edited by RUSSELLH. BAT
bullertin'board
Kenyon College Gambier. OH 43022
Teaching Quantum Mechanics with Theorist G. L. Breneman and 0. J. Parker Department of Chemistry and Biochemistry, Eastern Washington University, Cheney, WA 99004
The algebra and graphics package for the Macintosh, Theorist, by Prescience,' is a n extremely useful tool in teaching basic quantum mechanics in undergraduate physical chemistry. While this package does a large n u n ber of different types of mathematical manipulations, the ease of graphing functions (lines or surfaces using several
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Figure 2. Two-dimensionalparticle in a square box (1 x I ) in the state 2.2 showing: (a) surface and contour plots of y2,(b) normaiization, (c)probability of being in an area inside x = 0 to 0.5 and y = 0 to 0.5, and (d) expectation value of the x coordinate.
Figure 1. The harmonic oscillator with a = rr showing: (a) the function in the v = 2 state and its plot, (b) normaiization (note that 2 is large enough to approximate infinity in this case), (c) positional probability in the range W . 5 , (d)expectation value of the position, and (e) orthogonality of the v = 0 and v = 1 states.
cwrdinate systems), multiple integration, and differentiation are especially useful in studying the simple systems encountered in beginning quantum mechanics. The use of Theorist applied to the particle in a one-, two-, and threedimensional box, the harmonic oscillator, the rigid rotor, and the hydrogen atom will be discussed. '939 Howard Street, San Francisco, CA 94103, (415)543-2252. (Conrinuedon page A61
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Journal of Chemical Education
the computer bulletin board The wave functions and squares of the wave functions (probability)for one- and two-dimensional systems can be plotted. The probability can be integrated over various ranges of the variables to demonstrate normalization and show how the probabilities of these systems being in certain locations or orientations vary with different states. Expectation values can be calculated for the positional variables by integration, and the maxima in wave functions, in their squares (or Y Y * ) ,and in the radial distribution functions for hydrogen can be determined. Orthogonality ofthe wave functions also can be demonstrated. One-Dimensional Systems The one-dimensional particle-in-a-box is one of the first systems studied because the wave equation is easy to solve and the system allows the demonstration of most of the principles of quantum mechanics mentioned above. The integrals for some ranges can be determined by inspection because of the symmetry of the functions, but Theorist allows the calculation to be done conveniently dver any range for any state. The harmonic oscillator is another one-dimensional system that is mmmonly discussed. The shapes of the wave functions and probabilities are not as intuitively obvioua for this system as they are for the one-dimensional particle-in-a-box. Almost none of the possible integrals are obvious. This is where a program such as Theorist can give the students an intuitive feel for the system by having them make a number of plots and calmlate a number of integrals on their own. Again the position of the maxima in the probability plots can be determined to show the most probable positions in the different states. Figure 1shows examples for this system. -
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Figure4. Hydrogen atom 2s orbital showing: (a)the radial distribution function and plot, (b) plot of the derivative.
b) c)
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YYu sin do) dp = 0058058
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sin 0 dB
Figure 3. Rigid Rotor ( I = 1 and m = +1 and -1 states) showing: (a)a spherical coordinate plot of W'. (b) probability of orientafionfor 8 = ' 0 to 45', (c)expectation value for 8. A6
Journal of Chemical Education
C)
CIT,
ty? d.1sin 0 dB) dcp = 3.173
FigLre 5. Hydrogen atom 2s orbita showing: (a)zooming in on deriv. at~veplot aroLnd r c n t corresponding to most probable d stance, (bl aetermining ttns root witn Fmd Rwtcornrnano. (c)expectation value of distance from nucleus (10 approximates infinity in the integral)
Two-Dimensional Systems The two-dimensional particle-in-a-box is a good extension from one-dimensional systems. Separation of variables can be discussed, and the equations are still obvious to solve; but now two quantum numbers are generated. Degenerate states can also be discussed. Using Theorist, surface and contour plots can be made of the wave functions and their squares, and normalization, probability in an area, expectation value, and orthogonality of different states can all be shown using double integrals. Some examples are given in Figure 2. The rigid m b r is the simplest system using spherical coordinates and will be used as part of the hydrogen atom solution. Surface plots of the functions and their squares can be made with respect to the angles 9 and Q usmgspherical coordinates. Normalization can be shown by integ-ratof the ing over 9 = 0 to n and $ = 0 to 271. The rotor being in different angular ranges can be shown, as can expectation values of 9 and $. Figure 3 shows some examples of these. Three-Dimensional Systems The particle-in-a-box can next be extended to three dimensions, generating more degeneracy and a third quan-
tum number. Plots cannot be made, but triple integrals can be evaluated demonstrating the same general principles. can be made separat& for For the hydrogen atom, the anrmlar part (same as rieid rotor) and radial part ofthe wave functi& The radiaidistribution functiok can be plotted to show how the probability of finding the electron varies with distance from the nucleus, averaged over all angles. The derivative of this function can be plotted and the mot, corresponding to the most probable distance, can be determined. The average distance of the electron from the nucleus can also be determined along with showing normalization, probability in a volume range, and orthogonality of the wave functions. See Figures 4 and 5 for some examples. Conclusion In our physical chemistry classes, we use Theorist on a Macintosh SE with an overhead pmjector panel to project the output on a large screen and demonstrate some of these wave functions and calculations. The computer is on a cart that is normally stored in the physical chemistry laboratory where students work additional exercises in quantum mechanics on their own using Theorist. We have found that this hands-on experience greatly reinforces what has been covered with the lecture demonstrations.
Volume 69 Number 1 January 1992
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