Classical orbits

In accordance with the one-dimensional periodic orbit theory, any orbit contributing to g E) is supposedly constructed from closed classical orbits in the well and subbarrier imaginary-time trajectories. These two classes of trajectories are bordering on the turning points. For the present model the classical motion in the well is separable, and the harmonic approximation for classical motion is quite reasonable for more realistic potentials, if only relatively low energy levels are involved. 

The essence of Schrodinger s treatment was to replace the classical orbit of Bohr s semi-classical (particle) model of the H-atom by a corresponding wavelike orbital (single-electron wavefunction) L. Instead of specifying the electron s... 

The concept of scarred quantum wavefunctions was introduced by Eric Heller (E.J. Heller, 1984) a little over 20 years ago in work that contradicted what appeared at the time to be a reasonable expectation. It had been conjectured (M.V. Berry, 1981) that a semiclassical eigenstate (when appropriately transformed) is concentrated on the region explored by a generic classical orbit over infinite times. Applied to classically chaotic systems, where a typical orbit was expected to uniformly cover the energetically allowed region, the corresponding typical eigenfunction was anticipated to be a superposition of plane... 

While most derivations focus on the equation of motion, an equally important aspect of the MFT method is the correct representation of the quantum-mechanical initial state. It is well known that the classical limit of quantum dynamics in general is represented by an ensemble of classical orbits [23, 24, 26, 204]. Hence it is not appropriate to use a single classical trajectory, but it is necessary to average over many trajectories, the initial conditions of which are chosen to mimic the quantum nature of the initial state of the classically treated subsystem. Interestingly, it turns out that several misconceptions concerning the theory and performance of the MFT method are rooted in the assumption of a single classical trajectory. 

Figure 4. The average change in the squared value of the classical orbital angular momentum ((A/. )) and the standard deviation of the A/. distribution

This independency is related to the fact that the semiclassical calculation of the scattering amplitudes involves classical orbits belonging to an invariant set that is complementary to the set of trapped orbits in phase space [56]. The trapped orbits form the so-called repeller in systems where all the orbits are unstable of saddle type. The scattering orbits, by contrast, stay for a finite time in the scattering region. Even though the scattering orbits are controlled... 

Fig. 9.8 Fourier transforms of H spectra obtained in a magnetic field of 5.96 T with resolution 0.07 cm-1 (a) initial state 2p m = 0, final state m = 0 even parity states (b) initial state 2p m = — 1 final state m = — 1 even parity states. The squared value of the absolute value is plotted in both cases. The circled numbers correspond to the classical orbits depicted in Fig. 9.9 (from ref. 23).

The fact that the resonances in the Fourier transform spectrum of Fig. 9.8 can be predicted by finding classical orbits which return to the origin suggests a better... 

Applying the semiclassical quantization condition to the action around a closed classical orbit yields25-27... 

Fig. 4.2 Classical orbits for planets or electrons. One circular orbit and two elliptical orbits are shown.

R is called the Rydberg constant, with the value 1.097 x 10s cm-1, and i and n2 are numbers taking the values 1, 2, 3,. . . A classical orbiting system would be able to absorb and emit radiation in a continuous range of frequencies there is no way of explaining the line spectrum, or a formula... 

Equation (5.2) is a combination of the two two-dimensional Hamiltonians (2.39) and (3.15) which describe the vibrational and rotational excitations of BC separately. The Jacobi coordinates R, r, and 7 are defined in Figures 2.1 and 3.1 and P and p denote the linear momenta corresponding to R and r, respectively, j is the classical angular momentum vector of BC and 1 stands for the classical orbital angular momentum vector describing the rotation of A with respect to BC. For zero total angular momentum J=j+l = 0we have 1 = — j and the Hamilton function reduces to... 

Du, M.L. and Delos, J.B. (1988a). Effect of closed classical orbits on quantum spectra Ionization of atoms in a magnetic field. I. Physical picture and calculations, Phys. Rev. A 38, 1896-1912. 

The orbital angular momentum quantum number / has other strange characteristics. Notice that / = 0 is allowed. However, for a classical orbit, circular or elliptical, we have L = mv R (Equation 5.19), which cannot be zero. The vector L points in the... 

A trajectory rtf) is obtained by integrating r between turning points, where the argument of the square root vanishes. Using dO = A, this formula for r provides an equation to be integrated for a classical orbit,... 

Although most of the sensitivity criteria are closely related to the interaction energy, they are basically of the response, geometrical character, reflecting the reaction stimulus. i.e., the interaction between the two molecules. The mode resolved CSA may also be considered as supplementing the classical orbital interaction approach of Fukui [11]. 

---