Dirac Definition and 900 Threads

Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century.Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger "for the discovery of new productive forms of atomic theory". He also made significant contributions to the reconciliation of general relativity with quantum mechanics.
Dirac was regarded by his friends and colleagues as unusual in character. In a 1926 letter to Paul Ehrenfest, Albert Einstein wrote of Dirac, "I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful." In another letter he wrote, "I don't understand Dirac at all (Compton effect)."He was the Lucasian Professor of Mathematics at the University of Cambridge, was a member of the Center for Theoretical Studies, University of Miami, and spent the last decade of his life at Florida State University.

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  1. H

    Divergence of inverse square field and Dirac delta

    \nabla \cdot \frac{\mathbf{r}}{|r|^3}=4 \pi \delta ^3(\mathbf{r}) What's the proof for this, and what's wrong with the following analysis? The vector field \frac{\mathbf{r}}{|r|^3}=\frac{1}{r^2}\hat{r} can also be written \mathbf{F}=\frac{x}{\sqrt{x^2+y^2+z^2}^3}\hat{x}+...
  2. P

    Deriving Dirac Hamiltonian with (+,---) Metric Signature

    Hi can anyone explain how to derive an expression for the Dirac Hamiltonian, I thought the procedure was to use \mathcal{H}= i\psi^{\dagger}\Pi -\mathcal{L}, but in these papers the have derived two different forms of the Dirac equation H=\int d^{3}x...
  3. anorlunda

    Holes=positrons in the Dirac Sea?

    Professor Susskind describes the Dirac Sea. He says remove a negative energy electron, and replace it with a positive energy electron and a hole. In other words an electron-positron pair. I'm having trouble equating holes with positrons because positrons have mass but holes don't.
  4. A

    Exploring the Properties of the Dirac Delta Function

    Prove that. \int_a^b f(x)g' (x)\, dx = -f(0) This is supposed to be a delta Dirac function property. But i can not prove it. I thought using integration by parts. \int_a^b f(x)g' (x)\, dx = f(x)g(x) - \int_a^b f(x)'g (x)\, dx But what now? Some properties: \delta...
  5. ShayanJ

    Is Dirac Delta a Function or a Distribution?

    In texts about dirac delta,you often can find sentences like "The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin". If we take into account the important property of dirac delta: \int_\mathbb{R} \delta(x) dx=1 and the fact that it is zero...
  6. 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?
  7. 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} +...
  8. L

    Uncertainty principle in terms of expectations values in Dirac notatio

    Homework Statement Show that (\Delta A)^{2} = \langle \psi |A^{2}| \psi \rangle - \langle \psi |A| \psi \rangle ^{2}\\ \phantom{(\Delta A)^{2} }=\langle \psi | (A - \langle A \rangle )^{2} | \psi \rangle , where \Delta A is the uncertainty of an operator A and \langle A \rangle is the...
  9. B

    Quick question about Dirac delta functions

    What does the square of a Dirac delta function look like? Is the approximate graph the same as that of the delta function?
  10. P

    Dirac equation, curved space time

    Hi when trying to derive this equation, i am stuck on: [\Gamma_{\mu}(x),\gamma^{\nu}(x)]=\frac{\partial \gamma^{\nu}(x)}{\partial x^{\mu}} + \Gamma^{\nu}_{\mu p}\gamma^{p} . This [\Gamma_{\mu}(x) term is the spin connection, if this is an ordinary commutator: a) is it a fermionic so +...
  11. V

    Dirac Equation: Gamma Matrices as 4-Vector Components?

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

    Taylor series expansion of Dirac delta

    I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...
  13. L

    Limit of integral lead to proof of convergence to dirac delta

    Hi, I try to prove, that function f_n = \frac{\sin{nx}}{\pi x} converges to dirac delta distribution (in the meaning of distributions sure). On our course we postulated lemma, that guarantee us this if f_n satisfy some conditions. So I need to show, that \lim_{n\rightarrow...
  14. S

    Probability Density and Current of Dirac Equation

    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...
  15. J

    A question about Dirac delta function

    Hello, Is this correct: \int [f_j(x)\delta (x-x_i) f_k(x)\delta (x-x_i)]dx = f_j(x_i)f_k(x_i) If it is not, what must the left hand side look like in order to obtain the right handside, where the right hand side multiplies two constants? Thanks!
  16. J

    Simple equations in Dirac Delta function terms

    Hi there, I'm trying to comprehend Dirac Delta functions. Here's something to help me understand them; let's say I want to formulate Newton's second law F=MA (for point masses) in DDF form. Is this correct: F_i = \int [m_i\delta (x-x_i) a_i\delta (x-x_i)]dx Or is it this: F_i = [\int...
  17. S

    Imposing Klein-Gordon on Dirac Equation

    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...
  18. P

    Understanding the Derivation of the Dirac Equation in Cosmology

    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...
  19. B

    Integrating the Dirac Delta function

    Homework Statement I am trying to integrate the function \int _{-\infty }^{\infty }(t-1)\delta\left[\frac{2}{3}t-\frac{3}{2}\right]dt Homework Equations The Attempt at a Solution I think the answer should be \frac{5}{4} because \frac{2}{3}t-\frac{3}{2}=0 when t=9/4. then (9/4-1) = 5/4...
  20. A

    The meaning of the delta dirac function

    Homework Statement For a function ρ(x,y,z) = cδ(x-a), give the meaning of the situation and describe each variable. Homework Equations As far as units go, I know that: ρ(x,y,z) = charge density = C/ m^3 δ(x-a) = 1/m and if those two are correct, then b must have units of (C/m^2)...
  21. elfmotat

    Dirac Delta Function: Does it Have a Residue?

    Does the Dirac delta fuction have a residue? Given the close parallels between the sifting property and Cauchy's integral formula + residue theory, I feel like it should. Unfortunately, I have no idea how they tie together (if they do at all).
  22. G

    Dirac Delta Integral: Compute \int_ {-\infty}^\infty e^{ikx}δ(k^2x^2-1)dx

    Homework Statement compute the integral: \int_ {-\infty}^\infty \mathrm{e}^{ikx}\delta(k^2x^2-1)\,\mathrm{d}xHomework Equations none that I have The Attempt at a Solution I don't actually have any work by hand done for this because this is more complex than any dirac delta integral I have...
  23. G

    Trying to Derive Dirac's equation following Dirac

    Hi everyone, I'm trying to follow how Dirac went about deriving his wave equation. I know there's a couple of different ways to "derive" this, but I'm trying to follow the original method that Dirac used. There's a lot of good stuff online for this, but there's a step that I get stuck at and...
  24. G

    Dirac delta function how did they prove this?

    Hi all, I'm familiar with the fact that the dirac delta function (when defined within an integral is even) Meaning delta(x)= delta(-x) on the interval -a to b when integral signs are present I want to prove this this relationship but I don't know how to do it other than with a limit...
  25. Z

    Is Dirac's Delta to the fourth power equal to 0.5?

    hello, Please attached snapshot. Does the integral 7.9 equal to 0.5 \inthμσ dzμ/dτ dzσ/dτdτ ? I'm confused as to the fourth power of Dirac's Delta. Then where does the derivative on x go ? For more on this, see MTW pg180 thanks
  26. K

    Significance of Dirac to Wave Mechanics Model

    If this seems like a homework question please forgive me, but it is merely for understanding and confidence building. I've been reading into the Bohr Model and the Wave Mechanics Model and I read through de Broglie and have proceeded to Schrodinger, Heisenberg and Dirac. At this stage, my mind...
  27. K

    K^2 = J^2 + 1/4 for the central force problem of the Dirac equation

    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 +...
  28. F

    Pions vs. Fermi Dirac Statistics and Bose-Einstein Statistics

    Hello! I have a small question, and I am not sure if I am missing something: Today I glanced at the wikipedia page for Pions, and saw this: Statistics: Bosonic Can anyone explain to me why a quark paired with a anti-quark obey Bose-Einstein Statistics? If quarks obey Fermi-Dirac statistics...
  29. F

    Understanding Dirac Delta Function: Time Derivative & Hankel Transformation

    Hi All, I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution. My question are: 1. How can I find the time derivative...
  30. F

    Is there a coordinate independent Dirac delta function?

    I have been wondering exactly how one would express the Dirac delta in arbitrary spaces with curvature. And that leads me to ask if the Dirac delta function has a coordinate independent expression. Is there an intrinsic definition of a Dirac delta function free of coordinates and metrics? Or as...
  31. jk22

    Exploring Dirac Diffusion: Gaussian Solutions?

    does anyone know how dirac diffusion looks like, i.e. The solution of dirac equation with no potential and initial condition for 1 spinor component psi(x,t0) being delta(x-x0) ? Are the solution gaussian like the schroedinger case ? Thanks.
  32. F

    Integrating with a dirac delta function

    Homework Statement I have to integrate: \int_0^x \delta(x-y)f(y)dy Homework Equations The Attempt at a Solution I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity: \delta(0)=\infty I have to express the integral in terms of function...
  33. D

    Dirac Distribution with 2 Diracs

    Hi, It is well known that \int f(x) \delta(x-a) dx = f(a) \quad\mathrm{and}\\ \int f(x) \delta^{(n)}(x-a) dx = (-1)^n f^{(n)}(a) Similarly, Is there a way to express a distribution integral with multiplicative Diracs in a compact form (e.g., a sum)? \int f(x) \delta(t-x-l_1)...
  34. N

    Laplace transform of the dirac delta function

    Homework Statement L[t^{2} - t^{2}δ(t-1)] Homework Equations L[ t^{n}f(t)] = (-1^{n}) \frac{d^{n}}{ds^{n}} L[f(t)] L[δ-t] = e^-ts The Attempt at a Solution My teacher wrote \frac{2}{s^{3}} -e^{s} as the answer. I got \frac{2}{s^{3}} + \frac{e^-s}{s} + 2 \frac{e^-s}{s^2} + \frac{2e^-s}{s^3}
  35. T

    Are Spinors Truly Invariant Under Coordinate Transformations Beyond Lorentz?

    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?
  36. T

    Ladder Operators and Dirac as the source.

    Hello, I've read that Dirac introduced the idea of the creation and annihilation operators in the solution to the quantum harmonic oscillator problem, but can anyone tell me where he did this? In a paper, or maybe in a book? I've had a little search online, but I've yet to discover...
  37. F

    Laplace transform, sum of dirac delta

    Homework Statement Homework Equations I really wish they existed in my notes! *cry*. All I can think of is that integrating or in other words summing the dirac delta functions for all t, would be infinite? None the less the laplace transform exist since its asked for in the question and i...
  38. L

    Question to a calculation rule -> Dirac formalism

    Hi everyone Homework Statement I have a question: Am I allowed to do the following, where the a's are operators and the alphas are quantum states. \langle \alpha \mid a^\dagger a^\dagger a a \mid \alpha \rangle = \langle \alpha \mid a^\dagger a^\dagger \mid \alpha \rangle \langle...
  39. C

    Dirac Notation, Observables, and Eigenvalues, OH MY

    Alright... So I'm in an 'introductory' Q.M class in college right now, it's the only one that this two-year college has, so I don't have an upper division Q.M Profs to talk to about this, and since my prof is equally confused, I turn to the internet. Okay, so everyone knows that <ψ|Aψ> = <a>...
  40. F

    Charge conjugation in Dirac equation

    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 .
  41. Jalo

    Simple Dirac Notation Problem: Dot Product of Two Vectors

    Homework Statement Imagine you have two vectors |a> and |b> such that: |c> = |a> + |b> Now imagine you want the dot product: <c|a> Is that the same as: <c|a> = [ <a|*+<b|* ] |a> = <a*|a> + <b*|a> where * represents the complex conjugate of the vector? Homework Equations...
  42. F

    Help with heat equation dirac delta function?

    Homework Statement The question was way too long so i took a snap shot of it http://sphotos-h.ak.fbcdn.net/hphotos-ak-snc7/397320_358155177605479_1440801198_n.jpg Homework Equations The equations are all included in the snapshotThe Attempt at a Solution So for question A I've done what the...
  43. J

    How to Simplify Dirac Notation for Tensor Products?

    How would you simplify this expression: <a|<b|a>|a> where ψ = |a>|b> and I'm finding ψ*ψ.
  44. S

    Dirac delta function / Gibbs entropy

    Homework Statement This is an issue I'm having with understanding a section of maths rather than a coursework question. I have a stage of the density function on the full phase space ρ(p,x); ρ(p,x) = \frac {1}{\Omega(E)} \delta (\epsilon(p,x) - E) where \epsilon(p,x) is the...
  45. C

    Is the Dirac Delta Function of x^2 Equivalent to Delta of x?

    Hi I would like to know what is the dirac delta function of x^2, I read somewhere it is equal to delta of x itself but why?
  46. R

    Calculating Operator Addition on Ket Vectors: Understanding the Dirac Problem

    If i have an operator (for example Sz) and then add it on two kets in a row i e. Sz|1>|0> How could you calculate this? should i first add it on the |1> and then the |0> ? help please
  47. P

    How Does the Scaling Property Affect the Derivative of the Dirac Delta Function?

    Hello! I should prove: \delta'(\lambda x) = \dfrac{1}{\lambda \vert \lambda \vert} \delta(x), where lambda is just a constant. If we make use of the scaling property and the definition of the distributional derivative, we find: \left( \delta'(\lambda x), f \right) =...
  48. F

    Can the Dirac Delta Function be Applied in Curved Space?

    The Dirac delta function is defined as: \int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1 Or more generally the integral is, \int_{ - \infty }^{ + \infty } {\delta (\int_{{x_0}}^x {dx'} )dx} But if the metric varies with x, then the integral becomes, \int_{ - \infty }^{ + \infty }...
  49. H

    Momentum term to be expanded in dirac gamma matrices

    Homework Statement I need help to expand some matrices Homework Equations \pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0 The Attempt at a Solution How do I expand i\hbar \gamma^0 the matrix in this term, I am a bit lost. All the help would be...
  50. E

    Reconciling DeBroglie and Dirac?

    So in 1927 Davisson and Germer showed that electrons shot at a crystal do indeed have a DeBroglie wavelength inversely proportional to their momentum (h/p)? That would mean that their wavelength is a function of their velocity, the voltage used to accelerate them, etc. But I seem to remember...
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