Radial wave packet

Radial wave packets are localized only with respect to the radial electronic coordinate r. They consist of a superposition, (7.28), with several different values of n but only a few values of I or m. 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

As an example of this formulation we consider pulsed photoassociation of a coherent wave packet of cold Na atoms [345], The colliding atoms are described by an (energetically narrow) normalized Gaussian packet of J = 0 radial waves ... 

Figure 13 Comparison of the experimental and a quantum mechanically computed (by exact wave packet propagation using an ab initio computed potential energy) spectrum of a nonrotating Na, molecule pumped to its B electronic state. (Courtesy of Experiment by S. Rutz, E. Schreiber, and L. Woste Computations by B. Reischl, all of the Free University of Berlin) (a) The short time dynamics Shown is the population of the excited state vs. time as determined by a pump-probe experiment and by the computation (points connected by a straight-line segments). The periodicity (about 320 fs) is due to the symmetric stretch motion, (b) A frequency spectrum. The long time dynamics (as reflected in the well-resolved spectrum) show the contribution of a different set of vibrational modes. The dominant peaks can be identified as the radial pseudorotation motion of Na,(B) while the splittings are due to the angular pseudorotational motion. (Adapted from B. Reischl, Chem. Phys. Lett., 239 173 (1995) and V. Bonacic-Koutecky, J. Gaus, J. Manz, B. Reischl, and R. de Vivie-Riedle, to be published.)...

Fig. 2. Time dependence of a hydrogen wave-packet with averaged principal quantum number h = 80 and FWHM An = 4. On top we show the autocorrelation junction l(tp(t)l ip(0)[) and on bottom the radial expectation value. Both values document the occurrence of periodic structures as well as revival and partial revival structures as indicated by vertical dotted lines.

This angular oscillation gives rise to the observed structure, e.g., the even odd behaviour at f = 774 V/cm is due to the close to the 1 2 beat of the Kepler and the angular period. Hence every 2nd Kepler period the wave-packet is mainly back to its initial position. Due to the spreading of the wave-packet and the small deviation from the ratio 2, this even-odd behaviour vanishes after a few periods. This could also be shown by computing the radial and angular momentum expectation value [22]. 

H.M. Nilsen, J.P Hansen, Generation of an angular and radial Rydberg wave packet in a single laser pulse, Phys. Rev. A 63 (2001) 011405 (R). 

Fig. 7.16 (a) Excitation of a radial Rydberg wave packet by a short pump pulse from the ground state /> into a Rydberg state at its inner turning point and (b) its detection by a delayed photoionizing probe pulse [891]... 

A. Wolde, I.D. Noordam, H.G. MiiUer, A. Lagendijk, H.B. van Linden, Observation of radially localized atomic electron wave packets. Phys. Rev. Lett. 61, 2099 (1988)... 

For each partial wave J and parity e, the Hamiltonian and wave packet are discretized in the BF frame in mixed representation [21, 64, 80, 89,160] discrete variable representation (DVR) is employed for the two radial degrees of freedom and finite basis representation (FBR) of normalized associated Legendre function i jK(O) for the angular degree of freedom. Thus the wave packet in the BF frame is written as... 

When the Rydberg levels n m) are excited by a short laser pulse with the duration r from the ground state i) where the electron is localized within a few Bohr radii around the nucleus, the excitation process is fast compared to the oscillation period of a radial Rydberg wave packet, provided that T This fast excitation corresponds to a vertical tran-... 

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