VIADUCTO BICENTENARIO - Viaducto Elevado

To analyse the qualitative features of the m/M transition, it is useful to first examine a ‘semi-oscillator’ model. The attractive potential is replaced by an oscillator potential:

where K is a stiffness coefficient. We lose some important feature of the actual systems, since the oscillator attraction always holds the particles together. The model does not have ionized, i.e. continuum, states, apart from the center-of-mass continuum states.

First consider the additional approximation of neglecting the repulsion, i.e. U(y)=0. The spectrum is:

where the nx…, vx…are integers (including zero). Since each of the sub-Hamiltonians is rotation invariant, we have alternative descriptions in terms of

principal, angular momentum and azimuthal quantum numbers. These are important in discussing perturbations of the systems by external electric and magnetic fields.

In the molecular limit (M m) there is a triply-degenerate high-frequency branch represented by vx, vy, vz, proportional to and with a weak

dependence on m/M. The nx, ny, nz branch is proportional to and is low frequency.

In the atomic limit (m M) the branches coalesce and are described by the single frequency

Next, consider the effects of the repulsion U(q1–q2) between the M particles. The modes of frequency:

are unaltered. This is the only place that the mass m enters. These modes are high frequency in both the molecular and atomic limits, with a minimum at

m=2M.

The branches corresponding to nx, ny, nz in absence of the repulsion are now described by the rotation invariant Hamiltonian:

An appropriate set of quantum numbers is the angular momentum l with a (2l+1) azimuthal degeneracy. The radial function Rv.l(η) also characterized by a

We find the vibration-rotation spectrum from this equation. One first considers the case l=0. The potential energy is positive and has a minimum at:

when U(η)=e2/η. Expanding about this point the energy is:

In the large M limit asymptotic analysis shows that one can ignore the half range of η coordinate and the boundary condition at η=0.

For l=0 one then has a one-dimensional harmonic oscillator and thus a vibrational spectrum with a vibrational quantum number v. The frequency is i.e. proportional to . The rotational spectrum is roughly estimated by taking η=η* in the term This gives a moment of inertia Thus the electronic, vibrational and rotational separations are in the ratios 1,

This is a crude first approximation. It rests on the fact that, for M 1, the minimum η* is far from the origin, i.e. A more precise analysis of the Rv,l equation yields a theory of the vibration-rotation interaction. If one moves away from the molecular limit to the transition region, where M and m

are comparable, we get increasing distortion and modification of the spectrum. The analysis appears to be complicated but feasible, provided one does extensive computer calculations. The results should be similar to standard analysis using Morse potentials.

(c) Semi-oscillator model-atomic limit (m M)

Let us return to the standard coordinates used in the discussion of the effect of nuclear motions on atomic spectra. Since m/M 1 we look on the term:

as a perturbation.

independent particle description with hydrogenic wave functions and with the reduced masses An even better starting point is the Hartree- Fock independent particle description.

However, when Z is reduced so that one reaches the helium atom (Z=2), this description is not very good. One needs wave functions that, even in the fixed nucleus limit, give a more accurate account of the correlated motion of the electrons. It is necessary to classify the levels into para and ortho-helium. The Pauli principle requires that the wave functions are antisymmetric under combined interchange of space and spin coordinates of the electrons. The perturbations arising from the finite nuclear mass give differential shifts of the energy levels.

By the time one reaches the H− _{ion (Z=1) the independent electron description is completely inadequate. If one does not have the extreme situation}

m/M 1, the fact that the momenta are coupled leaves us with an unclear picture of the behavior. For the semi-oscillator model the description in terms of

, , and coordinates appears to be better in the intermediate situation, since the complications are transferred to the coordinates alone, i.e. one has a one-body problem. But this is not true for the actual V(r)=−e2/r problem.

(d) The

transition-actual potential

To solve the molecule problem for the case where we have the adiabatic (Born-Oppenheimer) approximation.6 _{We look for the}

electronic wave functions for fixed nuclear coordinates. In the , descriptions:

This is a two-center problem with the parameter only involving the nuclear coordinates. It separates in elliptical coordinates. In the simple adiabatic approximation enters as part of the potential for the nuclear wave functions

where n corresponds to additional quantum numbers for the rotation vibration spectrum (l and ml =−l,…0,…l). In fact εv depends only on the absolute

and the adiabatic approximation is exact. In the actual molecular ion the electronic wave function is governed by a potential that is when η→0. So the εv(η→0) are hydrogenic levels for Z=2 with a reduced mass μ.

On the other hand, for large nuclear separations we have a spectrum which is 1/4 of the η→0 spectrum. So the εv(η) curve is slowly varying and

proportional to The potential for the nuclear wave function also involves U(η)=e2/η which →∞ as η→0 and to 0 as η→∞. If we fix the units so

that m=1, we see that we lose the minimum as M→0, i.e. in the atomic limit. In addition the adiabatic approximation has broken down by the time M~1. Of course in the atomic region it is better to proceed by doing the perturbation analysis with the appropriate set of coordinates.

It is clear that the destruction of the highly organized level structure as m/M decreases occurs in a very non-uniform manner. Even in the extreme molecular limit the large quantum number rotational and vibrational states merge into the continuum. As m/M is reduced, only the lowest states retain identity. They are still satisfactorily described in terms of the collective coordinate. Finally all of these disappear and one is left with the H− _{ion. In the}

present problem, where there are only a few degrees of freedom, all of these properties emerge from rather simple familiar mathematical equations. There is nothing really surprising, but perhaps we obtain a more explicit understanding of the strengths and limitations of the use of collective coordinates. A very important point is that the failure to satisfy the boundary condition at η=0 becomes very serious as one moves away from the domain where the collective coordinate yields a good approximation. We would like to emphasize that, while it may be possible to give a theory of the ground state energy valid over the entire m/M range, it is much more difficult to give an account of the level structure.