Quantum Harmonic Oscillator Eigenvalues and Wavefunctions: Short derivation using computer algebra package Mathematica Dr. Kalju Kahn, UCSB,

Transcription

1 Quantum Harmonic Oscillator Eigenvalues and Wavefunctions: Sort derivation using computer algebra package Matematica Dr. Kalju Kan, UCSB, Tis notebook illustrates te ability of Matematica to facilitate conceptual analysis of matematically difficult problems. Quantum armonic oscillator is an important model system taugt in upper level pysics and pysical cemistry courses. In cemistry, quantum armonic oscillator is often used to as a simple, analytically solvable model of a vibrating diatomic molecule. Te model captures well te essence of armonically vibrating bonds, and serves as a starting point for more accurate treatments of anarmonic vibrations in molecules. Te classical armonic oscillator is a system of two masses tat vibrate in quadratic potential well (V k x ) witout friction. Te system can be caracterized by its armonic vibrational frequency n, force constant k (te second derivative of energy wit respect to distance), and te reduced mass m. Tese tree caracteristics are related to eac oter te frequency depends on te force constant (oscillators wit stiff bonds ave ig frequencies) and te reduced mass (oscillators wit larger reduced masses vibrate at lower frequencies). Te classical frequency is given as 1 k Our first goal is to solve te Scrødinger equation for quantum armonic oscillator and find out ow te energy levels are related to te armonic frequency. Tus, we need to rewrite te armonic potential in terms of te frequency and te reduced mass. In[]:= In[3]:= Remove"Global`" Express te force constant in terms of te reduced mass and armonic frequency karm Solve 1 k,kflatten Out[3]= In[4]:= k 4 Classical armonic potential for te armonic oscillator in terms of te force constant k is: Vquad k x In[5]:= Classical armonic potential for te armonic oscillator in terms of te reduced mass and frequency is: Vo Vquad. karm Out[5]= x Te Scrødinger equation contains te Hamiltonian, wic is a sum of te quantum mecanical kinetic energy operator and te quantum mecanical potential energy operator. Te quantum mecanical kinetic energy operator in one dimension can be easily derived from te quantum

2 QUantHO_Waven.nb mecanical momentum operator (p = -i p ) by recalling te tat te relationsip between te x kinetic energy and te momentum is: E kin = m v = p. In te case of armonic oscillator, te m action of a quantum mecanical potential operator is identical to te multiplication wit te classical potential. In[6]:= Hamiltonian for te Quantum Harmonic Oscillator: H H kin H pot Hf Dtf, x, Vo f 8 Out[6]= f x Dtf, x, 8 In[7]:= Solving te Vibrational Scrødinger Equation: H E VibrWF DSolve Hx Energyv x, x, x Out[7]= Energyv x C ParabolicCylinderD, x Energyv C1 ParabolicCylinderD, x In[8]:= Consider solutions wit real variables only solnherm FunctionExpandx. VibrWF. C 0 Out[8]= Energyv 4 x Energyv C1 HermiteH, x In[9]:= Obtain allowed energies by restricting Hermite polynomials to integer orders Env Solve Energyv 0 v, Energyv EnHO TableEnergyv.Env, v, 0, Flatten Out[9]= Energyv 1 1 v Out[10]=, 3, 5 We see tat te concept of quantized vibrational energy states (v = 0, 1,, 3... ) arises naturally from te discrete spectrum of pysically realistic eigenvalues of te solution to te vibrational Scrødinger equation. Tis spectrum can be experimentally probed using infrared spectroscopy. In[11]:= General vibrational wavefunction v, x SimplifysolnHerm. Env Flatten Out[11]= v x C1 HermiteHv, x

3 Cell[TextData[{Cell[TextData[{ValueBox["FileName"]}], "Header"], Cell[" ", "Header", CellFrame -> {{0, 0.5}, {0, 0}}, CellFrameMargins -> 4], " ", Cell[TextData[{CounterBox["Page"]}], "PageNumber"]}], CellMargins -> {{Inerited, 0}, {Inerited, Inerited In[1]:= Integration constant is determined by requiring tat x 1 c0v, x : SolveIntegrateLastv, x, x,,, Assumptions 0 1, C1 vx v, x : v, x. Lastc0v, x In[14]:= Some of te wave functions are v x FullSimplifyTablev, x. Lastc0v, x, v, 0, Flatten gv GridPartitionTablev, v, 0,, 1, Spacings 0, gwf GridPartition v x, 1 go GridPartitionEnHO, 1, Spacings 0, GridPartitiongv, gwf, go, 3, Frame All 0 x Out[18]= 1 4 x 54 x 14 3 x x 5 In[19]:= Verify tat te ground state wavefunction is indeed te same as expressed via in te traditional treatment 0 x 0, x Out[19]= x Lastc00, x. x x. 4

4 4 QUantHO_Waven.nb In[0]:= Next we plot te vibrational energy levels and associated wavefunctions for carbon monoxide molecule. Because of its cemical stability and permanent dipole moment, te vibrational spectrum of CO is experimentally well caracterized. For consistency wit te traditional infrared nomenclature, te energy on te y-axis is expressed in wavenumber (cm -1 ) units. Te internuclear distance on te x-axis is in meters wit te equilibrium internuclear distance set to zero. PysicalConstants` Planck Constant in appropriate units is: Kilogram 100 cm PlanckConstant. Joule Second Second cm Kilogram Reduced mass of carbon monoxide in kilograms is: ProtonMass 1 Kilogram Experimental armonic frequency in wavenumber units is: waven 168 cm 1 cm Harmonic frequency in appropriate cm units is: waven SpeedOfLigt. Meter 100 cm Second cm Square of te speed of ligt in appropriate cm units is: cc SpeedOfLigt. Meter 100 cm Second cm Bond force constant in appropriate units is: fc k. karm Te first five energy levels and wave functions are sown below. Note tat te magnitude of eac of te wavefunctions is scaled arbitrarily to fit below te next energy level. Te spacing between te energy levels is not scaled and corresponds to te experimental armonic frequency ( 168 cm 1 ). In[9]:= PlotEvaluate Append Table3 10 vx v, x v 1 x, , , Filling Tablev v cc fc x waven, v, 0, 4,, waven, v, 1, 5, AxesLabel x,"e" Out[9]=

5 Cell[TextData[{Cell[TextData[{ValueBox["FileName"]}], "Header"], Cell[" ", "Header", CellFrame -> {{0, 0.5}, {0, 0}}, CellFrameMargins -> 4], " ", Cell[TextData[{CounterBox["Page"]}], "PageNumber"]}], CellMargins -> {{Inerited, 0}, {Inerited, Inerited Te first five energy levels and squares of associated wave functions are sown below. Note tat te magnitude of eac of te squared wavefunctions is scaled arbitrarily to fit below te next energy level. Recall tat te square of te wavefunction gives te probability te plot below tus sows probability distributions in different vibrational states. For example, te most probable bond distance in te ground state CO corresponds to te equilibrium distance at te bottom of te potential well. In[30]:= Plot Evaluate Append Table vx v, x vx v, x v 1 waven, v, 0, 4, cc fc x, x, , , Filling Tablev v 1 waven, v, 1, 5, AxesLabel x, "E", PlotRange All Out[30]=