Wavepacket radial

A mixed DVR (discrete variable representation)43 for all the radial coordinates and basis set representations for the angular coordinates are used in the wavepacket propagation.44... 

The general philosophy is in fact identical to that used for the radial variables. The reader is referred to Ref. 87 for further details of the manner in which the operation of the Hamiltonian operator on the wavepacket is accomplished. 

Irrespective of whether the photon is considered as a plane wave or a wavepacket of narrow radial extension, it must thus be divided into two parts that pass each aperture. In both cases interference occurs at a particular point on the screen. When leading to total cancellation by interference at such a point, for both models one would be faced with the apparently paradoxical result that the photon then destroys itself and its energy hv. A way out of this contradiction is to interpret the dark parts of the interference pattern as regions of forbidden transitions, as determined by the conservation of energy and related to zero probability of the quantum-mechanical wavefunction. 

These questions appear to be understandable in terms of both photon models. The wavepacket axisymmetric model has, however, an advantage of being more reconcilable with the dot-shaped marks finally formed by an individual photon impact on the screen of an interference experiment. If the photon would have been a plane wave just before the impact, it would then have to convert itself during the flight into a wavepacket of small radial dimensions, and this becomes a less understandable behavior from a simple physical point of view. Then it is also difficult to conceive how a single photon with angular momentum (spin) could be a plane wave, without spin and with the energy hv spread over an infinite volume. Moreover, with the plane-wave concept, each individual photon would be expected to create a continuous but weak interference pattern that is spread all over the screen, and not a pattern of dot-shaped impacts. 

Another somewhat different example of coherence of electronic states is a radial electronic Rydberg wavepacket whose dynamics can be studied in pump-probe experiments. 

Figure 1. Intramolecular vibrational density redistribution IVR of Na3

We show how one can image the amplitude and phase of bound, quasibound and continuum wavefunctions, using time-resolved and frequency-resolved fluorescence. The case of unpolarized rotating molecules is considered. Explicit formulae for the extraction of the angular and radial dependence of the excited-state wavepackets are developed. The procedure is demonstrated in Na2 for excited-state wavepackets created by ultra-short pulse excitations. 

In the above example, the spatial localisation produced was in the radial coordinate. It is also possible, by using Rydberg states, to produce localisation in the angular coordinates, although the method of prepare tion of the wavepacket is then different. 

Figure 9.9 The radial probability distribution of an n 30 wavepacket in a hydrogen atom. The populations and binding energy of the eigenstates in the wavepacket are shown in the inset. Note that the figure shows the to probability distribution. A plot of the to wavepacket amplitude would show a single dominant feature in the region of the innermost lobe of the initial state (Is), with small wiggles at larger r (from Smith, et al., 2003b).

The center of the Rydberg wavepacket, (r)t, undergoes periodic radial oscillations, analogous to the planetary motions (Kepler orbits). The Kepler period of (r)t for the Rydberg wavepacket with center-n, n enter, is... 

The latter are displayed in Fig. 36. At first, the radial density is very compact and performs one vibrational oscillation. Then, at a time of about 1.5 ps, a bifurcation takes place where one part of the former localized wavepacket moves into the fragmentation channel (which is reached, by definition, for distances larger than... 

An explanation of the bifurcation can be found from the angular density (lower panel of Fig. 36). Because we start from the rotational ground state, the first excitation step prepares a wavepacket with the rotational quantum number 7=1. Then, the density, initially, is proportional to Tio(0,0) cos (0) 2. It is seen that the density changes with time and that a depletion at angles smaller than 7i/2 occurs, which goes in hand with a concentration of density at a value of n. It is now straightforward to find a classical interpretation of the (radial) density bifurcation in regarding the classical force which stems from the external field interaction and acts in the radial direction ... 

On the other hand, when the intrashell couplings are not included in the calculation, the intense field radial wavepackef is qualifafively identical with the ones obtained with weak fields. In this case, the wavepackets for weak as well as for strong fields oscillafe in the same way. 

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