vec{\sigma}'...
  37. S

    Question about the solutions of the dirac equation

    I am working through Greiner's text on relativistic quantum mechanics and I am confused about what appear to be two somewhat contradictory ways of presenting the solutions of the Dirac equation. In chapter 2, he just treats the equation as a system of coupled differential equations and solves...
  38. N

    How to Solve the Dirac Equation for a Two-Particle System?

    Hello I am trying to solve the dirac equation. I want to solve the dirac eq say for 2 particle system. therefore i request you to please suggest me the book or some material. Thank you
  39. LarryS

    Spin vs Dirac Equation: Understanding Physical Spin

    I believe I understand the mathematical derivation of the Dirac equation. I understand how the four 4X4 matrices, and their relation to the 2X2 Pauli Matrices, arise from that derivation. I understand that the 3 spin observables for Fermions are ALSO represented by the 3 Pauli Matrices...
  40. Matterwave

    Questions on the Dirac equation

    I must admit that I have never had a great familiarity with the Dirac equation. No matter how many times I study it, I get bogged down in the algebra and never seem to get a good understanding of it. So here's a few questions in my mind at the moment. I am referring here to the Dirac equation as...
  41. D

    Klein Paradox - Dirac equation with step potentional

    Hi all! I was reading up on the Klein paradox in Itzykson & Zuber's Quantum Field Theory (but I think this is a pretty standard part that's probably present in most QFT textbooks) and on page 62 they have a pretty straight forward solution to the Dirac equation with a step potential. I've...
  42. snoopies622

    Interpreting the Dirac equation

    Why does the \psi of the Dirac equation return four complex numbers instead of one, as in the Schrodinger equation? I know it has something to do with spin, but I'm not finding a clear answer to this question in my sources. What do these four complex numbers represent?
  43. snoopies622

    Potential energy in the Dirac equation

    Why does the Dirac equation not have a potential energy term? The Schrödinger equation does, and the Dirac equation is supposed to be the special relativity version of the Schrödinger equation, no?
  44. N

    A question about Dirac equation.

    It seems that notions of quantum field and wave function are utterly different from each other.Then is Dirac equation being equation for field or for relativistic wave function or for the both?
  45. L

    Undergraduate-level explanation of Dirac Equation?

    I am interested in learning about how the Dirac Equation was derived, how it allowed special relativity and QM to be unified, and how it predicted the existence of animatter. The explanations I have found so far are too advanced for me mathematically, and I was wondering if anybody could...
  46. P

    Curved Dirac equation, Spin connection

    (1,a^2,a^2,a^2)) from the action; \mathcal{S}_{D}[\phi,\psi,e^{\alpha}_{\mu}] = \int d^4 x \det(e^{\alpha}_{\mu}) \left[ \mathcal{L}_{KG} + i\bar{\psi}\bar{\gamma}^{\mu}D_{\mu}\psi - (m_{\psi} + g\phi)\bar{\psi}\psi \right] I can show that, i\bar{\gamma}^{\mu}D_{\mu}\psi -...
  47. R

    Dirac equation continuity issue

    So I definitely believe that the continuity of the Dirac equation holds, there is one thing that annoys me, which is that c \alpha . (-i \hbar \nabla \psi ) = c (i \hbar \nabla \psi^\dagger ) . \alpha from the first part of the Dirac Hamiltonian because the momentum operator should be...
  48. P

    Dirac Equation in (-,++++) Notation

    Just to clarify in the dirac equation (i\gamma^{\mu}\partial_{\mu} -m)\psi=0 Is it equal to (-i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i} -m)\psi=0 in (-,++++) notation?
  49. K

    The dirac equation of the hydrogen atom

    What potential would one use when evaluating the Dirac equation of the hydrogen atom? Would it simply be in the form used when examining the hydrogen atom-Schrodinger equation or does it need modification?
  50. T

    Solving the gauged Dirac equation perturbatively

    Homework Statement Given the gauge invariant Dirac equation (i\hbar \gamma^\mu D_{\mu} - mc)\psi(x, A) = 0 Show that the following holds: \psi(x, A - \frac{\hbar}{e} \partial\alpha) = e^{i\alpha}\psi(x, A) Homework Equations The covariant derivative is D_\mu = \partial_{\mu} +...
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