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by: Ashley Kunze


Ashley Kunze
GPA 3.61


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Class Notes
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This 4 page Class Notes was uploaded by Ashley Kunze on Thursday October 22, 2015. The Class Notes belongs to CHEM 112L at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 54 views. For similar materials see /class/226945/chem-112l-university-of-california-santa-barbara in Chemistry and Biochemistry at University of California Santa Barbara.

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Date Created: 10/22/15
Quantum Harmonic Oscillator Eigenvalus and Wavefunctions Short derivation using computer algebra package Mathematica Dr Kalju Kahn UCSB 20072008 I This notebook illustrates the ability of Mathematica to facilitate conceptual analysis of mathematically difficult problems Quantum harmonic oscillator is an important model system taught in upper level physics and physical chemistry courses In chemistry quantum harmonic oscillator is often used to as a simple analytically solvable model of a vibrating diatomic molecule The model captures well the essence of harmonically vibrating bonds and serves as a starting point for more accurate treatments of anharmonic vibrations in molecules The classical harmonic oscillator is a system of two masses that vibrate in quadratic potential well v gxz without friction The system can be characterized by its harmonic vibrational frequency v force constant k the second derivative of energy with respect to distance and the reduced mass u These three characteristics are related to each other the frequency depends on the force constant oscillators with stiff bonds have high frequencies and the reduced mass oscillators with larger reduced mass vibrates with lower frequency The classical frequency is given as v 2 Our first goal is to solve the Schrodinger equation for quantum harmonic oscillator and find out how the energy levels are related to the harmonic frequency Thus we need to rewrite the harmonic potential in terms of the frequency and the reduced mass n209 Remove quotGlohal quot n210 Express the force constant in terms of the reduced mass and frequency 1 k kharm Solve v k Flatten u 2 Pi om21o1 k a 4 n2 1 v2 n211 Classical harmonic potential for the harmonic oscillator in terms of the force constant k is 2 k unad x 2 n212 Classical harmonic potential for the harmonic oscillator in terms of the reduced mass and frequency is Vho unad kharm 0ut212 2 n2 x2 u v2 I The Schrodinger equation contains the Hamiltonian which is a sum of the quantum mechanical kinetic energy operator and the quantum mechanical potential energy operator The quantum mechanical kinetic energy operator in one dimension can be easily derived from the quantum 2 QuantHO Waven nb mechanical momentum operator f i 2quot Six by recalling the that the relationship between the 2 2 kinetic energy energy and the momentum is Ekin 5 In the case of harmonic oscillator the action of a quantum mechanical potential operator is identical to the multiplication with the classical potential Hamiltonian for the Quantum Harmonie Oscillator ll kin hip Hf E h2Dtfx2Vho f u 1 hZDtf x 2 0ul213 2 ffrz x2 u v2 a 2 8 7r l1 Solving the Vibrational Schrodinger Equation ll E E E VihrW39E DSolve Hmx Energyv Ex Ex x rhvr2Energyv 21V2 Xxxll Vv om214 me gtC2 ParabolicCyllnderD 2 h v vh rhv2Energyv Cl ParabollcCyllnderD 2 h v h n215 Consider solutions with real variables only solnHerm FunetionExpandEX VihrW39E C2 gt 0 irhvozEneleVl 273va 7h v 2 Energyv 2 x Vii V v om215 2 4M e ih ClHerm1teH 2 h v V n216 Obtain allowed energies by restricting Hermite polynomials to integer orders 2Energyv hv 2hv EnHO Table Energyv Env v 0 2 Flatten Env Solve o v Energyv Oul216 Energyv gt 2 h l 2 v v h v 3 h v 5 h v Oul217 7y 2 y I We see that the concept of quantized vibrational energy states v 0 1 2 3 arises naturally from the discrete spectrum of physically realistic eigenvalues of the solution to the vibrational Schrodinger equation This spectrum can be experimentally probed using infrared spectroscopy n21a General vibrational wavefunetion mv x SimplifysolnHerm En v Flatten 272x2w ZNXVM Vv om21a 2quot 2 e39 h ClHerm1teHv Vh QUantHO Waven nb 3 W19 Integration constant is determined by requiring that fl zdlx 1 V eOv x SolveIntegrateLastmv x2 x m m Assumptions gt 0 1 Cl n vaV X Ev x LasteOv x n221 Some of the wave functions are mvx FullSimplifyTableEv x LasteOv x v 0 2 Flatten gv Grid PartitionTable V V 0 2 l Spacings 0 2 gwf GridPartitionEvx 1 gho Grid PartitionEnHO 1 Spacings gt 0 2 GridPartitiongv gwf gho 3 Frame gt All hv 27am 14 i O 14 a 2 W2 e h n lt h gt many 4e h IF xVu lv M 3m Oul225 l T 2 Shv 2 Verify that the ground state wavefunetion is indeed the same as expressed via a in the traditional treatment 2 u 7r2 x2 v a x2 uv a Eox EOx LasteO0x gt gt n 2 h 472 X2 V 14 0m226 We Trrm 4 QUantHO Wa ven nb 1 W247 Plot Evaluate Append Table3 10 2 IVX v x v waven v 0 4 2 Out247 Next we plot the vibrational energy levels and associated wavefunctions for carbon monoxide molecule Because of its chemical stability and permanent dipole moment the vibrational spectrum of CO is experimentally well characterized For consistency with the traditional infrared nomenclature the energy on the yaxis is expressed in wavenumber cm 1 units The internuclear distance on the xaxis is in meters with the equilibrium internuclear distance set to zero ltlt PhysicalConstants Planck Constant h in appropriate units is Kilogram 100 cm 2 Second h PlanckConstant Joule gt 39 Second2 01112 Ki logram Reduced mass of carbon monoxide in kilograms is k 12 l6 1 FLU I39 12 16 Kilogram k Experimental harmonic frequency in wavenumber units is waven 2168 cmquot1 cm k Harmonic frequency in appropriate cm units is Second v waven SpeedOfLight Meter gt 100 cm cm k Square of the speed of light in appropriate cm units is CC 2 2 Second SpeedOfLight2 Meter gt 100 cm cm k Bond force constant in appropriate units is fc k kharm The first five energy levels and wave functions are shown below Note that the magnitude of each of the wavefunctions is scaled arbitrarily to fit below the next energy level The spacing between the energy levels is not scaled and corresponds to the experimental harmonic frequency 2168 cm391 0006 cc fc x2 2 1 x 15 10 9 15 10 9 Filling gt Tablev gt v waven v 1 5 AxesLabel gt x E 2 l l l I l l 1 l 1 l 1 l l l 1 l 1 l l l l I l l x 15gtlt10399 1gtlt10 9 5gtlt10quot 5gtlt10quot 1gtlt10 9 15gtlt10quotquot


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