Quantum anharmonic oscillator

Consider an anharmonic oscillator described by the Hamilton operator H = Hq + H, where 

According to perturbation theory, the anharmonic contribution to the energy level E is given by 

Hint Express H in terms of creation and annihilation operators a and a, respectively, using the relation 

3 Grlineisen Parameter, Thermal Expansion and Frequency Shift of a Monoatomic fee Crystal 

Consider an fee crystal with one atom in the primitive unit cell and with an interatomic potential (p(r) which extends only to neareast neighbours. Neglecting the tangential force constant, i.e., (y q) = there is then 

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The free energy of a Ginzburg-Landau field describing a system of weakly coupled chains in a plane is identified with the ground-state energy of a linear array of quantum anharmonic oscillators. The equivalent Hamiltonian is simplified for both the real and complex fields using a truncated basis of states. Results for both the real and complex fields will be discussed. In addition, the behavior of the specific heat and inverse correlation length for finite numbers of weakly coupled chains will be discussed. 

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

In the estimation of Acon(t), only the first two terms are considered, neglecting the higher-order terms. (Q - Goo) and (Q m - Goo) 810 die quantum mechanical expectation values of the anharmonic oscillator. They can be calculated using perturbation theory and is given by... 

H° and HFree are, respectively, the Hamiltonians of the fast and slow modes viewed as quantum harmonic oscillators, whereas Hint is the anharmonic coupling between the two modes, which are given by Eqs. (15), (21), and (22). Besides, He is the Hamiltonian of the thermal bath, while Hint is the Hamiltonian of the interaction of the H-bond bridge with the thermal bath. 

Calculation of Matrix Elements for One-Dimensional Quantum-Mechanical Problems and the Application to Anharmonic Oscillators. 

Garrett CGB. Nonlinear optics, anharmonic oscillators, and pyroelectricity. IEEE J Quantum Electron 1968 QE-4(3) 70. 

While the classical model of an anharmonic oscillator describes the effects of non-linearity, it cannot provide information on molecular properties. Calculation of molecular properties requires a quantum mechanical model. Application of perturbation theory (Boyd, 2003) leads to the following expression ... 

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. 

The author examines with success the efficiency of the methods by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Poschl-Teller potential in quantum mechanics. 

It seems intuitively clear that such an achievement is due to the control of quantum dissipative dynamics through the application of a suitably tailored, time-modulated driving field. Indeed, some interesting examples of suppression of quantum decoherence by the modulation of system parameters have been considered in [Viola 1998 Vitali 1999], An improvement of sub-Poissonian statistics of an anharmonic oscillator by the application of amplitude-modulated pump field have been demonstrated in [Kryuchkyan 2002 Kryuch-kyan 2003],... 

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y -> 0 limit of the Kramers model we are dealing with energy relaxation of a classical anharmonic oscillator. One may justifiably question the use of Markovian classical dynamics in this part of the problem, and we will come to this issue later. For now we focus on the solution of the mathematical problem posed by the low friction limit of the Kramers problem. 

In order to illustrate the relationship between the quantum mechanical Heff and the classical mechanical Ti derived from it by applying Heisenberg s (1925) version of the Correspondence Priniciple, we return to the problem of two coupled identical anharmonic oscillators (see Section 9.4.12 and Xiao and Kellman, 1989). The quantum mechanical HlqCAL is,... 

Herman M F, Kluk E and Davis H L 1986 Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator J. Chem. Phys. 84 326... 

Most molecular vibrations are well described as harmonic oscillators with small anharmonic perturbations [5]. Por an harmonic oscillator, all single-quantum transitions have the same frequency, and the intensity of single-quantum transitions increases linearly with quantum number v. Por the usual anharmonic oscillator, the single-quantum transition frequency decreases as v increases. Ultrashort pulses have a non-negligible frequency bandwidth. Por a 1... 

Here we describe exact quantum calculations on a model many-dimensional Fermi resonance system, using the methods explained in Chapters 1 and 3 (11). As in the previous section of this chapter, the Hamiltonian H = H0 + V consists of an unperturbed portion and a coupling term. As a simple means to model 5 = 6 anharmonic oscillators the normal... 

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