Wavefunction interference

Phonon Induced Delocalization in disordered solids, localization can result when a wavefunction interferes with itself due to elastic scattering and forms a standing wave. Phonon scattering can destroy this interference effect and cause the wavefunctions to be more extended. 

To improve our model we note that s- and /7-orbitals are waves of electron density centered on the nucleus of an atom. We imagine that the four orbitals interfere with one another and produce new patterns where they intersect, like waves in water. Where the wavefunctions are all positive or all negative, the amplitudes are increased by this interference where the wavefunctions have opposite signs, the overall amplitude is reduced and might even be canceled completely. As a result, the interference between the atomic orbitals results in new patterns. These new patterns are called hybrid orbitals. Each of the four hybrid orbitals, designated bn, is formed from a linear combinations of the four atomic orbitals ... 

FIGURE U5 When two ls-orbitals overlap in the same region of space in such a way that their wavefunctions have the same signs in that region, their wavefunctions (red lines) interfere constructively and give rise to a region of enhanced amplitude between the two nuclei (blue line). 

The former phase, external control tool that can be tuned to vary the interference term and hence the reaction outcome. The latter phase, 5(E), serves as an analytical tool that provides a route to the phases of the scattering wavefunctions. 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

To understand the oscillatory dependence of AE on d, it is necessary to look more closely at the interaction energy AW because, as (8.66) shows, AE is the sum of the two single-atom chemisorption energies (which are independent of d) plus AW. Hence, any effect of d on AE must arise due to AW. Alternatively, one may consider the situation in terms of the adatom wave-functions, which, as they spread out from each adatom, interfere in either a constructive or destructive fashion, thus creating oscillations in the electron density that are mirrored in the interaction. Since the wavefunctions are in or out of phase, depending on d, AE itself becomes a function of d. As d increases, the overlap of the wavefunctions decreases, and AE tends towards A eP. 

It should now be clear that the act of carrying out an experimental measurement disturbs the system in that it causes the system s wavefunction to become an eigenfunction of the operator whose property is measured. If two properties whose corresponding operators commute are measured, the measurement of die second property does not destroy knowledge of the first property s value gained in the first measurement. If the two properties do not commute, the second measurement does destroy knowledge of the first property s value. It is thus often said that measurements for operators that do not commute interfere with one another. ... 

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. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

The LCAO-MO in Eq. 1 has a lower energy than either of the atomic orbitals used in its construction. The two atomic orbitals are like waves centered on different nuclei. Between the nuclei, the waves interfere constructively with each other in the sense that the total amplitude of the wavefunction is increased where they overlap (Fig. 3.29). The increased amplitude in the internuclear region means that there is a greater probability of finding the electron in the orbital in locations where it can interact with both nuclei. Because an electron that occupies the molecular orbital is attracted to both nuclei, it has a lower energy than when it is confined to an atomic orbital on one atom. A combination of atomic... 

Fig. 2.9 The wavefunction given by eqns 2.37 and 2.38, representing the superposition of two waves of different length. Note the heats , periodic variations in amplitude arising from the Interference of the two sinusoidal waves.

It should be born in mind that our discussion now centers on extracting the resonance position Er and the total width T from the much richer scattering information that the S and Q matrices contain. Multichannel continuum wavefunctions are usually calculated for more general purposes of obtaining the scattering amplitudes and the cross sections for various state-to-state processes and of unraveling the dynamics of the whole continuum system including both resonance and nonresonance mechanisms and the intricate interference between them. 

The time-dependent wavepacket accumulates in the inner region of the PES while it oscillates back and forth in the shallow potential well as illustrated in Figure 7.8. This vibrational motion leads to an increase of the stationary wavefunction in the inner region, however, only if the energy E is in resonance with the energy of a quasi-bound level. If, on the other hand, the energy is off resonance, destructive interference of contributions belonging to different times causes cancelation of the wavefunction. 

Ionization at a given photon energy may proceed in several channels. For example, the dipole selection rule, A l- 1, permits an initial electronic state of angular momentum / to decay into two degenerate ionization channels, the / +1 and / -I channels in which the photoelectrons have angular momenta (/ + 1) h and (/ - 1 )h. Since the parameters a and P contain the radial matrix elements for ionization into the two channels, and since these elements are proportional to the overlap of the electronic wavefunctions for the initial and final states of the ionization process, it follows that a and P are functions of these overlaps. Secondly, since the two photoelectron waves have different phase and nodal structures, they may interfere this interference is also determinative of o and p values. For atomic photoionization and LS coupling, one finds ... 

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