Quantum momentum distribution function

F(pj) is the quantum momentum distribution function. The normalization follows from (1.26) ... 

Ignoring quantum-mechanical features such as spreading and interference, the essential classical physics then are described by the electron-momentum distribution function [17]... 

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

Alternatively, we can work in momentum-space with the momentum distribution given by the square of the modulus of the momentum wavefunc-tion. However, because of Heisenberg s uncertainty relation it is impossible to specify uniquely the coordinates and the momenta simultaneously. Either the coordinates or the momenta can be defined without uncertainty. In classical mechanics, on the other hand, the coordinates as well as the momenta are simultaneously measurable at each instant. In particular, both the coordinates and the momenta must be specified at t — 0 in order to start the trajectory. Thus, we have the problem of defining a distribution function in the classical phase-space which simultaneously weights coordinates and momenta and which, at the same time, should mimic the quantum mechanical distributions as closely as possible. 

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... 

Fig. 4.10. Electron momentum distributions for neon ( 75oi = 0.79 a.u. and /. 02 = 1.51 a.u.) subject to a linearly polarized monochromatic field with frequency ui = 0.057 a.u. and intensity I = 3.0 x 1014W/cm2, as functions of the electron momentum components parallel to the laser-field polarization. The left and the right panels correspond to the classical and to the quantum-mechanical model, respectively. The upper and lower panels have been computed for a contact and Coulomb-type interaction Vi2, respectively. In panels (a) and (d), and (h) and (e), the second electron is taken to be initially in a Is, and in a 2p state, respectively, whereas in panels (c) and (/) the spatial extension of the bound-state wave function has been neglected. The transverse momenta have been integrated over...

Both in the quantum-mechanical and in the classical calculations, only the first return of the electron to the ion has been considered. Due to wave function spreading, the contributions of the longer orbits are suppressed. The most remarkable result of these investigations is the surprisingly high sensitivity of the (pi, p2 )-momentum distribution to variations of the CE phase. In principle, this lends itself to a very precise determination and control of this parameter. 

In Eq.(23), J and K are the rotational quantum numbers for the total angular momentum and the component projected to the molecular principal axis, p is the anisotropy of the Raman polarization tensor, pj is the thermal rotational distribution function in the initial state, and N specifies the selection rule of the rotational Raman transitions. (J)=0 if J<0 and 4(J)=1 or all other values of J,... 

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... 

For the preceding examples classical distribution functions are sampled in choosing initial values for the coordinated and momenta. However, for some simulations it is important to sample a quantum mechanical distribution in selecting coordinates and momentum. The classical probability distribution for a vibrational mode peaks at the classical turning points (Fig. 

In quantum chemistry, the state of a physical system is usually described by a wave function in the position space. However, it is also well known that a wave function in the momentum space can provide complementary information for electronic structure of atoms or molecules [1]. The momentum-space wave function is especially useful to analyse the experimental results of scattering problems, such as Compton profiles [2] and e,2e) measurements [3]. Recently it is also applied to study quantum similarity in atoms and molecules [4]. In the present work, we focus our attention on the inner-shell ionization processes of atoms by charged-particle impact and study how the electron momentum distribution affects on the inner-shell ionization cross sections. 

In the present work, we showed applications of the momentum distribution only to K- and L-shell ionization processes. However, it would be interesting to apply the present study in the BEA for outer-shell electrons, where the wave-function effect is more important and quantum-mechanical calculations of the ionization cross section become more tedious. 

Some years later, aided by considerably more rapid computers than available to Wall and co-workers, Karplus, Porter, and Sharma reinvestigated the exchange reaction between H2 and H [24]. As with the earlier work, the twelve classical equations of motion were solved. In addition, discrete quantum-mechanical vibrational and rotation states were included in the total energy so that the trajectories were examined as a function of the initial relative velocity of the atom and molecule and the rotational and vibrational quantum numbers j and v of the molecule. The more sophisticated potential energy surface of Porter and Karplus was used [7], and the impact parameter, orientation and momentum of the reactants, and vibration phase were selected at random from appropriate distribution functions. This Monte Carlo approach was used to examine 200-400 trajectories for each set of VyJ, and v. The reaction probability P can be written as... 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

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