Harmonic oscillator Hooke’s law

In practice, however, these calculations are rather intractible for many-electron molecules, so that, using the Bom-Oppenheimer approximation (see below), it is assumed that the electronic motion can be factored off from Eq. (1.1), yielding an effective potential-energy field in which the nuclear motions (vibrations) take place and which couple with the rotatory motion of the molecule. Normally perturbation theory is used in the first and second approximations to give the vibrational and rotational eigenvalues of H. Instead of a theoretically-obtained potential-energy function (see below), it is normally assumed that the vibratory motion follows Hooke s law (harmonic oscillator) with... 

Show that, for a simple one-dimensional Hooke s-law harmonic oscillator having mass m, the three equations of motion yield the same results. 

For a Hooke s-law harmonic oscillator, the (nonvector) force is given by... 

The potential energy, denoted V, of a Hooke s-law harmonic oscillator is related to the force by a simple integral. The relationship and final result are... 

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

Three 10,0-g masses are connected by springs to fixed points as harmonic oscillators showui in Fig, 3-12, The Hooke s law force constants of the springs ai e 2k. k, and k as showui, where k = 2.00 N m, What are the pei iods and frequencies of oscillation in hertz and radians per second in each of the three cases a, b, and e ... 

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

In short, near-infrared spectra arise from the same source as mid-range (or normal ) infrared spectroscopy vibrations, stretches, and rotations of atoms about a chemical bond. In a classical model of the vibrations between two atoms, Hooke s Law was used to provide a basis for the math. This equation gave the lowest or base energies that arise from a harmonic (diatomic) oscillator, namely ... 

Fig. 6.2. Energy curve for Hooke s law versus Quantum Model of harmonic oscillator.

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

Fig. 3.1 Born-Oppenheimer vibrational potentials for a diatomic molecule corresponding to the CH fragment. The experimentally realistic anharmonic potential (solid line) is accurately described by the Morse function Vmorse = De[l — exp(a(r — r0)]2 with De = 397kJ/mol, a = 2A and ro = 1.086 A (A = Angstrom = 10 10m). Near the origin the BO potential is adequately approximated by the harmonic oscillator (Hooke s Law) function (dashed line), Vharm osc = f(r — ro)2/2. The harmonic oscillator force constant f = 2a2De...

Let us consider a diatomic molecule and assume that it behaves as a harmonic oscillator with two masses, nii and m2, connected by an ideal (constant-force) spring. At equilibrium, the two masses are at a distance Xq by extending or compressing the distance by an amount X, a force F will be generated between the two masses, described by Hooke s law (cf equation 1.14) ... 

Vibrational energy and transitions As seen in Fig. 3.2a, the bond between the two atoms in a diatomic molecule can be viewed as a vibrating spring in which, as the internuclear distance changes from the equilibrium value rc, the atoms experience a force that tends to restore them to the equilibrium position. The ideal, or harmonic, oscillator is defined as one that obeys Hooke s law that is, the restoring force F on the atoms in a diatomic molecule is proportional to their displacement from the equilibrium position. 

M for a simple harmonic oscillator where v, the vibrational quantum i AUmber has values 0, 1.2, 3, etc. The potential function F(r) for simple < harmonic motion as derived from Hooke s law is given by... 

Assignments for stretching frequencies can be approximated by the application of Hooke s law. In the application of the law, two atoms and their connecting bond are treated as a simple harmonic oscillator composed of two masses joined by a spring. The following equation, derived from Hooke s law, states the relationship between frequency of oscillation, atomic masses, and the force constant of the bond. 

PROBLEM 2.16.6. Solve by Laplace transform methods the classical linear harmonic oscillator differential equation mdzy/cHz= —kHy(t), where kH is the Hooke s law force constant, with the initial condition dy/dt = 0 at t = 0. Note Use p for the Laplace transform variable, to not confuse it with the Hooke s law force constant kH ... 

The harmonic oscillator in one dimension, with Hooke s law constant kH, obeys the Schrodinger equation ... 

PROBLEM 3.4.7. (i) Compute the classical energy for the harmonic oscillator of mass m, Hooke s law force constant kH, frequency v = (1 /2n)(kH/m)[/z, maximum oscillation amplitude a0, and displacement x. (ii) Next, compute the classical probability that the displacement is between x and x + dx. (iii) Compare this result with the quantum-mechanical probability for the harmonic oscillator of the same frequency v. 

Indeed, the relationship between the frequency of the bond vibration, the mass of the atoms, and the strength of the bond is essentially the same as Hooke s law for a simple harmonic oscillator. 

Thus, a simple diatomic molecule A-B has (3 x 2) - 5 = 1 vibrational mode, which corresponds to stretching along the A-B bond. This stretching vibration resembles the oscillations of two objects connected by a spring. Thus, to a first approximation, the model of a harmonic oscillator can be used to describe this vibration, and the restoring force (F) on the bond is then given by Hooke s law ... 

The quantum-mechanical treatment of molecular vibrations leads to modifications of the harmonic oscillator model. While the Hooke s law treatment presented above would indicate a continuum of vibrational states, the molecular vibrational energy levels are quantised ... 

When a broad-band source of IR energy irradiates a sample, the absorption of IR energy by the sample results from transitions between molecular vibrational and rotational energy levels. A vibrational transition may be approximated by treating two atoms bonded together within a molecule as a harmonic oscillator. Based on Hooke s law, the vibrational frequency between these two atoms may be approximated as ... 

Fairly good agreement exists between the calculated value of 1682 cm-1 and the experimental value of 1650 cm1. Direct correlation does not exist because Hooke s law assumes that the vibrational system is an ideal harmonic oscillator and, as mentioned before, the vibrational frequency for a single chemical moiety in a polyatomic molecule corresponds to the vibrations from a group of atoms. Nonetheless, based on the Hooke s law approximation, numerous correlation tables have been generated that allow one to estimate the characteristic absorption frequency of a specific functionality (13). It becomes readily apparent how IR spectroscopy can be used to identify a molecular entity, and subsequently physically characterize a sample or perform quantitative analysis. 

A related measure of the intensity often used for electronic spectroscopy is the oscillator strength,/ This is a dimensionless ratio of the transition intensity to that expected for an electron bound by Hooke s law forces so as to be an isotropic harmonic oscillator. It can be related either to the experimental integrated intensity or to the theoretical transition moment integral ... 

For simplicity we assume that each of the molecular vibrations is a simple harmonic vibration characterized by an appropriate reduced mass jU and Hooke s law constant k. The wave functions are determined by a single quantum number v, the vibrational quantum number. The energy of the oscillator is... 

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