Quantum dynamics of the harmonic oscillator

The flux is defined up to the constant A. Taking A = (a single particle wavefunction normalized in the volume Q) implies that the relevant observable is 2VJ(r), that is, is the particle flux for a system with a total of N particles with N Sometimes it is convenient to normalize the wavefunction to unit flux, J = 1 by choosing A = 

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Quantum dynamics of the harmonic oscillator 2.9.1 Elementary considerations... 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

Thus, with this result for the kinetic energy and Eq. (E.10) for the potential energy, we conclude that the quantum dynamics of the normal modes is just the dynamics of n uncoupled harmonic oscillators that is,... 

In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field (Chapter 3) and provides good starting points for describing nuclear motions in molecules and in solid environments (Chapter 4). It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach (see Section 6.5.4). In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9. 

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

In the following we show that a simple description of the (quantum or classical) dynamics can be obtained in a multidimensional system close to a stationary point. Thus, the system can be described by a set of uncoupled harmonic oscillators. The formalism is related to the generalization of the harmonic expansion in Eq. (1.7) to multidimensional systems. 

Here, pj is the Boltzmann density operator of the H-bond bridge viewed as a quantum harmonic oscillator, pe is the Boltzmann density operator of the thermal bath, and (t) are effective time-evolution operators governing the dynamics of the H-bond bridge depending on the excited-state degree k of the fast mode. They are given by Eq. (110), that is,... 

The simplest example of a classical or quantum dissipative system is a particle evolving in a potential V(x) and coupled linearly to a fluctuating dynamical reservoir or bath. If the bath is only weakly perturbed by the system, it can be considered as linear, described by an ensemble of harmonic oscillators. Starting from the corresponding system-plus-bath Hamiltonian and using some convenient approximations, it is possible to get a description of the dissipative dynamics of the system. 

In the Fourier method each path contributing to Eq. (4.13) is expanded in a Fourier series and the sum over all contributing paths is replaced by an equivalent Riemann integration over all Fourier coefficients. This method was first introduced by Feynman and Hibbs to determine analytic expressions for the harmonic oscillator propagator and has been used by Miller in the context of chemical reaction dynamics. We have further developed the approach for use in finite-temjjerature Monte Carlo studies of quantum sys-tems, and we have found the method to be very useful in the cluster studies discussed in this chapter. 

In order to extend the linearization scheme to non-adiabatic dynamics it is convenient to represent the role of the discrete electronic states in terms of operators that simplify the evolution of the quantum subsystem with out changing its effect on the classical bath. A way to do this was first suggested by Miller, McCurdy and Meyer [28,29[ and has more recently been revisited by Thoss and Stock [30, 31[. Their method, known as the mapping formalism, represents the electronic degrees of freedom and the transitions between different states in terms of positions and momenta of a set of fictitious harmonic oscillators. Formally the approach is exact, but approximations (e.g. semi-classical, linearized SC-IVR, etc.) must be made for its numerical implementation. 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

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