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. J

    Calculations with Dirac deltas and position/momentum operators

    Homework Statement Consider the matrix elements of \hat{x} in momentum space. That is, evaluate \langle p | \hat{x} | \psi(t) \rangle in terms of the momentum space wave equation \langle p | \psi(t) \rangle . Homework Equations \langle x | p \rangle = \frac{1}{\sqrt{2 \pi \hbar}}...
  2. C

    Solution to the dirac equation and the square root of a matrix?

    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...
  3. O

    Proof involving Dirac Delta function

    Prove that x \frac{d}{dx} [\delta (x)] = -\delta (x) this is problem 1.45 out of griffiths book by the way. Homework Equations I attempted to use integration by parts as suggest by griffiths using f = x , g' = \frac{d}{dx} This yields x [\delta (x)] - \int \delta (x)dx next I tried...
  4. P

    Derivative of Dirac Delta - Fourier Transform - Time Differentitation Property

    Homework Statement I am using the time differentiation property to find the Fourier transform of the following function: Homework Equations f(t)=2r(t)-2r(t-1)-2u(t-2) The Attempt at a Solution f'(t)=2u(t)-2u(t-1)-2δ(t-2) f''(t)=2δ(t)-2δ(t-1)-?? Can somebody explain what the...
  5. M

    Dirac notation expressions as integrals

    Can anyone point me to how to interpret Dirac notation expressions as wave functions and integrals beyond the basics of     <α| = a*(q)     |β> = b(q)     <α|β> = ∫ a* b dq For example in the abstract Dirac notation the expression     |ɣ> (<α|β>) can be evaluated as     (|ɣ><α|) |β>  ...
  6. J

    Eigenstates of Dirac Potential in Momentum Space

    Homework Statement Consider a particle moving in one dimension and bound to an attractive Dirac δ-function potential located at the origin. Work in units such that m=\hbar=1. The Hamiltonian is given, in real (x) space, by: H=-\frac{1}{2}\frac{d^2}{dx^2}-\delta (x) The (non normalized)...
  7. B

    Dirac delta function, change of variable confusion

    The Dirac delta "function" is often given as : δ(x) = ∞ | x = 0 δ(x) = 0 | x \neq 0 and ∫δ(x)f(x)dx = f(0). What about δ(cx)? By u=cx substitution into above integral is, ∫δ(cx)f(x)dx = ∫δ(u)f(u/c)du = 1/c f(0). But intuitively, the graph of δ(cx) is the same as the graph of...
  8. F

    Must the Fourier transform exist for Dirac delta functions?

    I originally asked this in the Calculus & Analysis forum. But perhaps this is better suited as a question in Abstract algebra. For the set of all Dirac delta functions that have a difference for an argument, we have the property that: \int_{ - \infty }^\infty {{\rm{\delta (x -...
  9. S

    Dirac Matrix Property? Possible Book mistake?

    Dirac Matrix Property? Possible Book mistake? Derive KG from Dirac I got a copy of QFT demystified and on pg. 89 he says he can write \gamma_{\nu} \gamma^{\mu} = g_{\nu \sigma} \gamma^{\sigma} \gamma^{\mu} = g_{\nu \sigma} \frac{1}{2} (\gamma^{\sigma} \gamma^{\mu} + \gamma^{\mu}...
  10. E

    Square integrable functions - Hilbert space and light on Dirac Notation

    Square integrable functions -- Hilbert space and light on Dirac Notation I started off with Zettilis Quantum Mechanics ... after being half way through D.Griffiths ... Now Zettilis precisely defines what a Hibert space is and it includes the Cauchy sequence and convergence of the same ... is...
  11. M

    Matrix operators Dirac notation

    I'm having trouble seeing how an operator can be written in matrix representation. In Sakurai, for an operator X, we have: X = \sum \sum |a''> <a''| X |a'> <a'| since of course \sum |a> <a| is equal to one. Somehow, this all gets multiplied out and you get a square matrix with the...
  12. M

    Dirac notation and conjugate transpose in Sakurai

    In Sakurai's Modern Quantum Mechanics, he develops the Dirac notation of bras and kets. In one part, he states (page 17): <B|X|A> = (<A|X^|B>)* = <A|X^|B>* where X^ denotes the Hermitian adjoint (the conjugate transpose) of the operator X. My question is, since a bra is the conjugate...
  13. Spinnor

    A Pauli and a Dirac electron, momentum density and spin.

    In the article, "What is spin", by Hans C. Ohanian we are shown how to take the wave-function for a Dirac electron with spin up and localized in space and then determine the momentum density in the Dirac field. The momentum density divides into two parts, a part that depends on the motion of the...
  14. S

    What's a thorough QM book besides Dirac?

    Hi, PF. I've got a question for you. Maybe this would be better posted in the science education or discussion sections, but it's directly related to QM. I'm just finishing up my undergrad coursework and I've taken QM using Griffiths. It's an okay book, but it does a bit of jumping around, and...
  15. S

    Is the delta in the commutation relations of QFT a dirac delta or a kronecker?

    If it's a dirac delta doesn't it mean it's infinite when x=y? Or is it a sort of kronecker where it's equal to one but the indices x and y are continuous? I'm confused.
  16. Spinnor

    How to graph Dirac equation, some complex numbers?

    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...
  17. F

    Variation of Dirac delta function

    Is it possible to take the variation of the Dirac delta function, by that I mean take the functional derivative of the Dirac delta function?
  18. J

    Brain freeze on Dirac EQ v. Dirac Hamiltonian

    Alright. So the Dirac Eq is (i \gamma^{\mu} \partial_{\mu} - m) \psi = 0 or putting the time part on one side with everything else on the other and multiplying by \gamma^0 , i \partial_t \psi = (i \gamma^0 \vec{\gamma} \cdot \nabla + \gamma^0 m) \psi I would think that this is the...
  19. L

    Derivative of Dirac Delta function

    Hello I'm trying to figure out how to evaluate(in the distribution sense) \delta'(g(x)). Where \delta(x) is the dirac delta function. Please notice that what I want to evaluate is not \frac{d}{dx}(\delta(g(x))) but the derivative of the delta function calculated in g(x). If anyone could post...
  20. F

    Algebraic structure of Dirac delta functions

    OK, the Dirac delta function has the following properties: \int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1 and \int_{ - \infty }^{ + \infty } {f({x_1})\delta ({x_1} - {x_0})d{x_1}} = f({x_0}) which is a convolution integral. Then if f({x_1}) = \delta (x - {x_1}) we get...
  21. Xezlec

    Why doesn't the energy come out right in the Dirac Equation?

    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...
  22. E

    Solving an equation with Dirac delta functions

    Hello, I'm dealing with the following equation: A e^{jat} + B e^{jbt} = C e^{jct} \forall t \in \mathbb{R} My book says: given nonzero constants A,B,C, if the above equation yelds for any real t, then the a,b,c constants must be equal. The above statement is prooved by taking the Fourier...
  23. K

    Dirac delta function with contineous set of zeros

    hi! i have a question regarding the delta function. if i have a delta distribution with an argument that is a function of multiple arguments, somthimg like: ∫δ(E-p^{2}_{i}/2m)dp^{N}, ranging over +-∞ now, the argument of the delta function vanishes on a sphere. i can evaluate the...
  24. N

    The Schrödinger equation as the non-relativistic limit of the Dirac equation

    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...
  25. N

    Charge conjugation for Dirac particles (error in problem?)

    Homework Statement Show that if \psi is a down-spin anti-electron, and we apply charge conjugation, then \psi^C is an up-spin electron. The Attempt at a Solution My calculations suggest that the anti-electron indeed becomes an electron; however, spin does not change for me. Is it possible...
  26. F

    Comparing Dirac and Schrodinger Equations for the Hydrogen Atom

    The Schrodinger equation solved for the hydrogen atom gave good agreement with spectral lines, except for line doublets. To account for these electron spin theory was grafted onto the theory, despite the problem of electron being a point particle. In 1928 Dirac gives his different answer...
  27. S

    Delta Dirac: Showing it's a Distribution

    I realize it's not a function in the classical sense, but how would one show that the delta dirac function is a distribution i.e. how do I show it's continuous and linear given that it's not truly a function?
  28. N

    Neglected solutions to the (free) Dirac equation?

    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...
  29. B

    Why does Dirac Equation describe spin 1/2 particles?

    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...
  30. T

    Can we interchange the Dirac Matrices?

    Ok, first off I will admit that I really am pretty much ignorant of proper QM, as I am a first year undergraduate at a UK university. Today our lecturer, in the final lecture of a Vibrations and Waves course, demonstrated how the Schrodinger equation is derived from applying the Energy and...
  31. snoopies622

    Looking for a Dirac book on qft

    I just happily bought a used copy of Dirac's Lectures on Quantum Mechanics from Amazon.com. I also want his Lectures on Quantum Field Theory but they don't carry it. Anyone know where I can find a copy?
  32. T

    Problem on integrating dirac delta function

    Hi there, I am trying to integrate this: http://imm.io/oqKi I should get the second line from the integral, but I can't show it. This should somehow relate to the Heaviside step function, or I am completely wrong. Any ideas? Sorry about the url, I fixed it.
  33. J

    Property of the dirac delta function

    Hello team! I saw the other day in a textbook that the Dirac delta function of the form d(x-a) can be written as d(a-x) but the method was not explained. I was wondering if anyone know where this comes from. I've been googling but can seem to find it out. Any help would be appreciated...
  34. L

    Dyade Dirac Notation: Why Last Equation?

    \{\vec{A},\vec{B}\}\cdot \vec{C}=\vec{A}(\vec{B}\cdot \vec{C}) \vec{C} \cdot \{\vec{A},\vec{B}\}=(\vec{C}\cdot \vec{A}) \vec{B} I want to write dyade in Dirac notation. (|\vec{A}\rangle\langle\vec{B}|)|\vec{C}\rangle= |\vec{A}\rangle\langle\vec{B}|\vec{C}\rangle...
  35. X

    What does couples as the 4th component of a vector mean in the Dirac equation?

    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...
  36. R

    Dirac formulation of QM, motivation for SE

    In Dirac's "The Principles of Quantum Mechanics", in chapter V on the equations of motion Dirac proceeds with a line of reasoning that is something along the following lines (I've modified it a bit to coincide with what's taught in the course I'm taking) 1. We assume that the motion...
  37. F

    Exploring the Dirac Equation: Positive & Negative Energy Solutions

    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...
  38. M

    Dirac Delta Function (electrodynamics)

    I'm having a hard time grasping when I should use this little "function". I'm using Griffith's Intro to Electrodynamics and either he doesn't touch on it enough or I'm missing the point. From what I think I understand I'm to use it when there would be a singularity in a result or calculation(?)...
  39. E

    Harmonic Oscillator in Dirac Theory

    Hello everyone, i'm looking for anypaper or such kind of thing that explain the resolution of the harmonic oscillator in the Dirac Theory. I have worked with the exact spin symmetry. I feel like a fish out the water and I'm sure that there are lot of bibliography about this area, but i...
  40. B

    Integrating the following delta dirac function should yield min(t,s), but how?

    Homework Statement I need to understand how to integrate \int_{0}^{t}\int_{0}^{s} \delta(\tau-\tau')d\tau d\tau' The solution is min(t,s) Homework Equations See aboveThe Attempt at a Solution min(t,s)
  41. J

    Dirac Notation: Am I doing this right?

    Homework Statement Find <P>. P = i√(mhw/2)(a†-a). Note a† and a are the ladder operators. P is the momentum operator of the harmonic oscillator. |ψ > = (1/sqrt(2))[ |1> - i |2>] The answer should be zero, can someone check my work?Homework Equations a† |n> = sqrt(n+1)|n+1> a |n> =...
  42. E

    Dirac equation and gamma factor

    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...
  43. R

    What are the properties of Dirac notation and operators?

    Homework Statement [A^{+}A]=1 A|a>=\sqrt{a}|a-1> A^{+}|a>=\sqrt{a+1}|a+1> <a'|a>=\delta_{a'}_{a} Homework Equations what is 1 <a|A|a+1> 4. <a+1|A^{+}|a> 3. <a|A^{+}A|a> 4. <a|AA^{+}|a> The Attempt at a Solution 1. <a|A|a+1> =<a|\sqrt{a+1}|a+1-1>=\sqrt{a+1}<a|a> since a=a and...
  44. P

    Covariant Bilinears: Fierz Expansion of Dirac gamma matrices products

    Homework Statement So my question is related somehow to the Fierz Identities. I'm taking a course on QFT. My teacher explained in class that instead of using the traces method one could use another, more intuitive, method. He said that we could use the fact that if we garante that we have the...
  45. B

    How to incorporate the neutral current into the Dirac equation

    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...
  46. J

    Proving the Delta Function Identity Using the Local Behavior of Functions

    Homework Statement See http://mathworld.wolfram.com/DeltaFunction.html I want to show (6) on that page. I can show it using (7), but we aren't supposed to do that. I already proved (5), and my prof says to use the fact that (5) is true to get the answer. Homework Equations The...
  47. andrewkirk

    Dirac Delta function as a Fourier transform

    It is fairly easy to demonstrate that the Dirac delta function is the Fourier transform of the plane wave function, and hence that: \delta(x)=∫_{-∞}^{∞}e^{ikx}dk (eg Tannoudji et al 'Quantum Physics' Vol 1 p101 A-39) Hence it should be the case that ∫_{-∞}^{∞}e^{ik}dk = \delta(1) = 0...
  48. T

    The Limitations of the Dirac Sea in Explaining Particle Motion

    I'm not sure if my interpretation is correct, but this Dirac Sea interpretaton does as far as I understand this, tell us that every energy level from -infinity to a certain energy level E<0 is filled with anti-particles. And this should be true for every single location in the universe. If...
  49. S

    Chiral currents of a Dirac plane wave

    I am currently trying to check the formula for the chiral current of the Dirac equation for a plane wave solution (found here ), that is, j_{R}^\mu = \psi_R^\dagger \sigma^\mu \psi_R j_{L}^\mu = \psi_L^\dagger \sigma^\mu \psi_L With \psi_R = I( \cosh(\frac{\theta}{2}) + \sigma^i...
  50. A

    Dirac delta and fourier transform

    In my book the dirac delta is described by the equation on the attached picture. This realtion is derived from the Fourier transform, but I'm not sure that I understand what it says. If u=t it is clear that one gets f(u) in the Fourier inversion theorem. But why wouldn't u=t? In the derivation...
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