What is Wigner: Definition and 38 Discussions

Eugene Paul "E. P." Wigner (Hungarian: Wigner Jenő Pál, pronounced [ˈviɡnɛr ˈjɛnøː ˈpaːl]; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist and also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".A graduate of the Technical University of Berlin, Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen. Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus. In 1930, Princeton University recruited Wigner, along with John von Neumann, and he moved to the United States.
Wigner participated in a meeting with Leo Szilard and Albert Einstein that resulted in the Einstein-Szilard letter, which prompted President Franklin D. Roosevelt to initiate the Manhattan Project to develop atomic bombs. Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first. During the Manhattan Project, he led a team whose task was to design nuclear reactors to convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical. Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. He became Director of Research and Development at the Clinton Laboratory (now the Oak Ridge National Laboratory) in early 1946, but became frustrated with bureaucratic interference by the Atomic Energy Commission, and returned to Princeton.
In the postwar period he served on a number of government bodies, including the National Bureau of Standards from 1947 to 1951, the mathematics panel of the National Research Council from 1951 to 1954, the physics panel of the National Science Foundation, and the influential General Advisory Committee of the Atomic Energy Commission from 1952 to 1957 and again from 1959 to 1964. In later life, he became more philosophical, and published The Unreasonable Effectiveness of Mathematics in the Natural Sciences, his best-known work outside technical mathematics and physics.

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  1. H

    I Bob and Alice, Wigner and his friend

    Hi Pfs, I wrote in another threas that when a source emits maximally entangled pairs of photons with nul global momentum and null global angular momentum, there is no local properies for the photons shared by Alice and Bob. i said that the the source only emits correlations. it has no sense to...
  2. S

    Spin-1 particle states as seen by different observers: Wigner rotation

    Summary: Suppose that observer ##\mathcal{O}## sees a ##W## boson (spin-1 and ##m > 0##) with momentum ##\boldsymbol{p}## in the ##y##-direction and spin ##z##-component ##\sigma##. A second observer ##\mathcal{O'}## moves relative to the first with velocity ##\boldsymbol{v}## in the...
  3. N

    A Measuring the Wigner Function

    Suppose i measure the phase and amplitude of some radiation, this might be a coherent state of an entangled state, how would i construct the Wigner function from these measurements?
  4. M

    A Diagonalization of 2x2 Hermitian matrices using Wigner D-Matrix

    Motivation: Due to the spectral theorem a complex square matrix ##H\in \mathbb{C}^{n\times n}## is diagonalizable by a unitary matrix iff ##H## is normal (##H^\dagger H=HH^\dagger##). If H is Hermitian (##H^\dagger=H##) it follows that it is also normal and can hence be diagonalized by a...
  5. A. Neumaier

    A Someone observing Wigner and his friend

    A human asking Wigner and his friend about the states they assign has certain and complete information about both Wigner's pure state of the lab and his friend's pure state of the device. How do these certainties show up (as certainties) in the pure state of ''lab + Wigner + his friend''...
  6. G

    Expectation values as a phase space average of Wigner functions

    Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...
  7. K

    A How to calculate the Wigner function for an entangled state

    I'm trying to calculate a Wigner Function of an entangled state, and I'm not quite sure how to proceed. I have created this state by sending in vacuum and a squeezed state into a 50/50 BS, where the output state has a density operator: $$\rho_{34}=S_{3}S_{4}S_{34}|00\rangle_{34}\langle...
  8. I

    Wigner function of cat state

    Homework Statement I am trying to calculate the wigner function for the even coherent state or cat state gives by, | \psi \rangle = N_+ \left( |\alpha \rangle + | -\alpha \rangle \right), where ##|\alpha \rangle ## is a coherent state and ## |N_+|^2 = \dfrac{1}{ 2 + 2e^{-2|\alpha|^2}}##. I am...
  9. Robsta

    Breit Wigner Curve: Finding Reaction Rate from FWHM

    Homework Statement I've got a given Breit Wigner curve of the number of decays at given energies. I've been told by several sources that the width (FWHM) of the curve gives the rate of the reaction. I can't see however how an energy here actually translates into a rate. Homework Equations I...
  10. lonewolf219

    CG coefficient in Wigner Eckhart Theorem-how to find

    Homework Statement The Clebsh Gordon coefficient is < 11;00 |11> = < l_1,m_1;l_2m_2 | kq > Homework Equations My professor determined the CG to be 1 The Attempt at a Solution How do we look up these numbers in the table? What is the direct product basis? 1x1? What is JM and what is m_1 and...
  11. Fedor Indutny

    Weinberg's QFT -- Two moving observers see a W-boson differently....

    Homework Statement Suppose that observer \cal O sees a W-boson (spin one and mass m \neq 0) with momentum \textbf{p} in the y-direction and spin z-component \sigma. A second observer \cal O' moves relative to the first with velocity \textbf{v} in the z-direction. How does \cal O' describe the...
  12. Raptor112

    A Quantum Optics Question and Wigner Functions

    I understand that Wigner function is a quasi-probability distibution as it can take negative values, but in quantum optics I see that the Q function is mentioned as often. So what is the difference between the Q function and the Wigner Function?
  13. naima

    Wigner function as average value of parity

    I found two definitions of wigner function on space time. the first uses a Fourier transform of ##\rho (q+ y/2,q-y/2)## the second uses the Weyl transformation and parity operator ## exp (i \pi \theta N)## where N is the occupation number operator. Could you give me a link which shows the...
  14. F

    Quantum Wigner master equation

    I have the quantum master equation: $$\frac{\partial\rho}{\partial t}=\frac{1}{i \hbar}[H_0,\rho]+\frac{\gamma}{i \hbar}[q,\{p,\rho\}]-\frac{D}{\hbar^2}[q,[q,\rho]]$$ And have to prove that the coordinates representation is like in the book of the link. I can't undertand how to obtain the...
  15. M

    A question about Weisskopf-Wigner approximation

    What is the physical interpretation of the Weisskopf-Wigner approximation, when it is applied in the neutral kaon system? I would say that the approximation means that a decay state has a small probability to suffer a transition (or be "transformed") into another decay state through weak...
  16. C

    Difference in phase convention for Wigner d-function

    I am looking for a way to connect the Condon-Shortley-Wigner to the Edmonds phase convention. Specifically I am writing a program to compute Wigner-d matrix coefficients From tabulated values (e.g. even Wikipedia) d^1/2_{1/2,-1/2}=(-1)^{-1/2-1/2}d^1/2_{-1/2,1/2}=-sin(theta/2) So...
  17. S

    Computing Wigner D-Matrices: Contradiction Found

    I am writing a program for computing the Wigner d-matrices and ran into an apparent contradiction: Specifically computing d^1/2_{-1/2,1/2} According to Edmonds, p.59, 4.1.27 this is given by (-1)**[1/2-(-1/2)][1!/(1! 0!)]**{1/2} sin(b/2)=-sin(b/2) Now for d^{1/2}_{1/2,-1/2} From...
  18. M

    Learn Wigner Rotation, Tensor Operator & Two-Particle Helicity State

    Hi, Is there any good books which explain/calculate Wigner rotation, tensor operator, two-particle helicity state and related stuff in detail? Thanks.
  19. Y

    Is the wigner D function a representation of SO(3)?

    Hello everyone, I'm reading a bit about the Wigner D matrix, defined by \mathscr{D}\left(\hat{n},\phi \right) = \exp[-\frac{i \phi}{\hbar}\vec{J}\cdot \hat{n}]. Now I'm wondering : is the map \pi : \text{SO(3)} \to \text{GL}\left( \mathscr{H} \right) given by R\left(\hat{n},\phi...
  20. K

    Wigner function of two orthogonal states: quantum harmonic oscillator

    The Wigner function, W(x,p)\equiv\frac{1}{\pi\hbar}\int_{-\infty}^{\infty} \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\, dy\; , of the quantum harmonic oscillator eigenstates is given by, W(x,p) = \frac{1}{\pi\hbar}\exp(-2\epsilon)(-1)^nL_n(4\epsilon)\; , where \epsilon =...
  21. A

    Evaluate Wigner Weyl Transforms for xp+px/2

    For Wigner transforming the function of operators x and p : (xp+px)/2 we need to evaluate something like: g(x,p) = ∫dy <x - y/2 | (xp+px)/2 | x+y/2> e(ipy/h) where h is h/2π. Now I am not sure how to evaluate <x - y/2 | (xp+px)/2 | x+y/2> . I mean what I did was think of |x+y/2> as a...
  22. A

    Wigner distribution in phase space

    This is about a specific property of the Wigner distribution in phase space. My professor mentioned the other day that the Wigner distribution treats all functions of momentum and space on the same footing as momentum itself or at least that's what I recall.He mentioned a specific problem where...
  23. P

    Breit Wigner Formula & Positron Annihilation

    Hey guys! Breit Wigner Formula describes the cross section for interactions that proceed dominantly via a intermediate particle (O*) A+B → O* → C + D: σ = \frac{2\Pi}{k^{2}}\frac{Γ_{i}Γ_{f}}{(E-E_{o})^{2} + (Γ/2)^{2}} A short question: Does the formula apply to situations when the...
  24. B

    Breit Wigner for photon intermediates

    Hey! I'm hoping someone can help me understand a basic problem I'm having with understanding the BW formula: \sigma(i,j) = \frac{\pi}{k^2} \frac{\Gamma_i \Gamma_j}{(E - E_0)^2 + \Gamma} In this, E_0 is the "characteristic rest mass energy of the resonance." I thought this meant the...
  25. S

    Definition of wigner D matrix

    Homework Statement I'm not sure if this is the appropriate board, but quantum mechanics people surely know about spherical harmonics. I need to implement the Wigner D matrix to do spherical harmonic rotations. I am looking at...
  26. M

    Particles and Wigner little groups

    Hello, from Weinberg's Quantum Field Theory book I am confused about the equation (2.5.5). I'll describe the problem briefly here, but in any case, here's that page from Weinberg's book (page 64)...
  27. P

    Wigner 3j symbol recursion relation

    Hi all! Homework Statement I have to show: \sqrt{(j \pm m ) (j \mp m+1} <j_1 j_2 m_1 m_2 | j_1 j_2 j m\mp 1 > = \sqrt{(j_1 \mp m_1 ) (j_1 \pm m_1+1} <j_1 j_2 m_1 \pm1, m_2 | j_1 j_2 j m > +\sqrt{(j_2 \mp m_2 ) (j_2 \pm m_2+1} <j_1 j_2 m_1 , m_2 \pm1 | j_1 j_2 j m > Homework...
  28. T

    Wigner Matrix or Wigner D-Matrix?

    Hello guys, I'm reading the attached article, and I found there the Wigner Matrix, the first equation in the second page... is that the Wigner D-Matrix? I really got lost in that. It doesn't look like the Wigner D-Matrix I see everywhere, and it's not the so-called random matrix... anyone...
  29. T

    Parity rule for wigner D-matrices

    Hi! Does anyone know how Wigner D-matrices transform under parity? Is it something like D^j_{m m'} (\pi - \theta, \phi + \pi) = (-1)^{j +m-m'} D^j_{m m'}(\theta, \phi)?
  30. S

    Primitive lattice vectors, reciprocal lattice, wigner seitz cell

    How can i develop a sketch of the lattice and reciprocal lattice from vector form a=i+4j b=3i i know how to draw the wigner site cell, but I am having problems developing a sketch from vectors. what is the method for working it out..please help
  31. W

    Wigner-Eckart Theorem: Rigorous Math Treatment

    I originally posted this in the Science Book and discussion forum but received no help. Am posting it here, hoping that I will. I was looking for material that would go over the Wigner Eckart theorem and mathematics of Angular Momentum in more rigor than the traditional texts do (in specific...
  32. W

    Jordan wigner transform and periodic boundary condition

    i think jordan wigner transform, when applied to open boundary system, can simplify a spin 1/2 system to a free fermion system but there is a difficulty in the case of periodic boundary condition in this case, we have to deal with terms like S_N^+S_1^-=(-)^{\sum_{k=1}^{N-1}n_k}...
  33. O

    Are Wigner Functions eigenfunctions of J^2 and Jz?

    Homework Statement I have a question related to representation of rotation operator R in the basis spanned by the eigenvectors of J2 and Jz. I am studying from Quantum Mechanics by Zettili. The development of Wigner D-matrix and its elements Dj (Wigner functions) is clear. But the book goes on...
  34. S

    Volume of a Wigner Sietz cell

    I have a few questions, first of all I'm trying to figure out how to find the volume of a 3 dimmensional wigner sietz cell. I have the 8th edition intro to solid state physics book by charles kitle and there is no where in the book that shows me how to find the volume of wigner seitz cell...
  35. E

    Understanding the Physical Meaning of Wigner Rotation

    i have read about wigner rotation but i think i can't understand it well , i want to know the physical meaning of it .
  36. A

    Wigner Eckart theorem / Electric dipole

    Homework Statement The eletric dipole of the atom D = qR is a vector op, ie transforms according to j = 1 rep of SU(2). Use wigner eckart theorem show <1, 0, 0|D|1, 0, 0> = 0 (<n',l',m'|D|n, l, m> = 0 Homework Equations...
  37. W

    What is the advantage of the truncated wigner approximation?

    In quantum optics and bose-einstein condensates, this is a well known technique however, i still cannot grasp its essense. in bec, what is its advantage over the gross-pitaevskii equation?
  38. N

    Wigner Effect in Metallic Lattices

    Hi, I'm a bit confused with the Wigner effect concept. This effect is normally associated to damages in moderator material typically graphite. But metallic cladding of the fuel element is also exposed to fast neutron, but Wigner effect is seldom being used as a term to explain the damages in...