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Unformatted text preview: Physical Chemistry Course Number: C362 8 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total quantum Hamiltonian looks like: H = − ¯ h 2 2 m d 2 dx 2 + 1 2 kx 2 (8.1) where k is the force constant for the Harmonic oscillator. [The k is not to be confused with the same symbol we used in the PIB.] Notice, this is similar to the PIB case and only differs in the potential term. The problem at hand de- termines what the potential should look like. We will now look at the solution to this problem, that is we will look for the wavefunction and energies when this Hamiltonian is used in the time-independent Schr¨odinger Equation. Or in other words we look now for the eigenfunctions and eigenvalues for this Hamiltonian. The HO problem forms the basis to a number of chemical problems, for ex- ample the solution to vibrational motion in molecules and in infra-red spec- troscopy. It also plays an important role in the quantum theory of solids. 2. As a prelim to the quantum mechanical treatment, let us discuss the classi- cal Harmonic oscillator (HO). The classical harmonic oscillator comprises a single mass attached to the end of a spring. When the spring is stretch the particle undergoes a simple harmonic motion. This is characterized by the particle moving from one end to the other and back and when there is no external disturbance this motion is perpetual. (See derivation on board.) Furthermore all values of energy are possible, since the particle can vibrate about its equilibrium position with any amount of energy. (The energy de- termines how much the spring stretches, that is the amplitude.) It is possible for the particle to have zero energy (no motion). There is no zero point en- ergy in classical mechanics. The simple pendulum is another example of the classical harmonic oscillator....
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This note was uploaded on 01/17/2012 for the course C 362 taught by Professor Amarflood during the Winter '11 term at Indiana.
- Winter '11