Cubic parabola potential

Fig. 19. Cubic parabola potential. Turning points are shown. The dashed line indicates the stable potential with the same well frequency.

For convenience of notation we accept from here on, that each frequency of the problem co has a dimensionless counterpart denoted by a capital Greek letter, so that co,- = coofl,. The model (4.28) may be thought of as a particle in a one-dimensional cubic parabola potential coupled to the q vibration. The saddle-point coordinates, defined by dVjdQ = dVjdq = 0, are... 

The optimal current fluctuations are plotted in the left panel of Fig. 3 for different conductors and If. The curves are symmetric owing to the symmetry of the cubic parabola potential. Common features are that they all reach the threshold current at maximum and their time spread is of the order of r. Still, the spread, shape, and most importantly, the integral of the current over time, varies significantly from curve to curve. This proves that the detector in use is dispersive and suffers from the first factor mentioned in the introduction. 

It follows from (A.3) and (A.15) that this influx is independent of x, which confirms that it corresponds with a genuine decay rate constant. For the cubic parabola potential (a = (w Xj)-173) it equals... 

Consider a potential V x) having a single minimum separated from the continuous spectrum by a sufficiently large barrier satisfying (1.1), e.g., a cubic parabola (fig. 19)... 

Let us now consider the more interesting case > 1. The simplest realization of such a barrier comprises N 1 Josephson junctions connected in series, this gives U() = NIcsin(/N), 4>o — N. However, this system is formally metastable the vortices can traverse the junction providing phase slips A = 27r. To eliminate this, one would increase the barrier for the vortex formation, for instance, by making several parallel chains of junctions. This would further complicate the concrete function U(4>). We notice that any function U() can be approximated by a cubic parabola if the tilting of the washboard potential is close to the critical value. This is why we choose the cubic parabola form... 

Fig. 1-5. Poleniial curve for a diatomic molecule actual potential (solid line), parabola (dashed line), and cubic parabola (dotted line).

Figure 3. Schematic depiction of first three terms in dimensional perturbation expansion of Eq.(12) for hydrogenic atom. For each the effective potential W(r) for the D oo limit is shown (solid curve). At left, the zeroth-order term corresponds to the electron at rest at the minimum of T (r). In the middle, the first-order term, proportional to 1/D, corresponds to harmonic oscillations (as if potential were replaced by the dashed parabola). At right, the second-order term corresponds to anharmonic vibrations (arising from cubic and quartic portions of the potential).

To the equation for the parabola, we add a cubic term to express the anharmonicity of the potential energy curve. Thus, we adopt a... 

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