(6) Margenau, H., and Murphy, G. M.."The Mathematicsof Chemistryand Physics: D. Van Nostrand.Princeton. NJ.1956.2nd Ed. ( I ) Daniel, d. W., and Msure. R. E., "Computation and Theory in Ordinary Differential Equations." W. H. heeman. San Frsnciscu. 1970. (8) Messiah, A,, "Quantum Mechanics: John Wiley, New York, 1961, vol. I, p. 346. : ~Or~w~Hlll, (9) Pauling, L., and Wibon. E.B., ~'Intttdddtiiit o Quantum M ~ ~ h h i i i M NewYork. 1935.p~.59-63. (10) Hildehiand, F. B., "introduction to Numericai Analysis: McGiaw~Hill,New York. 1956. Formulae6.12.18and6.12.19.SseslsoCuuley, J. W., Math. Cornput. 15,361 (1961).
Illustration of Quantizationand Perturbation Theory Using Microcomputers Christian Kubach Laboretoire des Collisions Atomiques et Moleculaires, Bit. 351, UniversitC de Paris Sud, 91405 Orsay, France Microcomputers allow students to become more familiar with the concepts of quantum mechanics and also to ro heyond prohlemshavin&alytic solutions. The purposesbf the present contribution are to exhibit the notion of quantization and to reinforce students' knowledge of the behavior of wavefunctions and the results of perturbation theory. Determination of vibrational energy levels of diatomic molecules, along with the associated wavefunctions, is of particular interest to chemists. This involves solution of a one-dimensional Schrodinger equation, which can he integrated either by linear variation or by numerical methods (see e.g., (1) and (2)). The Noumerov algorithm, which pertains to numerical methods (see e.g., ( 3 ) )has been selected here for its simplicity. Outline of the Noumerov Algorithm The one-dimensional Schrodinger equation may be reduced in suitable units to:
In the Noumerov algorithm the solution of eqn. (1)is calcnlated for equally-spaced values of x defined by r,+l-x,=x,-x,-l=...=h
(2)
Using a Taylor expansion off through order hS one obtains: where fk = f(xh ). Differentiating eqn. (3) twice gives fl, since: h2(f,+,+ fA-l) = 2h2< + h4%+ 0(h6)
(4)
Finally using eqn. (1) one obtains:
Equation (6) allows the determination of f,+l if the solution is known for the two previous points: x, and x , _ ~ . Application to the Potential Well For hound states of a potential well there are two classically forbidden regions defined by V(x) > E. The physical solutions of the Schrodinger equation vanish at points far into these regions. This property is used to begin the construction off a t a point sufficiently far into the first classically forbidden
' The solution of the Schrodinger equation is not normalized, and so may have any value compatible with the precision of the microcomputer used. f
212
Journal of Chemical Education
region that f may be chosen to have zero value. More preciselv.
bidden region. Selection of Parameters The harmonic potential (V(x) = x2),along with the known eigenvalues E, = 2(u + 1/2), u = 0 , l . . ., is used both to determine the integration parameters (xo,h . . . ) and to be a first exercise for the students. The following energy-dependent parameters have been selected to ensure sufficient precision for the construction of the lowest energy functions of the harmonic oscillator (v < 6) within a small computational time.
Eauation (7) shows that a leneth of 2 units of x is chosen as a starting point in the first c l a k a l l y forbidden region, (the quantity 2 a represents the length of the classicdy allowed region). Equation (8) gives the point X M where the construction of the fnnction f is achieved and is checked for convergence (the points xo a n d x are ~ symmetric with respect to the location of the potential minimum). Finally, eqn. (9) gives, for a fnnction having N nodes, 10 (N+ 1) integration points for the determination of the solution in the classical allowed region (since E, = 2 (u 112) for a wavefunction with u nodes). Other scaling parameters have also been determined in order to ensure the visualization of both the function and the potential on the screen of a Radio Shacka TRS-80 for a large range of input energy 0.1 < E < 15. The parameters of eqns. (7)-(9) can also he used for nonharmonic potentials, provided the ootentials chosen are nearlv harmonic.
+
" .
-"
E defining the integration parameters eqns. (7)-(9). The results aw~earinr .. - on the screen show. for each calculated value of x , the solution f(x) of eqn. (11, r h potential ~ V(x) and the location of the classicallv allowed reeion. In eeneral the mathematical solution f isdivergirig in the seconl classically forbidden region (x 5 XM). . .. excewt for articular values E,, (eigenvalues) where wavefunctions converge in this region. The eigenvalues may be bracketed using the relation between the signs of the divergence and of E - E, (see Fig. 1). Problems Given to the Students The harmonic wotential is of articular interest since it is a first approximation to the lowest vibrational levels of a diatomic molecule, and since its eieenvalues and behavior of eigenfunctions are usually knownby the students. With this potential the user observes a clear difference between physical (non-diverging) and only mathematical (diverging) solutions of eqn. (I), illustrating the quantization concept (see Fig. 1). One also visualizes the lowest energy wavefnnctions and verifies the law E, = 2(u 112) with a precision better than n 79" The addition of a perturbation term to the harmonic oscillator offers two benefits as an instructional exercise: i t allows the student to review perturbation theory, and it provides a closer approximation to the potential energy curves of diatomic molecules. A comparison between the results obtained for the two following potentials:
+
Figure 1, Illustration of the quantization concept for the energy level v = 1 of the harmonic oscillator. The solutions of the Schrodinger equation are shown for trial energies 3 (solid line). 2.9 (dashed line), and 3.1 (dotted line).
where or is chosen to ensure that the last term of the two potentials is a small correction (for example ol = 0.11, illustrates a main result of perturhation theory. The student finds that when the lowest eigenvalue of the unperturhed potential is used as a trial value for these perturbed potentials, a much lareer divereence occurs with eon. (11)than with e m . (101 (see u Fig. 2). This occurs because the first-order correction of the enerev. -" which is iust the exwectation value of the werturbation term over the unperturbed wavefunction, has, for reasons of parity, zero value for the potential given hy eqn. (10). When the lowest eigenvalue of the perturbed potential given by eqn. (11) has been determined by trial and error, the student may compare the necessary energy correction with a calculation of the first-order correction, using the known analytic wavefunction of the ground state of the harmonic oscillator. The illustration of first-order effects is provided by the particular form of the potential given in eqn. (11);however, the form of the potential given in eqn. (10) more closely approaches the physical situation found in diatomic molecules. The lowest (LI = 0 to 2) eigenvalws of this potential, found by the present method, verify that the corrections in energy are proportional to -(u 1/2)2. Some properties of radial wavefunctions of a repulsive electronic potential can he illustrated easily with a simple modification of the harmonic potential: introduction of V ( x ) = 0 for z > 0. The user verifies that there is no longer any
+
Figure 2. Solutions of the Schrodinger equation for potentials: V(x) = 9 (solid lines), V(x) = x2 - 0.1 9 (doned line). and V(x1 = rZ - 0.1 1x3[ (dashed line). In each case the trial energy is the eigenvaiue v = 0 of the harmonic oscillator.
quantization (i.e., diverging functions are no longer obtained). One also verifies that the wavefunctions behave like sine waves in the potential-free region. Conclusion
The present program has been made to cover a large range of . wrowerties and known results in the field of molecular physics. Obviously the program lends itself to further modifications of the potential and other applications, some of them being of greater difficulty. For example, the Morse potential mav he introduced in order to reuresent a diatomic eotential in a larger range of internuclear distances. T h e use of a potential with a harrier will permit the investigation of quasihound states and the study of resonances. Other possibilities include calculations of properties involving radial wavefunctions, e.g., Franck-Condon factors.
.
Acknowledgment
The author wishes to thank Drs. T. Baer and M. Graff for assistance during the preparation of the manuscript. Literature Cited
Audiovisual Program on DNA Available An audiovisual program on DNA and recombinant DNA technology for classroom use has been completed by the American Chemical Society. Entitled "DNA: Master Molecule of Life,"the program is designed to assist teachers in explaining the complex structure and functions of the deuxyribonucleicacid molecule and the powertul new gene splicing terhnology. It should be especially helpful for students of general science,biology and chemistry, at the junior high through early college levels. Distributed in kit form, "DNA: Master Molecule of Lif'e"eonsists of three, GO-frame each, filmstrips,tapednarration, 24-page teacher guide, activity sheets, suggested tests, suggestions for class discussions, class projects, resource list, and film s t r i ~ scrints.
Volume 60
Number 3
March 1983
213
