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. A

    Why is the Dirac delta function written as δ(x-x') instead of just δ(x')?

    What's the reason that you write δ(x-x') rather than just δ(x') both indicating that the function is infinite at x=x' and 0 everywhere else? For me that notation just confuses me, and in my opinion the other notation is easier.
  2. T

    Limit involving dirac delta distributions

    Hey All, I am trying to evaluate the limit: \lim_{x\to 0^{+}} \frac{\delta''(x)}{\delta''(x)} Where \delta'(x) is the first derivative of the dirac distribution and \delta''(x) is the second derivative of the dirac distribution. I thought about the fact that this expression...
  3. T

    The Dirac equation and its conjugate

    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...
  4. V

    Demonstration of Dirac equation covariance

    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...
  5. S

    A simple application of dirac delta shift theorem help

    A "simple" application of dirac delta "shift theorem"...help Homework Statement show that for a, b, c, d positive: δ(a/b-c/d) = bdδ(ad-bc) Homework Equations ∫f(x)δ(x-a)dx = f(a) The Attempt at a Solution Ok so I start with ∫δ(a/b-c/d)f(x)dx But I am not sure how to apply the shift...
  6. L

    Exploring the Dirac Fields in Srednicki (Ch88)

    Hi, In srednicki (ch88) he starts off considering the electron and associated neutrino, by introducing the left handed Weyl fields l, \bar{e} in the representations (2,-1/2), (1,+1) of SU(2)XU(1). The covariant derivaties are thus...
  7. M

    Global U(1) invariant of Dirac Lagrangian

    Does anybody know what interpretation the invariant corresponding to the global U(1) invariance of the Dirac Lagrangian is? I have always had it in my head that it's charge, but then I realized that uncharged free particles such as neutrinos satisfy this equation too! Any thoughts much...
  8. N

    The Dirac delta in squere root of the absolute value

    Dear Forum Users, I have got more math question rather then the physics question. Does someone know if: \mid d(x)\mid^2 equals just d(x), here d(x) is just the Dirac delta ? best regards, nykon
  9. J

    Generating Noether charges for Dirac Lagrangian

    I have been calculating the currents and associated Noether charges for Lorentz transformations of the Dirac Lagrangian. Up to some spacetime signature dependent overall signs I get for the currents expressions in agreement with Eq. (5.74) in http://staff.science.uva.nl/~jsmit/qft07.pdf . What...
  10. A

    Stupid question on Dirac alpha and beta matrices

    Homework Statement Dirac proposed that a relativistic wave equation that is linear in both space and time (unlike the Klein-Gordon equation, which is second order) has the form i\frac{\partial}{\partial t}\Psi = (\mathbf{\alpha} \cdot \mathbf{p)+\beta m)\Psi After squaring this, we'd like it...
  11. X

    Angular Momentum Problem in Dirac Notation

    Homework Statement http://img857.imageshack.us/img857/2079/dirac.png Homework Equations H|ψ> = E|ψ> L^{2}|ψ> = l(l+1)\hbar^{2}|ψ> L_{z}|ψ> = m_{l}\hbar|ψ> The Attempt at a Solution I know this problem is very simple since I've seen a very similar problem a while ago but I've completed forgot...
  12. I

    Dirac algebra (contraction gamma matrices)

    I would like to have a general formula, and I am quite sure it must exist, for: \gamma^{\mu}_{ab}\gamma_{\mu \,\alpha\beta} but I didn't succeed at deriving it, or intuiting it, I am troubled by the fact that it must mix dotted and undotted indices.
  13. N

    Dirac delta wave function impossible?

    Hello, I was under the impression that a dirac delta was a "legitimate" state for a particle: maybe not mathematically, but least physically. But I was recently told by a post-doc in QM that if your particle is in a dirac delta state at one moment, the very next moment the particle is...
  14. C

    Is the Kronecker Delta Related to the Dirac Delta in Quantum Mechanics?

    Hello PF, When I was studying Quantum mechanics, I realized that this equality should be true, <{\psi}_{n} \mid {\psi}_{m}>=\int {\psi}_{m}^*{\psi}_{n}dx={\delta }_{mn} So {\psi}_{m}^*{\psi}_{n} must be equal to dirac delta function so that we provide the kronecker delta as a solution of...
  15. J

    The Principles of Quantum Mechanics (Dirac)

    How advanced is this text? The only exposition I've had to quantum mechanics is through "The Quantum Universe" by Brian Cox and Jeff Forshaw. That book was a nice introduction but now I'm looking for something a lot more in depth. Would someone with a strong mathematical background but only a...
  16. M

    What is the interpretation of the Dirac equation and its current operator?

    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...
  17. E

    Dirac Gamma Matrices: Is Invariance Under Lorentz Transformation?

    Hi! I can define \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3 I know that the four gamma matrices \gamma^i\:\:,\;i=0...3 are invariant under a Lorentz transformation. So I can say that also \gamma ^5 is invariant, because it is a product of invariant matrices. But this equality holds: \gamma...
  18. A

    Does the dirac delta function have a Laplace transform?

    If yes, how can we find it?
  19. A

    What is the <lz> Expectation Value for Given Wave Function?

    Homework Statement Find <lz> using \Psi, where \Psi=(Y11+cY1-1)/(1+c^2)). Ylm are spherical harmonics, and <lz> is the angular momentum operator in the z direction. Homework Equations <lz> Ylm = hmYlm The Attempt at a Solution The brackets around <lz> are throwing me off...
  20. T

    Energy-momentum tensor for the Dirac spinor

    Hi there, I'm having a problem calculating the energy momentum tensor for the dirac spinor \psi (x) =\left(\begin{align}\psi_{L1}\\ \psi_{L2}\\\psi_{R1}\\ \psi_{R2}\end{align}\right)(free theory). So, with the dirac lagrangian \mathcal{L}=i\bar{\psi}\gamma^\mu\partial_\mu\psi-m\bar{\psi}\psiin...
  21. J

    Effective mass of Dirac electron increased by electrostatic potential?

    The Dirac electron in the Higgs vacuum field v and an electromagnetic field with vector potential A_\mu is described by the following equation: i \gamma^\mu \partial_\mu \psi = g v \psi + e \gamma_\mu A^\mu \psi where g is the coupling constant to the Higgs field and e is the coupling...
  22. E

    Dirac Notation in building Path Integrals

    Alright, so I was wondering if anyone could help me figure out from one step to the next... So we have defined |qt>=exp(iHt/\hbar)|q> and we divide some interval up into pieces of duration τ Then we consider <q_{j+1}t_{j+1}|q_{j}t_{j}> =<q_{j+1}|e-iHτ/\hbar|q_{j}>...
  23. J

    Mass of Dirac Electron increased by Electromagnetic field?

    The Dirac electron in the Higgs vacuum field v and an electromagnetic field with vector potential A_\mu is described by the following equation: i \gamma^\mu \partial_\mu \psi = g v \psi + e \gamma_\mu A^\mu \psi where g is the coupling constant to the Higgs field and e is the coupling...
  24. H

    The Discontinuity of Wave Functions in a Dirac Delta Potential

    consider a particle in one dimention. there is a dirac delta potential such as V=-a delat(x) the wave functions in two sides(left and right) are Aexp(kx) and Aexp(-kx) respectively. so the differential of the wave functions are not continious at x=0. what is the justification here?
  25. J

    Dirac equation for electron in EM and Higgs fields?

    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...
  26. L

    Use of Dirac delta to define an inverse

    I was wondering which are the properties of functions defined in such a way that ∫dx f(y-x) g(x-z) = δ(y-z) where δ is Dirac delta and therefore g is a kind of inverse function of f (I see this integral as the continuous limit of the product of a matrix by its inverse, in which case the...
  27. S

    How Do the Commutator Relations Lead to Equation 26 in Quantum Optics?

    Homework Statement http://quantum.leeds.ac.uk/~almut/section2.pdf Please note i am referring to the above notes I basically don't get how the maths works to get (eq(25))(eq(22))(eq(24)) = eq(26) am i missing something interms of the commutator relations ? Homework Equations The Attempt at a...
  28. maverick280857

    Dirac Principle Value Identity applied to Propagators

    Hi, How is \frac{1}{\displaystyle{\not}{P}-m+i\epsilon}-\frac{1}{\displaystyle{\not}{P}-m-i\epsilon} = \frac{2\pi}{i}(\displaystyle{\not}{P}+m)\delta(P^2-m^2) ? This is equation (4-91) of Itzykson and Zuber (page 189). I know that \frac{1}{x\mp i\epsilon} =...
  29. N

    Learning Dirac Notation: Writing Hamiltonian for 3 States

    I am new to quantum physics. My question is how to write the Hamiltonian in dirac notation for 3 different states say a , b , c having same energy. I started with Eigenvaluee problem H|Psi> = E|psi> H = ? for state a? SO it means that indvdually H= E (|a><a|) for state a and for all three...
  30. M

    How Many Bound States Does a Radially Symmetric Delta Potential with l=0 Admit?

    Dirac "bubble potential" Homework Statement Consider a radially symmetric delta potential V(r) = −Vo * δ(r − a) with l=0. How many bound states does this system admit? The Attempt at a Solution With l=0, the radial equation reduces to the one dimensional TISE. So, solving the 1D TISE with a...
  31. L

    Does Dirac manipulate his Delta function sensibly?

    In the Principles of Quantum Mechanics, Dirac derives an identity involving his delta function: xδ(x)=0. From this he concludes that if we have an equation A=B and we want to divide both sides by x, we can take care of the possibility of dividing by zero by writing A/x = B/x + Cδ(x), because...
  32. sunrah

    What is the Application of Dirac Delta in Charge Constellations?

    Homework Statement We have to give the total charge, dipol and quadrupol moments of a charge constellation, but I seem to be falling at the first hurdle. Q = \frac{1}{4\pi \epsilon_{0}} \int_{vol} \rho(\vec{r}) d^{3}\vec{r} whereby the charge density of the group of particles is...
  33. C

    Square of modified Dirac equation

    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...
  34. V

    How to Represent a Wavefunction in Dirac Notation for an Infinite Square Well?

    Homework Statement For the infinite square well, a particle is in a state given by \psi = \frac{1}{\sqrt 2}(\psi_1 + \psi_3) , where \psi_1 and \psi_3 are energy eigenstates (ground state and the second excited state, respectively). Represent this state as a column matrix \psi> in...
  35. K

    Calculate Expectation Value of Hamiltonian using Dirac Notation?

    Homework Statement I have the state: |\psi>=cos(\theta)|0>+sin(\theta)|1> where \theta is an arbitrary real number and |\psi> is normalized. And |0> and |1> refer to the ground state and first excited state of the harmonic oscillator. Calculate the expectation value of the Hamiltonian...
  36. sunrah

    Energy density (electrodynamics/ Dirac delta etc)

    So I have the following velocity vector of a charged particle in an EM field \dot{\vec{r}} = (v_{0x}cos(\alpha t) - v_{0z}sin(\alpha t), \frac{qEt}{m} + v_{0y}, v_{0z}cos(\alpha t) + v_{0x}sin(\alpha t)) and I have to state the energy density, which is defined as follows: \tau =...
  37. B

    Hermitian conjugate of plane wave spinors for Dirac equation

    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...
  38. N

    Dirac Notation and Magnitude of Bra's Help

    Hi all, I was diving into my 3rd year quantum assignment and I saw the following which I have to use for the rest of the question to prove the Cauchy-Schwarz inequality: Homework Statement || a|x> + b|y> ||^2 I only really learned a bit about Dirac notation last year, so please...
  39. P

    Dirac Equation for H atom - what's the small r behaviour?

    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...
  40. J

    Dirac delta function in reciprocal function

    From dirac, if A=B, then \frac{A}{x}=\frac{B}{x}+c\delta(x) (1) How this formula is derived? Since \frac{dlnx}{dx} = \frac{1}{x}-i\pi\delta(x) We can get \frac{A}{x} = A\frac{dlnx}{dx}+Ai\pi\delta(x) \frac{B}{x} = B\frac{dlnx}{dx}+Bi\pi\delta(x) So if A=B, \frac{A}{x}=\frac{B}{x}...
  41. U

    Normalised Energy Eigenfunction (Probability with Dirac Notation)

    Homework Statement Normalised energy eigenfunction for ground state of a harmonic oscillator in one dimension is: 〈x|n〉=α^(1/2)/π^(1/4) exp(-□(1/2) α^2 x^2) n = 0 α^2=mω/h suppose now that the oscillator is prepared in the state: 〈x|ψ〉=σ^(1/2)/π^(1/4) exp(-(1/2) σ^2 x^2)...
  42. J

    What actually is the Dirac Point ?

    What actually is the "Dirac Point"? I'm trying to find out what actually is the "Dirac Point"?! I've Googled it and searched around on the internet, looked through books, but haven't actually been able to find a definitive definition and explanation, just general references to it within the...
  43. J

    Dirac delta function in reciprocal function

    From dirac, if A=B, then \frac{A}{x}=\frac{B}{x}+c\delta(x) (1) How this formula is derived? Since \frac{dlnx}{dx} = \frac{1}{x}-i\pi\delta(x) We can get \frac{A}{x} = A\frac{dlnx}{dx}+Ai\pi\delta(x) \frac{B}{x} = B\frac{dlnx}{dx}+Bi\pi\delta(x) So if A=B, \frac{A}{x}=\frac{B}{x}...
  44. H

    How Do Different Dirac Matrix Choices Impact RQM Equations?

    Hello! I'm trying to write an essay on RQM. The problem I have encountered is the diffrent choices of matrices for the dirac equation. The two choices that I´m mixing up in my equations are: \begin{eqnarray} \gamma^0 = \left( \begin{array}{cc} I & 0 \\ 0 & -I \end{array} \right), \quad...
  45. J

    Delta dirac function times zero

    Let δ(x)=∞ at x = 0, and zero elsewhere. Then δ(x)(1-exp(x)) = ? It seems the above expression is zero. But isn't it zero times infinity at x = 0?
  46. B

    Can capacitors act as short circuits when first turned on?

    Let u(t) = \begin{Bmatrix} 1, & t \geq 0 \\ 0, & t<0 \end{Bmatrix} and let's have a simple circuit. Solo capacitor, connected to a DC voltage U0, a switch S exists. For purposes of this problem, I can mark the voltage across the capacitor as Vc(t) Vc(t)=u(t)*U0 Current...
  47. A

    Dirac Notation - Position and Momentum

    Homework Statement Show that \left\langlex|p|x'\right\rangle = \hbar/i \partial/\partialx \delta(x-x') 2. The attempt at a solution \left\langlex|p|x'\right\rangle = i\hbar \delta(x-x')/(x-x') = i\hbar \partial/\partialx' \delta(x-x') = \hbar/i \partial/\partialx \delta(x-x') For...
  48. W

    The Alternate form of the Dirac Delta Function

    Hello, I am trying to show that: \delta(x) = \lim_{\epsilon \to 0} \frac{\sin(\frac{x}{\epsilon})}{\pi x} Is a viable representation of the dirac delta function. More specifically, it has to satisfy: \int_{-\infty}^{\infty} \delta(x) f(x) dx = f(0) I know that the integral of...
  49. R

    Wavefunction collapse and dirac delta functions

    What is the experimental evidence that a wavefunction will collapse to a dirac delta function, and not something more 'smeared' out?
  50. M

    Dirac Delta from Continous Eigenfunctions

    In the equation for determining the coefficients of eigenfunctions of a continuous spectrum operator, I have trouble understanding the origin of the Dirac delta. a_f = INTEGRAL a_g ( INTEGRAL F_f F_g ) dq dg a is the coefficient, F = F(q) is an eigenfunction. From this it is shown that...
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