Analog Computer Solution Michael T. Marron University of W~nconsin-Porks~de Kenosha, Wisconsin 53140
of a Particle in a Finite Well A physical chemistry experiment
Several recent articles (1-5) in this J o u r m l have discussed the use of analog computers as a-teaching aid for discussion of chemical problems which can he characterized by ordinary differential equations. Two of these articles (2, 4 ) have focused on the quantum mechanical prohlem of a particle in a finite well as a n example of a prohlem which is ideally suited for solution by a n analog computer. We have incorporated the analog solution of this prohlem into the second semester physical chemistry laboratory. Our experience over the past two years estahlishes the experiment a s a n unqualified success. In this communication we wish to point out some of the pedagogic benefits which warrant its inclusion in the regular curriculum. Experiment Students are given a written discussionl of analog computers and their various components (pots, amplifiers, summers and integrators) which must he read hefore entering the lab. Characteristics of a n operational amplifier are stated; circuits and symbols are given for pots, summers, and integrators. Relationships between input and output voltages are explicitly derived using equations of current balance (Kirchoff's Law) and Ohm's law. Thus when the student first confronts the computer2 he has only theoretical knowledge of how the computer operates. He is given a brief "how-to-do-it" operation guide, and the first 3-hour lab period is devoted to experimenting with the computer: patching summers, integrators, solving simple f i t - o r d e r differential equations, and displaying results on a voltmeter, a n x-y plotter, and oscilloscope. In the second lab period the student is given a prepatched hoard which contains the programs for the Schroedinger equation, a time hase for the x-y plotter and a hold-delay circuit for halting computation when the particle reaches the wall. The hold-delay circuit is required so that the potential wall may he switched o n 3 The programs are recorded schematically and discussed in detail in the write-up. The student is asked to check the patching t o insure that it is done correctly. Prepatching the program saves considerable time and also avoids "trivial" patching errors which tend to needlessly frustrate students. We use the oromam described hv Ohline ( 4 ) for the Schroedinger equaiionwith a slightly modified hold-delay circuit which automaticallv halts comoutation when the wall is reached. The student is asked'to investigate the following items for a potential well 2 Bohrs wide and 30 atomic units deep (1) find all of the hound-state eigenvalues (to +1 in the third
significant figure and plot the corresponding wavefunctions) (2) plot several continuum wavefunctions (3) comment on penetration of the particle into the well wall as a function of energy level (4) comment on the effect of changing the width and/or depth of the well (5) relate the continuum wavefunction wavelength to particle momentum
The exact solution of the prohlem is provided in the form of a transcendental equation, together with a short program for a digital computer so that the student may ohtain exact results for comparison with his analog results.' The absolute error in the analog solution may thus he determined. Results Computer plots of wavefunctions have been described
bv Tahhutt (2). In terms of relative consistencv. - . results
aie reproducible to within the accuracy of the computer components, i.e., slightly less than 1% variation in the eigenvalues is observed. The absolute error is generally meater owing to student-errors in settine ootential well parameters. A number of students have invkrted application of the exact-solution o r o n a m hv attemotinz to fit their analog results, varying t h e weil depth'and width until the exact results agree with their analog results. Computer instability in the region of a n eigenvalue, noted hy Ohline (4) is of no real concern since we seek only to hracket the eigenvalues to three significant figures. Copies of the write-up, operation guide and sample output are available from the author upon request. Our laboratory is equipped with a Systron-Donner Model 3300 analog computer which sells for $3350. Computers adequate for this problem can he purchased and assembled for less than $300. [cf. Ref. f4)]. 3 Our computer is equipped with a HOLD jack which provides for electronic control of the HOLD function, i.e., suspension of computing operations with all voltages remaining unchanged. The hold-delay circuit is a time hase coupled to a comparator which activates the HOLD mechanism at a time specified by patentiometer, settings. Tahbutt (2) accomplished this manually by physically depressing a microswitch at a specified time. 'The analytical solution of a particle in a finite well is a nantrivial prohlem. The solution to the more general problem of a well with walls of two different heights may be found in A. Messiah's "Quantum Mechanics" (6). For the symmetric well problem described here, the energy levels may be found as solutions to the transcendental equation nr
- €(2!4'iZL = 2 sin-'(€)
--
where 6 (E/V)'IZ; V is the potential well depth, e.g., 30 a,".; n is the quantum number for the level; and L is the box width, e.g., 2 Bohr. There are several methods for solving this equation (Messiah does it graphically). Our students are asked to solve this equation using a PDP-8 computer and the method suggested is to set n and guess an E based on their experimental results, evaluate the right and left hand sides of the equation and campare. The choice of E is a solution when the right and left hand sides agree. Of course many values of E must be tried, and the students are shown how to vary E automatically via DO loops. Given sufficient patience a student can find a solution limited in accuracy only he computer word size. This exercise in itself is instructive as to the meaning of an "exact solution" (what is the "exact" value of sin 26" and how is it determined?). This method of solution is admittedly heuristic; better methods exist. Students with some background in analysis will often solve this equation by Newton's method or a method of successive approximations. Volume 50, Number 4. April 1973
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Student Response
Student reaction to this experiment seems to follow a pattern. The initial response after reading the experimental write-up and confronting the computer for the first time, is one of helplessness; a feeling that they'll never be able to understand the rationale behind connecting wires, resistors and capacitors on the patch board. The initial reaction slowly develops into confidence as they spend the first lab period working through simple circuits and problems outlined in the operations guide. By the end of the first period they have at least a general, if not specific, understanding of the various components and their relationship to one another. The second period, which is devoted to finding eigenvalues and plotting wavefunctions, leads to the state where they feel they are in command of the computer. It is interesting to note that students who have completed the experiment, often find time to return in following weeks to coach those who are performing it for the first time. In our experience, students teaching students is extremely effective. Though complaints over complexity are voiced from time to time, there are usually a few extra students clustered around the experimental station coaching and kibitzing. Conclusion
This experiment is instructive in three areas: electronics,
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differential equations and quantum mechanics. While the students have some familiarity with simple electronic circuits through other experiments, the analog computer provides them with a more specific exercise, particularly in working with operational amplifiers and feedback circuits. For example, they have encountered time bases hefore as black boxes used to drive an x-y recorder. In this experiment they actually construct one. Perhaps the most valuable experience they gain is obtained hy viewing "numerical" integration of a differential equation graphically in real time. The connection hetween boundary conditions and the solution is vividly displayed. The student's understanding of the quantum mechanics of the prohlem is greatly enhanced by examining solutions as they vary the energy. It is a simple exercise to determine how changing well depth or width will affect energy levels and the depth the particle can penetrate the wall. Student familiarity with the analog computer provides a rich source for independent study projects, especially in the area of kinetics (3, 5). Literature Cited (11 (21 (3) (4) (51 16)
Corrin, M.L., J. CHEM. EDUC..43,579(1966). Tabbutt. F.D.. J.CHEM.EDUC.. 44,64(1967l;44,486~19671. Griswald. R.,and Haugh. J. F., J. CHEM. EOUC. 45.576 (1968). Ohline, R.W.. J. CHEM. EDUC. 47.651 (1910). Hamari. E., J.CHEM.EDUC. 49,39(1972). Messiah. A,. "Quantum Mrhanien." John Wilw & Sam. Inc. New York, 1361. Vol. I. p 90.
