Vanishing particle widths

Typically, the vanishing particle width occurs at e = —pTlT1/2 which is related to the width of the broad intruder and the strength of the coupling. This relationship, and the associated effects are discussed in section 8.29. 

The vanishing particle width is perhaps a surprise, since it involves stabilising a level in the continuum by introducing a further perturbation. More... 

Fig. 8.21. The vanishing particle width effect in a doubly-excited series of the Ca spectrum. Note the narrowing of the n — 6 member, although the remainder of the series members are broad. The perturber (hardly visible on account of its breadth and weakness) is indicated by an X in the figure (after U. Griesmann et al. [390]).

In principle, the particle width is not the only one which needs to be considered, especially if the autoionisation width is small. Narrow autoionising lines are well suited for study by laser spectroscopy. Since they imply the existence of long lived excited states, they can be investigated directly in atomic beams (see, e.g., [396]). However, the earliest examples appear to have been found by Paschen [397], White [398] and Shenstone [399] - the pioneer to whom we owe the very name autoionisation. In specific cases, where vanishing widths may occur (see section 8.29) or,... 

With just one particle channel open, one can also show that the q reversals occur at two poles, one of which corresponds to a zero in the particle widths, while the other does not. Thus, one of the poles only is associated with a vanishing width (cf section 8.29 and the spectra in fig. 8.18). 

The third part of the spectral function Oj =Oj(p) is the pathological portion of the spectral function 0(p). It is continuous but almost nowhere differentiable as the trajectories of a classical Brownian particle or of a random walker in the limit of vanishing step width (Ax 0) and time... 

Returning to the one-dimensional box of constant width, if the potential does not increase suddenly to infinity at one of the walls then the wave-function does not vanish there. Due to the continuity requirement of the wavefunction, it decreases exponentially to zero inside the wall of finite height. Therefore, there is a non-zero probability that the particle will penetrate the wall, although its kinetic energy is lower than the potential barrier (Fig. 2.6). This effect is called quantum-mechanical tunnelling. 

Figure 2.7 shows the situation encountered if one of the barriers has both a finite height and a finite width. Now, the wavefunction does not vanish inside the barrier and the particle will be described in the outer region, where F = 0, as it is inside the box. 

Since the barrier has a finite width, the wave function does not vanish completely within the barrier range. Thus, after leaving the barrier region we again have a wave with the same frequency but with a smaller amplitude than that before the barrier range. This means that there is a non-zero probability that the particle... 

For some samples (e.g., Fe particles in a polymer matrix,mag-nesioferrite particles in MgO, and for iron-containing microclusters in borate and silicate glasses ) a coexistence of the broad line with a narrow line near g = 2 is observed (Fig. F.7.1). With decreasing temperature the broad line grows, the width increasing and the shape becoming more and more distorted, and the narrow line tends to vanish (Fig. F.7.2). 

As discussed above, when an infinite graphene sheet is cut to form a quasi-one-dimensional graphene nanoribbon with a finite width and infinite length, the 7T-electrons wave function is confined along the direction perpendicular to the axis of the ribbon and is forced to vanish at large distances along this direction. These particle in a box like boundary conditions induce... 

The right-hand side of the above equation describes the error introduced by the Hartree approximation. The error vanishes if the Hamiltonian is separable, and it becomes small if the functions W i and W2 are almost constant over the width of the single-particle functions

time-independent Hartree approach. The time-dependent wave packet is more or less localized, whereas the eigenstates are usually very delocalized. 

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