First excited state harmonic oscillator

As has been discussed in Section 111, the initial phase-space distribution pyj, for the nuclear DoF xj and pj may be chosen from the action-angle (18) or the Wigner (17) distribution of the initial state of the nuclear DoF. To specify the electronic phase-space distribution pgj, let us assume that the system is initially in the electronic state v i ). According to Eq. (80b), the electronic state vl/ ) is mapped onto Ne harmonic oscillators, whereby the nth oscillator is in its first excited state while the remaining Nei — 1 oscillators are in their ground state. The initial density operator is thus given by... 

In this last equation, the right-hand side matrix elements are those of the IP time evolution operator of the driven damped quantum harmonic oscillator describing the H-bond bridge when the fast mode is in its first excited state ... 

Figure 16. The effects of the parity operator C2 on the ground and the first excited states of the symmetrized g and u eigenfunctions of the g and u quantum harmonic oscillators involved in the centrosymmetric cyclic dimer. (The subscripts 1 and 2 refer, respectively, to the a and b moieties of the centrosymmetric cyclic dimer).

This is the quantity found in equation (29). Often, the reciprocal of the velocity-averaged transition probability, PTo = Z10, is reported in the literature. It is referred to as the average number of collisions required to deactivate a molecule from the first excited state to the ground state. Other authors have simply used Z, the so called collision number, which is the product Sir. In some instances, these two quantities, related for a harmonic oscillator by... 

Equation 3.5, where v is the vibrational quantum number, means that only transitions between nearest vibrational states can directly occur in the case of the harmonic oscillator. This means that the 1R spectrum is generally mostly constituted hy fundamental transitions, that is, those associated with excitation from the fundamental state to the first excited state. This condition, however, is relaxed in the case of anharmonic oscillators, so that not only fundamental transitions but also overtone and combination modes (also called the harmonics, i.e. modes associated with the excitation from the fundamental state to a second or third excited state) can be sometimes observed, although they are usually weak. 

Drawing from our experience with the particle in a box, we might surmise that the first excited state of the harmonic oscillator would be a function similar to Eq (5.20), but with a node at x = 0, say,... 

The force constant for HF is 966 N m. Using the harmonic oscillator model, calculate the relative population of the first excited state and the ground state at 300 K. 

The oscillator strength f is defined as the ratio of the probability of a given transition to that of a harmonic oscillator able to absorb the same electromagnetic energy between its ground state and its first excited state. Quantum-mechani-cally, the transition probability is proportional to the square of the so-called transition moment . 

The prominence of these quantum dynamical models is also exemplified by the abundance of theoretical pictures based on the spin-boson model—a two (more generally a few) level system coupled to one or many harmonic oscillators. Simple examples are an atom (well characterized at room temperature by its ground and first excited states, that is, a two-level system) interacting with the radiation field (a collection of harmonic modes) or an electron spin interacting with the phonon modes of a surrounding lattice, however this model has found many other applications in a variety of physical and chemical phenomena (and their extensions into the biological world) such as atoms and molecules interacting with the radiation field, polaron formation and dynamics in condensed environments. 

The 03 dependence of the force constant k(Q) = rnoP iQ) creates an anharmonic coupling between both modes q and Q. It is at the origin of the exceptional width of bands of H-bonds, and of their shifts towards lower wavenumbers when compared to bands of the same X H molecules, when they do not establish H-bonds. These most intense bands are 0 1 transitions, or transitions between the ground vibrational state of q and its first excited state. The other possible transitions, 0 n with n > 1, have intensities equal to 0 in the case of an harmonic oscillator in q (no terms in q, (f, etc. in V q, Q)). Overtones of v, particularly those corresponding to 0 2 transitions in the 6000-6500 cm region, however often... 

For the anharmonic oscillator with Hamiltonian (9.3), evaluate for the first excited state, taking the unperturbed system as the harmonic oscillator. 

The first terms are kinetic energies. They are obtained either directly or by noting that for a harmonic oscillator the zero-point kinetic energy is and the first excited state kinetic energy is Ihco. The energy difference is... 

What about the excited states in the Born-Oppenheimer approximation We consider here the optimal case m - m. This implies that it is much more economical to excite the relative motion of the heavy quarks than the motion of the light quark around them. For instance, in the harmonic oscillator model, there is a ratio exactly [2m/(2m + m)]" between the corresponding excitation energies. Then, for the first excited states, the light-quark wave function remains in the lowest adiabatic f p. A), the binding energy and the qq distribution are obtained from the low-lying excited states of eq. (7.9) or (7.6). 

Light of wavelength 4.33 x 10 m excites a quantum-mechanical harmonic oscillator from its ground to its first excited state. Which one of the following wavelengths would accomplish this same transition if i) the force constant only was doubled ii) the mass only was doubled ... 

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