Harmonic-oscillator

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

The harmonic oscillator is used as a simple model for the vibrational motion of atoms along bonds in molecules. This will in turn be used to model infrared absorption spectroscopy in the next chapter. 

The Schroedinger equation for this system can now be readily written 

The solution to this differential equation is well known. The energy 

The functions A (z) in Equation 5-5 are polynomials in z known as the Hermite polynomials. The Hermite polynomials can be generated from the following formula  

The Hemiite polynomials for the v + 1 state (the next state) can also be obtained from the following recursion relationship given that Ao(r) = 1. 

We solve the time-independent SE for a harmonic oscillator. The vibrational wave function is denoted as T. In one dimension. Equation 4.7 may be written as 

Equation 4.53 is the time-independent SE for the vibrational motion. The boundary conditions are P - 0 exponentially when X 00. A possible solution of this type is 

We insert this expression into Equation 4.53 and obtain 

This polynomial in X can only be equal to zero for all X if the coefficient before X is = 0 and the terms without X also are equal to zero. This implies that 

FIGURE 4.2 Energy levels for the harmonic oscillator and the two first eigenfunctions. 

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The classical mechanical RRKM k(E) takes a very simple fonn, if the internal degrees of freedom for the reactant and transition state are assumed to be hamionic oscillators. The classical sum of states for s harmonic oscillators is [16]... 

Marquardt R and Quack M 1996 Radiative excitation of the harmonic oscillator with applications to stereomutation in chiral molecules Z. Rhys. D 36 229-37... 

The vibrational part of the molecular wave function may be expanded in the basis consisting of products of the eigenfunctions of two 2D harmonic oscillators with the Hamiltonians ffj = 7 -I- 1 /2/coiPa atid 7/p = 7p - - 1 /2fcppp,... 

The coordinates of interest to us in the following discussion are Qx and Qy, which describe the distortion of the molecular triangle from Dy, symmetry. In the harmonic-oscillator approximation, the factor in the vibrational wave... 

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrddinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian assuming the fonn [95]... 

Mavri, J., Berendsen, H.J.C. Dynamical simulation of a quantum harmonic oscillator in a noble-gas bath by density matrix evolution. Phys. Rev. E 50 (1994) 198-204. 

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

Now consider a system of N one-dimensional harmonic oscillators with the Hamiltonian... 

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

Fig. 1. Optimization of the Onsager-Machlup action for the two dimensional harmonic oscillator. The potential energy is U(x,y) = 25i/ ), the mass is 1...

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

Consider for the nth time step the linear harmonic oscillator... 

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

This com poTicii 1 is oficn approximated as a harmonic oscillator and can be calculated using Hooke s law. 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

Tlris is the Schrodinger equation for a simple harmonic oscillator. The energies of the system are given by E = (i + ) x liw and the zero-point energy is Hlj. 

The hamionic oscillator (Fig. 4-1) is an idealized model of the simple mechanical system of a moving mass connected to a wall by a spring. Oirr interest is in ver y small masses (atoms). The harmonic oscillator might be used to model a hydrogen atom connected to a large molecule by a single bond. The large molecule is so... 

The atomic harmonic oscillator follows the same frequency equation that the classical harmonic oscillator does. The difference is that the classical harmonic oscillator can have any amplitude of oscillation leading to a continuum of energy whereas the quantum harmonic oscillator can have only certain specific amplitudes of oscillation leading to a discrete set of allowed energy levels. 

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