# Sep 5 - Quantum Theory of Motion Translational Motion...

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Quantum Theory of Motion Translational Motion The "particle in a box" problem Concept of quantization of energy imposed by spatial constraints tunneling Vibrational Motion The harmonic oscillator problem Model for vibrational motion of bonds and molecules Relationship to IR spectroscopy Rotational Motion The rigid rotor problem Model for molecular rotation Relationship to microwave spectroscopy

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The Harmonic Oscillator A particle undergoes harmonic motion and is said to be a harmonic oscillator, if it experiences a restoring force proportional to its displacement. Examples: 1. A pendulum 2. A coiled spring 3. A chemical bond [ to a first approximation] Mathematically: F = -k f x 8B.1 Note k f is the force constant –the stiffer the "spring" the greater the value of k f . Also, the force F is a restoring force, so the sign of k f x is negative
The Harmonic Oscillator To solve the harmonic oscillator problem, we need to think also about potential energy V(x) = ½ kx 2 8B.2 This is a parabolic potential energy

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The Harmonic Oscillator Next, we need to solve the Schrodinger Equation for the harmonic oscillator Notice that now we have to include the potential energy term as well [As opposed to the particle in a box problem, which we solved "by inspection", this makes the harmonic oscillator Schrodinger equation somewhat more challenging to solve!] 8B.3 It is possible to solve this Schrodinger equation exactly, and obtain the eigenfunctions (x), but we will not attempt to do it in this course! If you are interested, the authors of our text book recommend Molecular Quantum Mechanics , Oxford University Press, 2011 Or get yourself a strong cup of coffee and google it……
The Harmonic Oscillator The solution to the Schrodinger Equation for the harmonic oscillator : N v = normalization constant H v (y) = Hermite polynomial exp(-y 2 /2) = Gaussian function 8B.8

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The Harmonic Oscillator The eigenvalues of the Schrodinger Equation for the Harmonic Oscillator represent the allowed quantum mechanical energies of the harmonic oscillator 8B.4  is the frequency of oscillation; this could also be written as  easy to confuse with v ]
PHYSICAL CHEMISTRY: QUANTA, MATTER, AND CHANGE 2E| PETER ATKINS| JULIO DE PAULA | RONALD FRIEDMAN ©2014 W. H. FREEMAN D COMPANY PHYSICAL CHEMISTRY: THERMODYNAMICS, STRUCTURE, AND CHANGE 10E | PETER ATKINS | JULIO DE PAULA ©2014 W. H. FREEMAN AND COMPANY CHAPTER 8: FIGURE 8B.2 Again, boundary conditions require that the system is quantized; only certain wavefunctions and energy levels are possible Again, there is a zero point energy. Unlike the classical harmonic oscillator, which can be a motionless pendulum or spring, the quantum mechanical harmonic oscillator must fluctuate incessantly around its equilibrium position

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PHYSICAL CHEMISTRY: QUANTA, MATTER, AND CHANGE 2E| PETER ATKINS| JULIO DE PAULA | RONALD FRIEDMAN ©2014 W. H. FREEMAN D COMPANY PHYSICAL CHEMISTRY: THERMODYNAMICS, STRUCTURE, AND CHANGE 10E | PETER ATKINS | JULIO DE PAULA ©2014 W. H. FREEMAN AND COMPANY
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