Parabolic potential

Figure S-14. (A) A parabolic potential barrier and a linear perturbation. (B) Sum of the parabolic and linear functions, showing shift in maximum in accord with the Hammond postulate. (C) Two parabolic potential wells aa and bb are equivalent to the parabolic barrier cc .

Consider a classical particle of mass w in a parabolic potential well. At time t the displacement x of the particle from the origin is given by... 

Consider a particle of mass wr in a parabolic potential well. Calculate the probability of finding the particle in the classically allowed region when the particle is in its ground state. 

What is the probability density as a function of the momentum p of an oscillating particle in its ground state in a parabolic potential well (First find the momentum-space wave function.)... 

Thus, to obtain the timescales d and Q( ) we need to find the solution Y(x, s) of Eq. (5.78) at appropriate boundary conditions and evaluate limits (5.82) and (5.83) for y - > 0. It is precisely that way by which the various timescales for piecewise-linear and piecewise-parabolic potential profiles were derived in Refs. 32 and 33. 

First, we should consider the time evolution of the mean coordinate in the monostable parabolic potential tp(x) = ax1 /2 (linear system), because for this... 

Semiempirical Model of Radical Reaction as an Intersection of Two Parabolic Potential Curves... 

SEMIEMPIRICAL MODEL OF RADICAL REACTION AS AN INTERSECTION OF TWO PARABOLIC POTENTIAL CURVES... 

Theoretical models include those based on classical (Newtonian) mechanical methods—force field methods known as molecular mechanical methods. These include MM2, MM3, Amber, Sybyl, UFF, and others described in the following paragraphs. These methods are based on Hook s law describing the parabolic potential for the stretching of a chemical bond, van der Waal s interactions, electrostatics, and other forces described more fully below. The combination assembled into the force field is parameterized based on fitting to experimental data. One can treat 1500-2500 atom systems by molecular mechanical methods. Only this method is treated in detail in this text. Other theoretical models are based on quantum mechanical methods. These include ... 

Bell, R. P. Tunnel effect corrections for parabolic potential barriers,Trans. Far. Soc. 55, 1 (1959). 

Fig. 9.20. The parabolic potential barrier. (Reprinted from J. O M. Bockris and S. U. M. Khan, Quantum Electrochemistry, Plenum, 1979, p. 250.)...

Exercise. Solve the diffusion equation in an inverted parabolic potential with initial delta distribution,... 

One-Particle Model with Parabolic Potential-Energy Wells... 

Figure 7.2 Parabolic potential-energy well for one-diinensional particle jumps. Unlike...

As an application of the method of file separation of variables, we consider the nonstationary behavior in the generalized, fractional version of the Brownian harmonic oscillator with the parabolic potential... 

It can also be seen in Fig. 6.10 that the solvent acetone has an almost parabolic -potential over a wide cr-range, but this turns into a strong descent in the donor region, because acetone has a strong lack of hb-donors. As can be seen in the -profiles, its hb-acceptors are less polar than those of methanol, and therefore acetone is less attractive for very strong hb-donors than is methanol. 

C. Nonrigid Oscillator Linear-Response Theory for the Parabolic Potential... 

If strong collisions are rare (e.g., if y 1), another important factor—an anharmonicity of a molecular ensemble—plays an important role. This notion, introduced in GT, Section VII with respect to a parabolic potential well (or the well close to the latter), means dependence of the period on the phase variables h and l. 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

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