Oscillators, 3-dimensional harmonic potential determination

Suppose that we wish to determine the potential energy when x = 1.5 nm and y = 0.5 nm for a two-dimensional harmonic oscillator in which k = 250 N m. The equation can be rewritten to show the implied multiplication as... 

To understand the complete role of vibration in determining electrical properties, it is useful to consider a diatomic molecule in the harmonic oscillator approximation, where the stretching potential is taken to be quadratic in the displacement coordinate. The doubly harmonic model takes the various electrical properties to be linear functions of the coordinate. This turns out to be most reasonable in the vicinity of an equilibrium structure, but it breaks down at long separations. Letting x be a coordinate giving the displacement from equilibrium of a one-dimensional harmonic oscillator, the dipole moment, dipole polarizability, and dipole hyperpolarizability, within the doubly harmonic (dh) model, may be written in the following way ... 

The one-dimensional quadratic potential V = kx2 has been used for the description of covalent binding. The ground-state wave functions for a simple harmonic oscillator, /t and iR, have been used to describe the proton in the left and right wells. The force constant k has been determined from the stretch-mode vibrational transitions for water occurring at 3700 cm-1. The ground-state energy for the proton is 0.368 x 10-19 J. The tunneling barrier is AE = 4 x 10-19 J. 

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

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