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Published byJulianna Thomas Modified over 9 years ago
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Quantum Springs
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Harmonic Oscillator Our next model is the quantum mechanics version of a spring: Serves as a good model of a vibrating (diatomic) molecule The simplest model is a harmonic oscillator:
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Harmonic Oscillator What does this potential mean? Let’s take a look at a plot: x = spring stretch distance V x 0 = “equilibrium bond length”
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Harmonic oscillator Let’s do the usual set up: The Schrodinger equation: Insert the operators Rearrange a little This is a linear second order homogeneous diff. eq., BUT with non-constant coefficients… Too hard to solve by hand, so we’ll do it numerically on the computer!
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Numerov technique Just as a matter of note, we have to use rescaled x, , and E for the numerical solution algorithm we’ll use: the Numerov technique. Get spit out of the Numerov alg. Scaling coefficients
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Numerov technique is the “reduced mass”: k is the “spring constant” Measures “stiffness” of the bond m1m1 m2m2 With the spring constant and reduced mass we can obtain fundamental vibrational frequencies
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Solve the Harmonic Oscillator
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E v = 0
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Solve the Harmonic Oscillator E v = 1 E v = 0
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Solve the Harmonic Oscillator E v = 2 E v = 1 E v = 0
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Solve the Harmonic Oscillator E v = 3 E v = 2 E v = 1 E v = 0
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Solve the Harmonic Oscillator E v = 4 E v = 3 E v = 2 E v = 1 E v = 0
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Solve the Harmonic Oscillator E v = 5 E = ħ E v = 4 E v = 3 E v = 2 E v = 1 E v = 0 v = {0, 1, 2, 3, …}
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Solve the Harmonic Oscillator Ground State
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Solve the Harmonic Oscillator First Excited State
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Solve the Harmonic Oscillator Second Excited State
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Solve the Harmonic Oscillator Third Excited State
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Solve the Harmonic Oscillator Fourth Excited State
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Solve the Harmonic Oscillator Fifth Excited State # nodes, harmonic oscillator = v
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Anharmonic Oscillator Real bonds break if they are stretched enough. Harmonic oscillator does not account for this! A more realistic potential should look like: Energetic asymptote
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Anharmonic Oscillator Unfortunately the exact equation for anharmonic V(x) contains an infinite number of terms We will use a close approximation which has a closed form: the Morse potential
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Anharmonic Oscillator Wave function dies off quickly when it gets past the potential walls Ground State # nodes, anharmonic oscillator = v
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Anharmonic Oscillator First Excited State Note how anharmonic wave functions are asymmetric
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Anharmonic Oscillator A energy increases toward the asymptote, eigenvalues of the anharmonic oscillator get closer and closer Energetic asymptote
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Anharmonic Oscillator Bond almost broken…
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Anharmonic Oscillator Energetic asymptote Bond breaks! D 0 = bond energy
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