Moving Harmonic Oscillator

To further illustrate the theory, we apply Jarzynski s identity to the analytically solvable example of a ID moving harmonic oscillator with Hamiltonian 

In the following, we will show explicitly that the correct result is obtained if Jarzynski s identity is used to evaluate the free energy difference, A(t) —, 4(0) = 

For given initial conditions, trajectories in phase space satisfying Hamilton s equation of motion, (5.17), are given by 

Now we specify an initial phase-space distribution as the Boltzmann distribution for the Hamiltonian at time t 0 

With this distribution being Gaussian in bothp(0) and c/(0), and the work being linear in p(0) and q(0), the distribution of the work obtained for Boltzmann-weighted initial conditions is Gaussian as well, with a mean and variance of 

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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... 

These harmonic-oscillator solutions predict evenly spaced energy levels (i.e., no anharmonicity) that persist for all v. It is, of course, known that molecular vibrations display anharmonicity (i.e., the energy levels move closer together as one moves to higher v) and that quantized vibrational motion ceases once the bond dissociation energy is reached. 

Curve C of Fig. 6-15(a) corresponds to /a 0.1, and curve C of Fig. 6-15(b) to p = 10. It is observed that in the first case the curve O is almost a circle (as in a harmonic oscillator) while in the second case it has very special features. In fact from very high velocity (y — sc), the solution drops very rapidly to zero (at the point where it cuts the x-axis). At this point the relaxation interval begins, the representative point moving near the x-axis with a very small velocity after that its velocity increases rapidly to a considerable value, and soon. 

In tests using the moving ID Hamiltonian harmonic oscillator, (5.25), a velocity Verlet integrator [24] combined with ttapezoidal integration of W (/.) performed well when compared to the analytic solution. An interesting analysis of how... 

Image current detection is (currently) the only nondestructive detection method in MS. The two mass analyzers that employ image current detection are the FTICR and the orbi-trap. In the FTICR ions are trapped in a magnetic field and move in a circular motion with a frequency that depends on their m/z. Correspondingly, in the orbitrap ions move in harmonic oscillations in the z-direction with a frequency that is m/z dependent but independent of the energy and spatial spread of the ions. For detection ions are made... 

This is, of course, the reciprocal of the time taken for an atom to move a lattice site distance into a vacancy. um can be estimated from the heat capacity of the crystalline material using the Einstein or Debye models6 of atoms as harmonic oscillators in a lattice. Combining Equations (2.30) and (2.31) gives the number of atoms moving per second as... 

Lets use these ideas to solve some problems focusing our attention on the harmonic oscillator a particle of mass m moving in a one-dimensional potential described by V(x) =... 

In order to reduce the number of variables, the location of the end point atoms A and C is fixed and only atom B is allowed to move. A condensed phase environment is represented by adding a harmonic oscillator degree of lieedom coupled to atom B. This can be interpreted as a fourth atom which is coupled in a harmonic way to atom B... 

However, as seen in Fig. 3.2, this idealized harmonic oscillator (Fig. 3.2b) is satisfactory only for low vibrational energy levels. For real molecules, the potential energy rises sharply at small values of r, when the atoms approach each other closely and experience significant charge repulsion furthermore, as the atoms move apart to large values of r, the bond stretches until it ultimately breaks and dissociation occurs (Fig. 3.2c). 

Note that if j = 1, (9.12) is formally identical with the classical expression (9.7) the classical multiple oscillator model, which will be discussed in Section 9.2, is even more closely analogous to (9.12). However, the interpretations of the terms in the quantum and classical expressions are quite different. Classically, o30 is the resonance frequency of the simple harmonic oscillator quantum mechanically 03 is the energy difference (divided by h) between the initial or ground state / and excited state j. Classically, y is a damping factor such as that caused by drag on an object moving in a viscous fluid quantum mechanically, y/... 

The harmonic oscillator is used as a simple model to represent the vibrations in bonds. It includes two masses that can move on a plane without friction and that are joined by a spring (see Fig. 10.3). If the two masses are displaced by a value x0 relative to the equilibrium distance / , the system will start to oscillate with a period that is a function of the force constant k (N m ) and the masses involved. The frequency, which is independent of the elongation, can be approximated by equation (10.2) where n (kg) represents the reduced mass of the system. The term harmonic oscillator comes from the fact that the elongation is proportional to the exerted force while the frequency i/yib is independent of it. 

For the c = 0 and v= 1 vibrational levels of CO, calculate the maximum departure of each nucleus from its equilibrium position in the principal-axis coordinate system if it is assumed the nuclei move classically. Assume harmonic-oscillator energy levels. 

An entirely new principle of the ion-trap mass spectrometers is the Orbitrap with a coaxial inner spindle-shaped electrode in an outer barrel-like electrode. The peripheral injected ions move due to their electrostatic attraction to the inner electrode on orbits around and swing simultaneously along the electrode. The frequency of these harmonic oscillations is inversely proportional to the square root of mJz. The detected signals are induced by the frequency of these swings and resulted... 

Near the equilibrium bond length qe the potential energy/bond length curve for a macroscopic balls-and-spring model or a real molecule is described fairly well by a quadratic equation, that of the simple harmonic oscillator (E = ( /2)K (q — qe)2, where k is the force constant of the spring). However, the potential energy deviates from the quadratic (q ) curve as we move away from qc (Fig. 2.2). The deviations from molecular reality represented by this anharmonicity are not important to our discussion. 

Important examples of chemical interest include particles that move in the central held on a circular orbit (V constant) particles in a hollow sphere V = 0) spherically oscillating particles (V = kr2), and an electron on a hydrogen atom (V = 1 /47re0r). The circular orbit is used to model molecular rotation, the hollow sphere to study electrons in an atomic valence state and the three-dimensional harmonic oscillator in the analysis of vibrational spectra. Constant potential in a non-central held dehnes the motion of a free particle in a rectangular potential box, used to simulate electronic motion in solids. 

In chapter 2 we showed how the wave equation of a vibrating rotator was derived through a series of coordinate transformations. We discussed the solutions of this wave equation in section 2.8, and the particular problem of representing the potential in which the nuclei move. We outlined the relatively simple solutions obtained for a harmonic oscillator, the corrections which are introduced to take account of anharmonicity, and derived an expression for the rovibrational energies. Our treatment was relatively brief, so we now return to this subject in rather more detail. 

Exact Differential Equation for Particle Density for N Particles Moving in Onedimensional Harmonic Oscillator Potential.—One of the major aims of density theory must remain direct calculation of the particle density p from one-body potential of the form of equation (147). So far, this has not proved possible exactly for any three-dimensional potential. 

There is one, admittedly elementary, example where an exact differential equation has been derived by Lawes and March.127 This is for N particles moving in a one-dimensional harmonic oscillator potential. The motivation of their argument was to study the functional derivative dtx/6p appearing in the Euler equation (49). Adapted to the linear harmonic oscillator, this reads, with suitable choice of units... 

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