The Two-Dimensional Particle in a Box G. L. meneman Eastern Washington University. Cheney WA 99004 The one-dimensional particle in a box has been used for years as an introductory example for quantum mechanics and riehtlv so. The differential eauation involved can be solveduby inspection, the requirements for the wave functions lead naturallv to auantum numbers, the vrobabilitv of the particle position is easy to plot and understand, t h e effect of very large and small masses and restricting the movement of the particle inside various size boxes on the energy level spacing can be shown, and even the uncertainty principle can be easily approximated. All in all it is a very complete and simple example for introducing quantum mechanics. The next usual example is the three-dimensional particle in a box t o demonstrate separation of variables and how the three one-dimensional solutions can be put together to form a three-dimensional solution. This example also
allows showing how three quantum numbers arise and introduces the idea of degeneracy of states a t the same energy level. What is lost in this examole is the abilitv to d o t the wave functions and probabilitie's in a straightforward, easyto-understand manner. Using the two-dimensional particle in a box has the same advantages as the three-dimensional box and in addition has several other advantages. Plots of the wave functions and probabilities are easy to make and understand. Linear combinations of the solutions can form new solutions. Plots of these new functions and their probabilities can be made. This process is similar to forming the sine and cosine forms of atomic orbitals that we usually depict from the exponential, doughnut-shaped orbitals that come directly from solving the equation for the hydrogen atom. In addition, forming
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Figure 1. Plots of Me wave function f w different 61ates of lhs particle in a square twdimensionai box. The quantum number, n, is listed across the top, and n, is dDwn Me lefl side.
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and plotting combined states, similar to hybrid atomic orhitats, can he easily accomplished. These points are discussed and illustrated in detail below. The solution t o the wave equation for the two-dimensional particle in a box is
Y = (2Io)'" sin (n,nlo)(2/b)"2 sin (n,uy/b) where a is the length of one side of the box and b is the other and n, and ny are integers with the allowed range from 1to m. If the box is square then a = b and
The functions for several combinations of the quantum numbers n, and n, for a square box are plotted from 0 to a in both the r and y directions in Figure 1. The coordinate system used is the same for all figures and is shown in Figure 5 with the origin in the far corner with x going down to the left and y going down t o the right. The state with n, = 1and ny = 1 (denoted as (1, 1) hereafter) reaches a maximum in the center of the box where x and y equal a12 and !T = (21a) sin ( ~ 1 2sin ) ( ~ 1 2 )The . ( 2 , l ) and (1,2) states each have one peak and one valley as one of the sine terms goes through a full cycle from 0 to 27r. These two states have the same energy for a square box and are therefore degenerate.
The probability of where the particle is located is just the square of Y since the function is real. Plots for all the states from (1, 1) through (4,4) are plotted in Figure 2. Of course these plots show only peaks and no valleys, reflecting the requirement that prohahility has to be positive. The (1, 1) state has the highest prohahility in the center of the box ( r = a/2, y = a/2), and thus this is where the particle is most likely to he found. The prohahility peaks a t r = a/4, y = a12 and at x = 3al4, y = a12 for the (2.1) state. There are 16peaks in the orohability by the time the (4,4) state is reached. Notice how these plot; approach zero at the edges of the box so as to be continuous with the zero prohahility outside the box. The (1, I ) state looks similar to the cross-sectional plot of an s hydrogen orbital with circular symmetry. The ( 2 , l ) and (1,2) states look similar to the cross section through p orbitalscutting through the two 1ohes.The (2,2) state is reminiscent of cross sections of some of the d orhitals, cutting through the four lobes. Some of these'similarities will he made use of in the following sections. The other states have less obvious similarities with hydrogen orbitals. The (2, 1) and (1, 2) states are degenerate and so linear comhinations of these states should also be solutions to the waveequation. Figure 3 shows plou of the probability of the states ohtained by comhining the two states together in two different ways.
where h is Planck's constant.
Figure 2. Plots of probability of Me location of the particle in a square twc-dimensional box. mese are the squares of the plots in Figure 1.)
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Figure 3. Linear combination states wlth [(Z.1)- (1. 2)12on the lefiand [(Z. 1) + (1. 2)I2onme right.
Figure 5. Hybrid states similar to sp2 hybrid atomic orbitals. (W2on lhe left, Y"Zon the right, and q2on the bottom.)
Figure 4. Hybrid states similar to sp hybrM atomic wbitais. IF2on the lefiand W2 on me right.)
This is analoeous to combinine the douehnut-sha~ed1 = 1. rn = i l orbit& for the hydrog& atom toobtain thk usualp; and p, orbitals. However, in the case of the two-dimensional particle in the box the "orbitals" keep the same shape but now are oriented 90° from the original - states (see (2. . . 1) . and (1,2) in Figures 1and 2). If the (1.1)state and (2.1) state are combined inamanner similar to the formation of sp hybrid orbitals from an s and a p orbital, the probability plots of the resulting functions k for these plots are Hppear as in ~ i g u r 4. e ~ h functions
These hybrid states have maximum probability in directions 180° from each other and a lower probability in the opposite
direction just as cross sections of sp hybrids would show. Figure 5 shows plots for combining the (1, I), (2,1), and (1, 2) states analogous to forming sp2hybrids. Again the correct shapes result with three probability peaks coming 120° apart.
The (1, I), (2, I), (1,2), and (2,2) states could he combined into a hybrid state similar to the square planar dspz hybrids also. Thus these various plots show that the two-dimensional particle in a hox can be a valuable extension of the onedimensional system in introducing quantum mechanics and helping students understand some of the principles behind the various atomic and hvbrid orbitals.
Iterations II:Computing in the Journal of Chemical Education Iterations II, e collection of 46 articles that appeared in the Computer Series between 1981 and 1986, has been carefully selected by the editors, Russell Batt and John W. Moore, to bring up to date the coUedion of computer applications that appeared in its predecessor volume Iterations. In addition to covering all aspects of instructional computing from introductory to graduate level, Iterations 11 provides an annotated bibliography of all computer-related articles that have appeared in the Journal from 1981 to 1986. 1981 paperback, 160 pp; US. $16.50; foreign $17.50 (postpaid).Send prepaid orders to Subscriptionsand Book Order Department, Journal of Chemical Education, 20th end Northampton Streets, Easton, PA 18042.
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