Double-slit case

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

Sofar the imaging results of Fig. 3.1 were discussed in very classical terms, using the notion of a set of trajectories that take the electron from the atom to the detector. However, this description does not do justice to the fact that atomic photoionization is a quantum mechanical proces. Similar to the interference between light beams that is observed in Young s double slit experiment, we may expect to see the effects of interference if many different quantum paths exist that connect the atom to a particular point on the detector. Indeed this interference was previously observed in photodetachment experiments by Blondel and co-workers, which revealed the interference between two trajectories by means of which a photo-detached electron can be transported between the atom and the detector [33]. The current case of atomic photoionization is more complicated, since classical theory predicts that there are an infinite number of trajectories along which the electron can move from the atom to a particular point on the detector [32,34], Nevertheless, as Fig. 3.2 shows, the interference between trajectories is observable [35] when the resolution of the experiment is improved [36], The number of interference fringes smoothly increases with the photoelectron energy. 

Case 1 In Eq. (28), Q = Q and C2 = C2 and nothing was placed between DS-1 and DS-2 before the recording screen. Then, both beam states are pretty much the same The conclusion is simple if this quantum state interacts with the double slit at the right hand side, interference pattern will show up at a recording screen. 

For a double slit source, the modulation due to the source being spatially resolved appears at lower frequencies, as expected, because the source is spatially bigger than the interferometric beam for any baseline length. In this particular case, this modulation due to the Stellar Interferometer almost reaches zero intensity, because we are interfering two almost identical sources and therefore destructive interference is achieved. The destructive interference is not complete due to the different optical elements in the system (i.e. transmissions of the optical elements, the spectral arm contains one extra mirror compared to the spatial arm). 

From the Zemax simulation presented in Chap. 3 it is known that the focal plane scale is 0.027 degrees/mm. The slit width is 1 mm, which converts to 97.2 arcsec. The slit separation for the case of the double slit is 5.5 mm, this is 534.6 arc sec. The FllnS spatial resolution is set to A6 = 25 arcsec, three times better than the required resolution at the maximum wavenumber, 35cm and with the maximum baseline, = 392mm. This spatial resolution represents 4 pixels per slit in the sky map. The slit separation converts to 21 pixels. Figure5.19 shows the spatial map simulated with the Sky Simulator included in FllnS for a single slit (left) and a double... 

Fig. 5.24 Ideal case results obtained for the double slit aperture in front of the MAL. Integration of the dirty image over wavenumber (left) and recovered spectral data for the 8 pixels that define the aperture (dailc green for right sUt, dark blue for left slit)...

For the case of the ideal double slit aperture presented in Fig. 5.24, the spatial position of the two apertures (left) is recovered. Again, the sources are unresolved and so the shape of the spatial map is the sum of the two dirty beams. The recovered spectra (right) for the aperture situated at the negative part of the FOV (blue) and for the aperture at the right part of the FOV (green) is consistent with the input spectrum. The amplitude variation between the 4 pixels that define each aperture is due to the interferometric dirty beam, as in the previous case. 

The results just presented correspond to a simulation where no errors have been introduced, which is the ideal case scenario. The next plots correspond to the same simulations (single slit aperture, double slit aperture and double slit with a low-pass Alter on one of the slits) when instrumental errors are introduced. 

For the three simulations shown in Figs. 5.26, 5.27 and 5.28 (single slit, double slit, and double slit with filter, respectively) it can be observed that the results are consistent both with the previous ideal simulations and with the testbed data. The residual differences have been calculated as the difference between the normalised detected spectra in the ideal case and the normalised detected spectra in the realistic... 

Equation (45) is to be preferred to the non-relativistic Eq. (36 b) in the case of heavy atoms with large spin-orbit slittings. zlf° now measures monopole relaxation and screening including relativistic effects and is extremely useful for the calculation of approximate binding energies of double and multiple vacancy levels (Eq. (38)-(40)) for which zlSCF results are very scarce. 

Boundary effects on the electrophoretic mobility of spherical particles have been studied extensively over the past two decades. Keh and Anderson [8] applied a method of reflections to investigate the boundary effects on electrophoresis of a spherical dielectric particle. Considered cases include particle motions normal to a conducting wall, parallel to a dielectric plane, along the centerline in a slit (two parallel nonconducting plates), and along the axis of a long cylindrical pore. The double layer is assumed to be infinitely thin... 

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