The absorption lineshape of a harmonic oscillator

The Langevin equation (8.31), with R t) taken to be a Gaussian random force that satisfies (i ) = 0 and R 0)R t) = ImyksTSit), is a model for the effect of a thermal environment on the motion of a classical harmonic oscillator, for example, the nuclear motion of the internal coordinate of a diatomic molecule in solution. 

This is a Lorentzian lineshape whose width is determined by the firiction. The latter, in turn, corresponds to the rate of energy dissipation. It is significant that the normalized lineshape (characterized by its center and width) does not depend on the temperature. This result is associated with the fact that the harmonic oscillator is characterized by an energy level structure with constant spacing, or classically-with and energy independent frequency. 

In Section 6.2.3 we have seen that a simple quantum mechanical theory based on the golden rule yields an expression for the absorption lineshape that is given essentially by the Fourier transform of the relevant dipole correlation function ( (0)/ti(0). Assuming again that yu is proportional to the displacement x of the oscillator from its equilibrium position we have 

Another point of interest is the close similarity between the lineshapes associated with the quantum damped two-level system, Eq. (9.40), and the classical damped harmonic oscillator. We will return to this issue in Section 9.3. 

This is another route to the corresponding absorption lineshape. 

It is important to note that Eq. (8.31), with a constant y, is valid only in the Markovian case. Its generalization to non-Markovian situations is discussed in Section 8.2.6. 

A standard experimental probe of this motion is infrared spectroscopy. We may use the results of Sections 7,5 and 8.2.3 to examine the effect of interaction with the thermal environment on the absorption lineshape. The simplest model for the coupling of a molecular system to the radiation field is expressed by a term —fi S in the Hamiltonian, where is the molecular dipole, and (t) is the oscillating electric field (see Section 3.1). For a one-dimensional oscillator, assuming that /r is proportional to the oscillator displacement from its equilibrium position and taking cos((uZ), we find that the coupling of the oscillator to the thermal environment and the radiation field can be modeled by Eq. (8.31) supplemented by a term (F/ni where F denotes the radiation induced driving force. We can 

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