Quantum numbers, continued vibrational

Electronic levels are spaced more closely together at higher quantum numbers as the ionization limit is approached, vibrational levels are evenly spaced, while rotational and translational levels are spaced further apart at high energies. The classical principle assumes continuous variation of all energies. 

Let us denote the excited state after initial excitation by x,) with quantum numbers denoted by the composite index i, which is assumed to be in a particular vibrational state v,. The continuum eigenstate of the full Hamiltonian H is denoted as whose quantum numbers are given by the composite index /. The probability per unit energy that the system will end up in the continuous state / is given by... 

The continuous function /(E) corresponds to the discrete distribution /(u) over vibrational quantum numbers, known as the Gordiets vibrational distribution (Gordiets et al.,... 

T = 47 jjK. In order to maintain such a low translational temperature, the ion has to be exposed continuously to the radiation field. Switching off the cooling laser leads to a heating rate, (driy/dt), of 6 vibrational quanta per sec, see Fig. 6.1. After 2s the mean expectation value of the vibrational quantum number is already (riy) = 12. This corresponds to a translational temperature of Tt = 1.7 mK of the 3-dimensional oscillator. For more details of this fundamental experiment see Ref. 23. 

It is clear that the process can be continued without adding anything significantly new. The higher orders describe the couplings between rotations, vibrations and electronic motions. Quantum numbers additional to those already introduced will not appear. 

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