WebJan 29, 2024 · As a result, the Wigner function is a mathematical construct intended to characterize the system’s probability distribution simultaneously in the coordinate and the momentum space - for 1D systems, on the phase plane [X, P], which we had discussed earlier - see Fig. 5.8. Let us see how fruitful this intention is. Fig. 7.2. WebApr 14, 2024 · The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this...
Quantum forced oscillator via Wigner transform
WebOct 21, 2024 · The Jaynes–Cummings model describes a simplified model of an atom coupled to an electromagnetic field in the rotating wave approximation [1,2,3,4].We can analytically treat the electromagnetic field as a harmonic oscillator while modelling the atom as a two-level system, representing the two states of an atom. WebMay 25, 2000 · The Wigner function introduced almost 70 years ago 1, 2 to calculate quantum corrections to a classical distribution function of a quantum-mechanical system … meaning of nettlesome
23.6: Forced Damped Oscillator - Physics LibreTexts
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal … See more A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in … See more The Wigner transformation is a general invertible transformation of an operator Ĝ on a Hilbert space to a function g(x, p) on phase space and is given by Hermitian operators map to real functions. The inverse of this … See more As already noted, the Wigner function of quantum state typically takes some negative values. Indeed, for a pure state in one variable, if $${\displaystyle W(x,p)\geq 0}$$ See more It has been shown that the Wigner quasiprobability distribution function can be regarded as an ħ-deformation of another phase-space distribution function that describes an ensemble of de Broglie–Bohm causal trajectories. Basil Hiley has … See more The Wigner distribution W(x,p) of a pure state is defined as where ψ is the wavefunction, and x and p are position and momentum, but could be any conjugate … See more 1. W(x, p) is a real-valued function. 2. The x and p probability distributions are given by the marginals: $${\displaystyle \int _{-\infty }^{\infty }dp\,W(x,p)=\langle x {\hat {\rho }} x\rangle .}$$ If the system can be described by a pure state, one gets See more The Wigner function allows one to study the classical limit, offering a comparison of the classical and quantum dynamics in phase space. It has been suggested that the Wigner function approach can be viewed as a quantum analogy to … See more Weboperator, the Wigner function is not necessarily a function of the Weyl transforms of the constants of the motion. We derive conditions for which this is actually the case. The Wigner functions of the energy eigen-states of a two-dimensional isotropic harmonic oscillator serve as an important illustration. WebApr 24, 2024 · Recall, from the text, these are all ingredients of Wigner functions f normalized to 1, whence the common normalization; trivially checkable for n=0, $$ 1=\int\!\! dxdp ~f(z)= \frac{\pi \hbar }{2}\int_0^\infty\!\! dz ~e^{-z/2} L_n (z)\frac{(-)^n}{\pi \hbar} ~. $$ But from n=0 and the Sheffer sequence recursion (7), you may readily check the ... meaning of network marketing