Wigner distribution function

Wigner distribution function ). Show that this is not a probability density since it may become negative - e.g., when (x) = (sin x)/x /n. 

The Wigner distribution function f(rpt) has many properties in common with the classical distribution function. Unfortunately, however, this distribution function is not positive definite, and, therefore, it cannot be interpreted as a usual distribution function. 

The quantum mechanical definition of a distribution function in the classical phase-space is an old theme in theoretical physics. Most frequently used is the so-called Wigner distribution function (Wigner 1932 Hillery, O Connell, Scully, and Wigner 1984). Let us consider a onedimensional system with coordinate R and corresponding classical momentum P. The Wigner distribution function is defined as... 

The Wigner distribution function for the vibrational ground state of the harmonic oscillator is the product of two Gaussians, one Gaussian in P-space centered at the equilibrium distance Re and one Gaussian in P-space localized at P = 0. 

Fig. 5.2. Contour plots of two representative Wigner distribution functions PW(R,P) for two harmonic oscillators in their ground vibrational states, Equation (5.15), in the two-dimensional phase-space (R,P). The widths in the R-and in the P-directions are inversely related.

If the parent molecule is described by normal modes with coordinates q and momenta pk j the multi-dimensional wavefunction is simply a product of uncoupled one-dimensional harmonic wavefunctions ip iQk) (Wilson, Decius, and Cross 1955 ch.2 Weissbluth 1978 ch.27) and the corresponding Wigner distribution function reads... 

The six-dimensional Wigner distribution function for the ABC molecule in its lowest state is then a product of six Gaussians,... 

Wigner distribution function for excited states has negative parts. On the other hand, the extension of the semiclassical theory, established for onedimensional systems, is not straightforward especially if the two degrees of freedom are appreciably coupled, t... 

Fourier analysis. Among several different kinds of time-frequency distributions, we employ the Wigner distribution function [12], defined as... 

Equations AlO and All allow us to regard Ufp,q) of Eq. A9 as the probability with which phonons can be found at coordinate vector q with momentum vector p (although it is not normalized yet). Since the coordinate and its conjugate momentum cannot be determined simultaneously at definite values in quantum mechanics [6], this probability must be approximate. In fact, it happens to have (nonphysical) negative values in some systems [22]. In phonon systems, fortunately, it is always positive, and it can be used as a semiclassical simultaneous distribution function for the coordinate q and the momentum p. After normalization, the Wigner distribution function thus obtained is given by... 

The distribution which is most widely used is the so called Wigner distribution function [15]. It is defined as... 

If we impose an assumption that the chroniophore molecule is excited by an ultrashort laser, then the pulse covers all the frequencies. Consequently, the whole initial wavefunction is promoted to the excited state, hi the experiment, however, a cw laser with a fixed wavelength is usually used. Within the Wigner trajectories approach we have, therefore, to filter the Wigner distribution function in such a way that the energetic relation... 

Figure 9 Wigner distribution function of the n = 10 eigenfunction of the harmonic oscillator. The picture shows the extent of the wave function in phase space which has nearly optimal sampling due to the balance between the representation of the kinetic and potential energy.

Davis and Oppenheim examined the validity of the IBC model from the point of view of quantum statistical mechanics using a generalized Wigner distribution function. They thus avoided the weak coupling approximation used by both Fixman and Zwanzig. They pointed out an inconsistency in Herzfeld s argument, since if a molecule undergoes a... 

Irving and Zwanzig that the theorem expressed by Eq. 1.11 also holds in quantum-statistical mechanics if the function f is the Wigner distribution function and if the dynamical variable a is a polynomial of a degree not higher than the second in the momenta. 

The Wigner distribution function /tr(p, i , f) is the quantum analogue of the classical Boltzmann distribution function for particles of spin (t with... 

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