IEEE TRANSACTIONS ON EDUCATION, VOL. E-23, NO. 1, FEBRUARY 1980
breadboards, simulates, and lays out his circuits but does notactually process them; he finally tests the finished chip. Thus,our students gain a realistic understanding of most of whatgoes on in the industrial IC design process.Also important is the potential this custom IC design tech-
nique provides for students to do more sophisticated specialindividual projects, as well as Master's thesis projects.Students who participate in both this bipolar IC design
course and the bipolar IC fabrication course get a compre-hensive understanding of the basics of the IC industry. Gradu-ates of our program are actively sought by regional industryand are highly regarded by their employers, in part due to thisnearly unique educational experience afforded them.
ACKNOWLEDGMENTAppreciation is expressed to Mr. Larry Wheaton for his im-
portant contributions to this project. Thanks are due to Dr.Alan Grebene and Mr. George Krautner of Exar IntegratedSystems, Sunnyvale, CA, for their technical assistance and helpin obtaining masterslice wafers.
REFERENCES[1] L. W. Nagel and D. O. Pederson, "Simulation program with inte-
grated circuit emphasis," Int. Symp. Circuit Theory Proceedings,Apr. 1973.
[2] S. R. Combs and J. D. Meindl, "KITCHIP-A generalized integratedcircuit for biomedical system design," Biotelemetry, voL 3, pp.91-94, 1976.
131 L. Forbes, "Information on integrated circuits instruction and fa-cilities at UCD," Dep. Elec. Eng., Univ. California, Davis, Apr.1977.
[4] K. Miyasako, L. B. Wheaton, D. J. Edeil, and K. W. Current, "Lay-out and processing manual for masterslice IC's," Integrated Cir-cuits Laboratory Manual, Dep. Elec. Comput. Eng., Univ. Cali-fornia, Davis, Sept. 1979. (Available upon request from author.)
K. Wayne Current (M'74) received the B.S.E.E.,the M.E., and the Ph.D. degrees in electricalengineering from the University of Florida,Gainesville, FL, in 1971, 1972, and 1974, re-spectively.In 1975 he joined the Microelectronics Cen-
ter of the TRW Systems Group, RedondoBeach, CA, as a Member of the Technical Staffdoing large-scale-integrated bipolar circuit de-sign, among other things. Since 1976, he hasbeen an Assistant Professor with the Depart-
ment of Electrical Engineering, University of Calffornia, Davis, CA.His interests include integrated circuits, device models, and computeraids to design.Dr. Current is a member of Tau Beta Pi, Eta Kappa Nu, Phi Eta
Sigma, and Sigma Xi, and is a Registered Professional Electrical Engi-neer in California.
Solving the Schrodinger Equation with aHand-Held Calculator
GUY FAUCHER
Abstrct-A rather simple method of numerical integration, the Eulermethod, is applied to the one-dimensional Schrodinger equation for apaticde bound to a rmite square well, a case which is exactly sovableby analytical methods. Insight is gained into the dominant role playedby the boundary conditions in the sdution of eigenvalue equations.Comparison is made with the exact soution of the algebraic methods,which leads to the discussion of improving the method, or switchingto more sophisticated procedures. Though the results presented camefrom calculations performed on a nonprogrammmable calculator, pro-grammable ones could be used. The same treatmnent can be appUiedto more complex potentials, for example two- and three-dimensionalones. The beauty of the method is that the solution seems to comefrom the calculating ringers rather than from an abstract equation.Emphasis is on the physics rather than on the calculator.
Manuscript received July 24, 1979; revised November 12, 1979.The author is with the Department of Engineering Physics, Ecole
Polytechnique, Universit6 de Montreal, Montreal, P.Q., Canada.
INTRODUCTIONTHE purpose of this article is obviously not to present a
method of numerical integration on a fast computer, butrather to show that the use of a pocket calculator can improvethe understanding of a mathematical solution.The engineering physics degree is offered on a four-year
basis at tcole Polytechnique. Students first encounter quan-tum mechanics in their fourth semester. We have adoptedWhite [1] as a textbook because it is especially suited for engi-neering students.' We have also prepared a solutions manual[3]. In fact, the teaching of quantum mechanics takes twosemesters. Prior to registering for the course, students had a
I Lindsay [21 could be classified in the same category, but at a lowerlevel.
IEEE TRANSACTIONS ON EDUCATION, VOL. E-23, NO. 1, FEBRUARY 1980
Vo vo
E2
-o +
Fig. 1. Potential well of width 2a and height VO. Only the fundamen-tal and first excited energy levels are shown.
lecture on programming and an elementary one on numericalanalysis. The finite square well is one of the few exactlysolvable problems in quantum mechanics, but we assign itsnumerical solution to students as homework.
ANALYTICAL SOLUTIONA particle ofmassm is bound inside a potential well ofheight
V0 and width 2a (see Fig. 1). The total energyE (kinetic +potential) of the particle is lower than the height of the po-tential barrier. The number of energy levels is finite anddepends on the value of the parameters V0 and a. To obtainE, we must solve Schrodinger's one-dimensional equation forstationary states (analogous to electrical resonances)
d2+r2 (E- V(x)) ,(1)
where 4 is the particle wave function, A = h/2vr is Planck's con-stant, and V(x) is the potential (O inside, VO, outside).The eigenfunction is periodic2 inside the well
4=A cos kx +B sin kx (2)
where
k= 12(m , (3)
while it is nonperiodic2 outside the well
4 = Cex +Deax
where
=2m(Vo - E))1/2ai = k 'i2
(4)
(5)
By applying the conditions of continuity for 4 and d/ldx atx = +a and by normalizing 4, we obtain five equations withfive unknowns: A, B, C, D, and E. We obtain two kinds ofsolutions. For symmetric states, like 4'I for the fundamentallevel (see Fig. 2), the energy is given by the transcendentalequation
(2mE)l12 =(VO E)uI2 (6)
For antisymmetric states, like 42 for the first excited level,
2Equations (2) and (4) bear some resemblance to expressions fortransmission lines: voltage or current distribution along lossless onesand purely resistive ones. However, the boundary conditions are dif-ferent. Both current and voltage must remain continuous at the junc-tion of the lines, but their slopes are unrestricted; in quantum mechan-ics, both the eigenfunction and its slope must be continuous.
the energy is given by another transcendental equation
(2mE)l1/2 (V E)/12 (7)
Most textbooks consider graphical solutions to the tran-scendental equations. With the value ofE obtained, the con-stants are calculated and the eigenfunction is determined in-side and outside the well. Note that the probability of findingthe particle outside the well is not zero contrary to classicalmechanics expectations.
NUMERICAL SOLUTIONThe students were requested to use Euler's method [41, not
only because it is simpler, but especially because its step-by-step integration illuminates the striking role of the boundaryconditions.We rewrite (1) in terms of finite differences following
Sherwin [51A (slope) -2m [E V(x)] ]
Ax -i~2 [-Vx4where the slope is d4/dx. This equation means that, in thegraph of 4 versus x, when one moves from x to x + Ax, the
) slope changes by the amount
-2m [E - V(x)] v Ax.
(8)
(9)
However, in order to proceed we must have starting valuesfor 4 and d/cdx. Calling these values 4 = 4' and d4/dx =(d/cdx), at x = 0 chosen arbitrarily, we define our integra-tion sequence
4 = 4o
l = Oo + )dx AX
at x0 =0
at x1 =Ax
initial slope-So
2=1+So - 2 [E- V(x)]_41'AxiAx
new slope-s I
at X2 = 2Ax
43 = 42 + {S1 -2 [E - V(x)J 2AX}AX
new slope-s2at x3 =3Ax. (10)
Note that in (8), (9), and (10) V(x) is quite general and couldbe, for example, the harmonic oscillator potential (V= 2 kx&).Fig. 3 shows the process of integration starting from arbitraryconditions at x = 0.The curvature of 4' is given by (8). IfE - V is positive, 4 is
curving towards the x axis (see Fig. 3); ifE - V is negative, 4'is curving away from the axis. E is therefore playing a domi-nant role on the curvature of 4 and different numerical values
34
FAUCHER: SCHRODINGER EQUATION AND HAND-HELD CALCULATOR
4,
-a
4,Vo
,00
so2. 4'a
60
40
20*X+a
Fig. 2. The eigenfunctions for fundamental level (~p1) and first excitedlevel (IP 2).
--SI/,'o-I--
Xo! 1 !3V'o *, Vr s3o
Ax 3AX0 2AX
Fig. 3. The numerical integration of Schrodinger's equation for an un-specified potential and arbitrary initial conditions at x = 0.
will generate different shapes for the curve. When E - V = 0,the curve is a straight line.
NUMERICAL EXAMPLE
Returning to the finite square well,3 we have to specify V,.Let us suppose that the well is characterized by the following
2ma' V (11*2
Schrodinger's equation (1) can now be written
d2= R-x2= 18(1-R)4 inside
= -18R 4 outside
withR =E/Vo.For selecting the starting values, we take advantage of the
symmetry of the problem. As Fig. 2 reveals, 4 for the funda-mental level is symmetric. Therefore, we may choose to in-tegrate only on the positive half-axis. Also, for the funda-mental level, the slope of 4 is 0 at the origin. We chooseAx = 0.1a (10 integration steps inside half the well).Instead of selecting any value forR (the ratio of energy to
well depth), we make the following guess. For an infinitelydeep well (VO = oo), the energy of the fundamental level is[1]
E1 = 8ma2a (13
-20
0.4 0.8 1.2
*x
.6Xa
R=0.110
Fig. 4. The numerical integration for the ground state of the finitesquare well of length 2a and parameter 2ma2 V0/*2. Only the +xaxis is shown since the wave function is symmetric.
which gives R = 0.137. Since E must be lower for the finitewell,4 we start with R = 0.090.Working with a Texas Instruments SR-50 calculator, the
author has solved the problem numerically. The value of4 = 100 at the origin is purely arbitrary and has no incidenceon the results shown in Fig. 4. The energy of the fundamen-tal level is given by R - 0.105. In fact, an infinite number ofsmall integration steps is required for an exact numerical solu-tion (4 at large distances is exactly zero). In practice, we findthe value ofE which leads to sufficiently weak values of 4 atlarge distances; we then show that, by varyingE slightly (E ±AE), A diverges quite rapidly. It is apparent in Fig. 4 thatout first guess,R = 0.090, meant a divergence to plus infinityfor 4 at small values of a.
) For the first excited level of the same well, 4 is antisym-metric and we need to work on only half the well. However,we have to make an auxiliary guess, the slope of 4 at x = 0.But we easily obtain results similar to those of Fig. 4.
TRANSCENDENTAL EQUATIONS') The exact solution can be found by solving the transcenden-
tal equations (6) and (7). For the fundamental level of ourexample, (6) becomes
tan/18R= --1 (14)
A few trials on the SR-50 lead finally to R - 0.0892.Our Euler calculations had, therefore, an R value which is
typically too high (for this example). Increasing the numberof steps lowered the value ofR.
DISCUSSIONA problem of the type described here was given as home
3) work to our students in Quantum Mechanics I for three con-
3We stress that the results of the preceding section are quite generaland apply to any one-dimensional potential.
4The particle has more space to go in a finite well since it can pene-trate in the classically forbidden region (lx > a). Taking a largervalue of a in (13) will lower E1.
35
IEEE TRANSACTIONS ON EDUCATION, VOL. E-23, NO. 1, FEBRUARY 1980
V.
Vo
i /2kX2
x
Fig. 5. Truncated harmonic potential.
V
oscillates sharply. We may even discuss other methods ofnumerical integration (Runge-Kutta, predictor-corrector,etc.) and the question of numerical stability. However,we are limited by the fact that, in numerical analysis coursesand literature, emphasis is on large fast computers.6 Theauthor [8] is working on other simple methods of numericalintegration of the Schrodinger equation.Due to the high similarity between the following equations:
Laplace's, HeLmholtz's, Schrodinger's, and the diffusion equa-tion,7 our work can be extended with modifications to otherparts of physics and engineering. Note that our method ap-plies to two- and three-dimensional potentials as well.
Finally, saepe repetita placent, we were more interested inthe physics than in the calculator, but appreciated both. Thepurpose ofcomputing is insight, not numbers.
[11[2]
[3]
-_ x
-a 0 b +aFig. 6. Finite square well with hole.
secutive years. The first year, nonprogrammable and pro-grammable calculators were used in the same proportion bystudents. With time, the latter have become more popular.The students now favor the programmable calculators (HP-25,HP-29C, TI-58, and TI-59, among others) in a proportion of80 percent, the computer 10 percent, and nonprogrammablecalculators 10 percent.We would like to make a few comments on our experience.After performing the numerical integration ofSchrodinger's
equation, the students understand better what a solution is,the role of the boundary conditions, and the symmetry of theproblem. The solution seems to come from their fingers andnot from an abstract equation. Moreover, they can use themethod for potentials, the analytical solution of which doesnot exist (e.g., truncated harmonic potential; see Fig. 5) orwhich is less readily available (e.g., finite well with hole;5 seeFig. 6).We may also ask the students to discuss the Euler method
which is a good approximation insofar as the slope of thewave function does not move rapidly. No doubt it wouldbe harder to apply the Euler method to high energy levelswhere the wave function, having many nodes inside the well,
sFor this case, perturbation methods are also useful.
[41
[51
[61[7]
[8]
REFERENCESR. L. White, Basic Quantum Mechanics. New York: McGraw-Hil, 1966.P. A. Lindsay, Introduction to Quantum Mechanics for Elec-trical Engineers. London, England: McGraw-Hil, 1967.G. Faucher, "Probl6mes de mecanique quantique I," Ph.D. disser-tation, Ecole Polytechnique, Montr6al, Canada, 1977; "Problemesde m6canique quantique II," Ph.D. dissertation, Ecole Polytech-nique, Montr6al, Canada, 1979.A. A. Bennett, W. E. Milne, and H. Bateman, The Numerical In-tegration ofthe DifferentialEquation. New York: Dover, 1956.C. W. Sherwin, Quantum Mechanics. New York: Holt, Rinehart,and Winston, 1959.J. M. Smith, Scientific Analysis on the Pocket Calculator. NewYork: Wiley, 1975.P. Henrici, Computational Analysis with the HP-25 Pocket Cal-culator. New York: Wiley, 1977.G. Faucher, "Simple numerical methods of integration ofSchrodinger equation," to be published.
Guy Faucher presented his doctoral disserta-tion on nuclear physics at Orsay, France.He is a Professor of quantum mechanics and
general physics with the Department of Engi-neering Physics, Ecole Polytechnique, Montreal,Canada. He worked for three years in the areaof elementary particle physics. He also taughtone year in Ethiopia and was an engineer atCanadair, Montr6al. His interests are in solarenergy, especially in the use of lumiducts forthe ilumination of the interior of buildings.
6Two notable exceptions in the literature are [6] and [7], but theydo not include new methods which tend to keep the number of func-tion evaluations as small as possible.7Recent work has shown that, under specific conditions, the Navier-
Stokes equation can be reduced to a Schr6dinger equation.
