the computer bulletin-board Investigating the Harmonic Oscillator Using Mathematica James J. Bruce and Bruce D. ~nderson'
Division of Biological and Physical Sciences Lander University Greenwood,SC 29649 Recently, computer algebra systems, such as MathCad, Mathematica, and Maple, have been adapted to facilitate learning in the quantum mechanics portion of physical chemistry. For example, the graphics capabilities have been used to examine wavefunctions and their probabilities (1,2)as well as atomic and molecular orbitals (3). The computational abilities of computer algebra systems have been used to explore overlap integrals (4) and the solution of the Sehriidinger equation involving the Morse potential (5, 6). Furthermore, Rioux (7)has shown how MathCad can be used throughout quantum mechanics to find numerical solutions to the Schrodinger equation and solve other types of problems as well. As a further example of integrating computer algebra systems into physical chemistry, the use of Mathematica2 to study the one-dimensional harmonic oscillator is discussed. Calculating the Zero-Point Energy for CO
To evaluate the harmonic oscillator, the students should be familiar with the SchrLidinger equation and preferably have experience with Mathematica. As a specific example, consider the calculation of the zero-point energy for CO. Entries for the Diatomic Molecule First, the students should enter the wnstants for the diatomic molecule, including quantum number n force constant k the mass of each atom in kilograms Planck's constant divided by 2n hb Next they enter the expression for the reduced mass of the molecule, and then the time-independent wave function w as shown below. w=U(ZAnn !IA.5(alPiIA.25HemiteH[n,zl Exp[-(axA2Y21 where
Kenyon College Gambier, OH 43022
where F is the energy of the system. The Output The output should be an expression in terms of x and the eigenvalue F. To calculate the eigenvalue, enter a value for x and solve. Because the eigenvalues are known to be independent of x , any value of x will produce the correct answer. For best results, however, exact values of the atomic masses and other constants must be used. For example, if the constants are used with only three significant figures, then values of x that deviate widely from the equilibrium bond length of the molecule will not produce accurate results. This problem can be alleviated if the students enter a reasonable bond length for x. At this point students can repeat the calculation for another value of n to verify that the energy levels of the harmonic oscillator are evenly spaced and nondegenerate. Additlonal Challenges
The problem described could be done as a homework assignment or a computational lab experiment in which each student does the calculations for a different diatomic molecule. In either case, it may be beneficial to combine the assignment with a graphical exploration of the wavefunctions as suggested elsewhere (1,2). Finally, similar computational procedures have been canied out successfully using Mathematica for both the rigid rotor and hydrogen atom problems. It is hoped that using this approach will enhance the students' understanding ofthe importance and validity of the Schrodinger equation. Acknowledgment
The authors would like to thank the Lander Foundation for the funds to purchase the software, and B. Anderson would like to thank the Chem 402 class for their work and their interest in this project.
An I-Strain Prediction via Computer Modeling Keith A. Bellomo, Russell C. Bush and Richard D. sands3
Alfred University
a = (k uP.5 h b
Alfred, NY 14802
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and where x is the internuclear separation, and u is the reduced mass. Finally, the Sehrtidinger equation is input as follows. -hbA2/(2u)D[w,(x,2ll+ k xA2w - F w == 0 'Author to whom correspondence should be addressed. 2WolframResearch, PO Box 6059, Champaign. lL 61826. 3Author to whom correspondence should be addressed. 4TriposAssociates, Inc., St. Louis, MO, 1988. A122
edited by
RUSSELLH. BAT^
Journal of Chemical Education
ALCHEMY 11.4 one of several molecular-modeline>urngrams now available, was used by an undergraduate student to studv the conformations and strains of the uossible intermediate ions involved in reactions in which Bve- and six-membered rings (Fig. 1) or six- and seven-membered rings (Fig. 2) must compete to determine the outcomes. Both reactions involve the conversion of a dihalide to a ketone. The mechanism has been conclusively established by Kakis (8).In each case the product depends on which of two possible product-controlling intermediates--2 or 6 and 11or 15--would have less steric energy and thus would be more likely to form. (Continuedon page A124)
