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Lecture 10 Analysis of the Harmonic Oscillator Wave Functions Zero-Point Energy Nodes and Kinetic Energy Spectroscopy of the QM Harmonic Oscillator—Infrared Spectroscopy October 2 2009 M&S pp. 157-173.

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The Harmonic Oscillator The QM harmonic oscillator wave functions are shown on the next page, as are their square moduli, which represent their probability distributions. There are several important things to note. We’ll discuss some of these in more detail below, but for now, let’s just list them (there is some overlap with the list from last lecture). 1) The zeroth level wave function is not at the zero of energy. It’s height above the zero of energy (the bottom of the potential well) is called the zero-point energy, and is equal to (1/2) h ν . 2) The zeroth level wave function has no nodes, and each higher level wave function has one more node than the one below it. 3) For all of the wave functions, the expectation value of the kinetic energy is exactly equal to the expectation value of the potential energy (this is not obvious from inspection, but is the subject of tomorrow’s homework).
The Harmonic Oscillator 4) From (3), it follows that the average potential and kinetic energies each increase by (1/2) h ν as we go from one level to the next. It seems obvious that the potential energy increases, since the wave function samples more distant regions of the potential, and we will look at the kinetic energy below. 5) The probability distributions of the wave functions penetrate the potential barrier. This is very much like what we saw with tunneling. It is possible to find bond lengths that correspond to the system having negative kinetic energy. [Note that this seems like a paradox only if you permit yourself to imagine that you can know the kinetic energy without measuring it. If you measure it, it will be positive, but then you won’t have been able to measure the bond length to an accuracy beyond that permitted by the uncertainty principle. ..]

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The Harmonic Oscillator 6) Although all of the energy levels have < x > = 0, only the probability amplitude of the ground state is peaked at x = 0 (indeed, there is a node there for all the levels with odd n !) For higher and higher levels, the maximum probabilities shift to the walls of the potential. The points x and x where the potential is equal to the total energy are called the ‘classical turning points . 7) Spectroscopic transitions between harmonic oscillator energy levels are permitted only for changes to the level immediately above or immediately below, i.e., n n ± 1. The energy of the photon required for this transition is E = h ν where the frequency of the photon is exactly the same frequency as that of the harmonic oscillator (since the separation in energy levels of the harmonic oscillator is also h ν ). We’ll look at spectroscopy more below.
Zero-Point Energy An interesting aspect of zero point energy is that, for the same potential energy function (which

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## This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

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lecture10_umn - Lecture 10 Analysis of the Harmonic...

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