Spherical Harmonics in a Cartesian Frame Todd McDermott and Giles Henderson Eastern Illinois University, Charleston, IL 61920 Undergraduate physical chemistry studenta probably first encounter mathematical representations of the de Broglie matter waves in the context of particles undergoing linear motion, e.g., the free particle, the localized free particle (particle in a box), the harmonic oscillator, etc. In these cases, plots of the corresponding eigenfunctions in the linear displacement space have self-evident wavelike character with nodal spacings and curvatures that relate to the velocity, momentum, and energy of the particle in the expected manner. However, as we consider particles undergoing angular motion, the student may find the descriptions of the matter waves moreabstract and may not recognize the waveproperties of the eigenfunctions plotted in the usual polar coordinates. We find it helpful and instructive to display and compare angular wave functions in both Cartesian and polar representations. Computer graphics are employed in this article to illustrate examples of the spherical harmonics in both frames of reference. Wave Functions for Angular Motion
The eigenfunctions for both the rigid rotor Hamiltonian and the angular part of the hydrogen atom's Hamiltonian are the well-known spherical harmonics:
These functions are a complete orthonormal set in two variables, 0 5 0 5 ?i and 0 5 @ 5 Zr,defined in Figure 1. The individual members of this set are specified by two indices (quantumnumbers); 1 = 0,1,.. . ,called the rank, and m = 0, f 1 , . . . , f1, called the order of the spherical harmonics. Most authors (1-7) distinguish the rigid rotor application from the hydrogen atom case by using J f o r the rank index of the former and 1 for the rank index of the latter. It is evident from eq 1that the spherical harmonics are products of three components. The first term in brackets is called the normalization coefficient and merely insures that the total probability of all orientations is unity:
wheresin R dodo is a differential element of solid angle. The second term in to 1 identifies the 0 component of the spherical harmonic as member of the set ofassociated ~ e g e n d r e polynomials. These functions make up a complete set of orthogonal polynomials in cos 0 and may be generated by well-known methods discussed in many mathematical and physical chemistry texts (8).The last term in eq 1is a simple harmonic function in the @ space. This complex function may be indexed with either a positive or negative rn value and can thereby describe either clockwise or counterclockwise rotations about the z axis. Representative examples of some normalized lower rank spherical harmonics are given in the table. After encountering a formal introduction to these functions as eieenfunctions to model Hamiltonians and angular momentum operators, our students are usually provided plots of the spherical harmonics (or their complex square) in polar coordinates. This frame of reference has the advantage that the probability of a given orientation may be deduced
Flgure 1. The instantaneous orientation of a linear molecule (oran electronand nucleus) may be specified by polar angles 0 and 4 where the molecular (atomic) fixed origin is located at the center ot mass.
directly from the distance from the origin to the ~,,Yi,, surface at that particular (O,@)value. If avector is construct, its ed from the origin to some point on the ~ , , Y I , surface, direction specifies an instantaneous orientation, and its magnitude gives us the relative probability of this orientation. However, the student may not find the curvature, nodal spacing, wavelength, and other wave properties of these functions self-evident in this format. We will now consider an alternative graphical representation in which the spherical harmonics are plotted as ordinates in a Cartesian space with orthogonal 0 and @ abscissas. Computer Graphlcs Polar plots of Y&YI,, (Fig. 2) were produced by obtaining
Some Normallzed Spherical Harmonlcs Y4, (8.6)
a
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Discussion Plots of the lowest ranked complex s uared spherical harmonic a Yo,o are given in the upper left corner of Figure 2. The upper polar plot reveals that for the nonrotating, 1 = 0 eigenstate a11 (9, $) orientations (Fig. 1) are equally probable; i.e., all orientation vectors are of equal magnitude and correspond to the radius of the sphere. This same conclusion is evident in the lower Cartesian representation since all points on the 17,,oYo,osurface are of equalelevatlon. This lack of information on the rotor's orientation is consistent with the uncertainty principle, i.e., we cannot have precise knowledge of the conjugate variables, momentum and position, simultaneously. In this example we know the angular momentum precisely:
3
--
t 2 the total angular
1 here is Inomentum
squared operator = hI2s. Since 1 = 0, both t;he angular velocity and angu1a r momentum are precisely 1:ero. Therefore, the spatial 0)correspond o eieenstates with ereater angular momentum. As we comFigure 2. Polar(upper)and Cartesian (1ower)representatlonsof some lowerranked complex squared spherical harmonics Pare these functions in frames of reference, the correTAB. 4). fi,M.$1. lations of nodal spacings and the roots to eq 3 m selected x, y and x , z planes using an the curvature of the wave functions to the angular momenalgorithm developed by R. P. Brent (9). tum are more apparent in the Cartesian representations; i.e., as 1 increases along with the rotors angular momentum, we note that the de Broglie and probability waves exhibit more Coordinates for these solutions were then plotted as a proclosely spaced nodes and the waves are folded more tightly in the 9 space, resulting in greater curvature. These observajection with hidden line features from a specified viewing tions may be compared with the corresponding effects of anele usine a FORTRAN routine develoned bv W. L. Jorge&en (107. increasing the velocity and momentum of a particle underCartesian plots of Z-Y,, going linear motion on its de Broglie matter wave. ",,,. .,... are illustrated in the lower parts of ~ i g u r e2. In these plots the value of q,,Yl,, are I t is instructive to compare a fixed, $,9 sweep as well as a depicted as an elevation above the O.$~lane.A hiehlv~robafixed 9, 4 sweep, for a given 1,m case in both frames of .. hle 6, m orientation corresponds to a high elevation un the reference. Consider any of the q,,Yl,, cases illustrated in Figure 2. If we fix @ a t some arbitrary value and sweep the Cartesian representation of the probability surface. 'l'hene plots were produced by FORTRAN codes previously develorientational probability vector from the 0 = 0 north pole of oped t o plot proiections of three-dimensional surfaces from the upper plot to the 8 = s south pole, the magnitude of this aspecified view& angle (11-13). Thew routines have been vector is modulated in precisely the same manner as a constant $, 0 cross section of the lower Cartesian representation. adapted to run on a Nicolet 1280prncessor driving a Henlett Packard 7470.4 x,y plotter. Alternatively, if we fix the latitude a t some arbitrary 9 in a L2nd h
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particular polar plot and then sweep the orientational probability vector about the z axis, we note that the magnitude of this vector is constant for all orientations in the q3 cone. This outcome correlates with the constant elevation of the Cartesian surface for a foreground to background q3 sweep a t a fixed 8. Once the connection between these frames and the methods of mapping the orientational probability has been recognized, our student should be saying "I see" rather than the wondering why the spherical harmonics are called waue functions. Literature Cited
Afkinr, P. W. Physied Chmiafry, 2nd ed.: W. H. Freeman: San Franeixo. 1982: pp 420.441. Levine. I. N. Physical Chemistry, %dod.:McCraw-Hill: NewYork, 1983:pp604,683. Bromkrg, J. P. Physical Chemistry, 2nd ed.: Allyn snd Bamn: Bmton, 1984; pp 501, 520. 3rd New York, ~ d A. W. A~Textbook ~ olPhysicol ~ Chemistry, ~ ~ ed.; Aesdemie: . 1986: pp 692,706. Alherty, H. A. Phy$ical Chemistry, 7th ed.; Wiley: New York. 1987: pp 371,497. No&, J. H.Physico1 Ch~misfry, 2nd ed.: Scott, Foreman: Boston, 1989; ~ ~ 6 8 5 . 7 6 7 . See. for example. Psuling, L.: Wilaan. E. B. lntroduclion to Quonlvm Mechanics uilh Applications to Chemistry; McGraw-Hill: New York. 1935: pp 124-136. Brent. R.P. AIgoriihms/or Minimization uilhoul Dlriuativea: prentice-Hdl: EnglewocdCliffs, NJ. 1913; Chapters3.4. Jorgensen, W. .I Psi2177: QCPE No. 340: Indiana University: BloomingMn, IN, 1917. Henderson. G. L. J. Chem. Educ 1977.54 57. Henderroo, G.L. J. Cham.Edur. 1979.56.631. Henderson, 0. L. Am. J.Phys. 1980,48.604.
I. Moore. W. J. PhysicaiChrmislry,lhed.;Prenfieo-Hall: EnglewoodClifis. NJ, 1 9 1 2 : ~ 628.
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