TUrning points, classical

Figure 1.13 Plot of potential energy, V(r), against bond length, r, for the harmonic oscillator model for vibration is the equilibrium bond length. A few energy levels (for v = 0, 1, 2, 3 and 28) and the corresponding wave functions are shown A and B are the classical turning points on the wave function for w = 28...

Each point of intersection of an energy level with the curve corresponds to a classical turning point of a vibration where the velocity of the nuclei is zero and all the energy is in the form of potential energy. This is in contrast to the mid-point of each energy level where all the energy is kinetic energy. 

As V increases, the two points where j/l, the vibrational probability, has a maximum value occur nearer to the classical turning points. This is illustrated for u = 28 for which A and B are the classical turning points, in contrast to the situation for u = 0, for which the maximum probability is at the mid-point of the level. 

The wavelength of the ripples in ij/ increases away from the classical turning points. This is more apparent as v increases and is pronounced for u = 28. 

The classical turning point of a vibration, where nuclear velocities are zero, is replaced in quantum mechanics by a maximum, or minimum, in ij/ near to this turning point. As is illustrated in Figure 1.13 the larger is v the closer is the maximum, or minimum, in ij/ to the classical turning point. 

Figure 7.21 illustrates a particular case where the maximum of the v = 4 wave function near to the classical turning point is vertically above that of the v" = 0 wave function. The maximum contribution to the vibrational overlap integral is indicated by the solid line, but appreciable contributions extend to values of r within the dashed lines. Clearly, overlap integrals for A close to four are also appreciable and give an intensity distribution in the v" = 0 progression like that in Figure 7.22(b). 

Figure 35. Electronic excitation of a LiH wave packet from the outer classical turning point 6ao) of the ground X S" state. The X —> B transition is considered. The initial wavepacket is the shifted ground vibrational state. Taken from Ref. [37].

Fig. 2.7. Coulomb barrier penetration by a charged particle, a is the range of the nuclear force and b the classical turning point.

In the framework of DECP, the first pump pulse establishes a new potential surface, on which the nuclei start to move toward the new equilibrium. The nuclei gain momentum and reach the classical turning points of their motion at t = nT and t = (n + l/2)T. The second pump pulse then shifts the equilibrium position, either away from (Fig. 3.10b) or to the current position of the nuclei (Fig. 3.10c). The latter leads to a halt of the nuclear motion. Because photo-excitation of additional electrons can only shift the equilibrium position further in the same direction, the vibrations can only be stopped at their maximum displacement [32]. 

A multiple-pump experiment on Te by Mazur and coworkers revealed that the reflectivity oscillation was enhanced to maximum or canceled completely when At was considerably shorter than nT or (ra+ 1/2)T [32]. In other words, the nuclear vibrations do not stop at their classical turning point, in contrast to the weak excitation case. This departure from a classical harmonic motion is the manifestation of a time-dependent driving force, whose physical origin... 

Figure 1. Kick potential as a function of q for F = 5.0 (a.u.) and u> = 0.52 (a.u.), for both exact and approximate (evaluated only in the vicinity of the classical turning points) kicking terms. The curves clearly show that a simple constant shift is the only difference. Note that F = 5 means that the external field is five times larger than the binding field.

At the classical turning point, E" = 0, which turns p" positive. 

Fig. 12.2 Left The ground (X, solid line), excited (6, dashed line) and dissociative [a1g(3II), dotted line] electronic state potentials of the iodine molecule. The arrow indicates the electronic excitation. The initial excited wave packet is located in the Franck-Condon region near to the inner classical turning point of the B state. The transition from the B to the a state is forbidden by symmetry in the isolated molecule but becomes allowed when the molecule is placed in a solvent.

The simplest way to extend the approach above is to recognize that the tunneling exponent, its in Equation 6.8 above, can be identified with the magnitude of the semi-classical action integral between the classical turning points... 

The nautre of the He-surface interaction potential determines the major characteristics of the He beam as surface analytical tool. At larger distances the He atom is weakly attracted due to dispersion forces. At a closer approach, the electronic densities of the He atom and of the surface atoms overlap, giving rise to a steep repulsion. The classical turning point for thermal He is a few angstroms in front of the outermost surface layer. This makes the He atom sensitive exclusively to the outermost layer. The low energy of the He atoms and their inert nature ensures that He scattering is a completely nondestructive surface probe. This is particularly important when delicate phases, like physisorbed layers, are investigated. 

Fig. 4.13. Classical turning point in atomic-beam scattering. When the repulsive potential on a He atom at the sample surface equals the kinetic energy of the He atom, as a classical particle, the He atom is turned back.

The detailed data from He-scattering experiments provide information about the electron density distribution on crystalline solid surfaces. Especially, it provides direct information on the corrugation amplitude of the surface charge density at the classical turning point of the incident He atom, as shown in Fig. 4.13. As a classical particle, an incident He atom can reach a point at the solid surface where its vertical kinetic energy equals the repulsive energy at that point. The corrugation amplitude of the surface electron density on that plane determines the intensity of the diffracted atomic beam. 

In this expression, Rq designates the greatest root of the argument of the square root appearing in the denominator Ro is the outermost classical turning point. Equation 5.70 represents the inverse function, t(R), of the desired solution R(vo,b t). With the knowledge of the time dependence of... 

For a certain range of values of impact parameter b and initial speed vo, the radicand in Eq. 5.70 may actually have three roots. By choosing the largest classical turning point Ro, we have limited our considerations to free, i.e., colliding particles. The other two turning points (if existent) are characteristic of bound states which may be similarly treated [168]. 

Figure 7.1 shows the vibrational wave functions for some vibrational levels of two electronic states that have different values of Re (as is usually the case). For t>= 0, the function vib is largest at or near Re and is small at the classical turning points of the vibration. For larger v, the vibrational wave functions are large near the classical turning points. (This is in accord... 

If P(e) extends to values e>e0+ (00), it can be concluded that in some part of the accessible R range, V+(R) is more attractive than V (R). Under normal conditions these parts will be close to the classical turning point for central collisions on K (R). Classically, the largest measured electron energy emax is then related to the value of the potential V+(R) at the turning point Rtp by... 

From their wavefunctions, we can infer some of the properties of Rydberg atoms. We begin by estimating the expectation values of ra where o is a positive or negative integer. The expectation values of ra, o > 0, are determined mostly by the location of the outer classical turning point, r = In2. Since the electron spends most of its time there, a good estimate for the expectation values of ra, o > 0, is... 

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