H. D. Schreiber and J. N. Spencer1 Lebanon Valley College Annville, Pennsylvania 17003
Contour Functions for the Particle in a 3-D Box
The wave mechanical treatment of the particle constrained to move within a three-dimensional box has long served as an introduction to quantum calculations. The use of the three-dimensional box problem to illustrate the concept of degeneracy and separability of the wave equation is familiar to all beginning students of quantum mechanics. Not so familiar, however, is the construction of probability contour functions for the particle in the three dimensional box.% The construction of these probability curves allows the student to better grasp the probability concept and provides an introduction to the use of contour functions for atomic and molecular systems. Consider a cubic box the sides of which have length a in the x, y, and z directions, the origin of the Cartesian coordinate system being at one corner of the box. The potential energy is zero within the box but a t the boundaries of the box and in all space without the box the potential energy is infinite. The wave equation for the system may then be written
where J, is a function of the coordinates, x , y, z. By assuming a product function +(X,Y,Z)
=
X(z)Y(.v)Z(z)
(2)
where X ( x ) is a function of x only, Y ( y ) a function of y only, and Z ( z ) a function of z only, the normalized solution to eqn. (1) may easily be shown to be +(x,y,z)
&?T
= ( 2 / ~ ) "sin ~ a z
n ?T sin L
sin a z
(3)
The square of $(x,y,z) gives the probability per unit volume that the particle exists at some point x,y,z. The quantum numbers n,, n,, and n, may assume any integral value 1, 2, 3,.. . . The energy levels for the particle in the cubical 3-D box may be found from E = (n-2
+ nVZ+ nS2)h2/Xma~
The number of planar nodes in the wave function is given by n, n, n, - 3. The contour lines, that is, lines following a path of constant probability, may be found by considering a plane cutting the z axis a t a/2n,and parallel to the xy plane. The probability density is given by the square of the wave function
+ +
To whom correspondence should be directed. BOCKKOFP,FRANKJ., "Elements of Quantum Theory," Addison-Weslev Puhlishine Comuxnv. Inc.. Readine. iM&?s.. 1969, pp. 140-5.
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The plotting of the contour lines will he somewhat simplified by the use of the ratio $2/$2,.,, where $2,,, the maximum value of $%,is 8/aa. The probability along each contour line will then represent $2/$2,.,. Since z has been taken to be a/2n., the sine term in z will be unity whenever n, = 1. Eqnation (4) for n, = 1 may then be written
For given values of $2/$2,., this equation was solved by successive approximations for x and ?J using an RCA Spectra 70/46 computer. The contour functions for 1~~= n, = n, = 1 are given in Figure 1. The similarity to the contour functions for a 1s atomic state or the c,,, molecular orbital is readily seen.3 Moving from the center of the box toward the walls, the probability density is seen to start a t its maximum value and decrease as the edge of the box is approached becoming zero a t the walls of the box, similar to a plot of $2 versus distance from the nucleus for a 1s atomic state. Analogous to the 1s atomic state there are no nodes in this wave function. Figures 2 and 3 show the contour functions for the degenerate wave functions for n, = n, = 1, n. = 2 and n, = n, = 1,n, = 2. These contours resemble those of a 2p atomic orbital4 or a sz,, molecular orbitaL5 The third member of the degenerate set for n, = n, = 1, n. = 2 differs from Figures 2 and 3 only in the relative position of the nodal plane similar to 2p atomic orbitals.
The number of nodes in t,hese functions corresponds to the total number of nodes for the second principle quantum level of atomic systems. However for atomic systems the requirement that the radial function be properly bounded leads to values of the angular momentum quantum number that range from zero t o n - 1 where n is the principle quantum number. Thus the second principle quantum state for atomic systems is split into the 2s and 2p sub-levels. No analogous situation exists for the particle in the 3-D box and there is only one quantum level with wave functions containing a single node. Figure 4 gives the contours for the wave function of the third energy level n, = n, = 2, n, = 1, for the particle in the 3-D box. The resemblance of these contour functions to those of the s*,,,molecular orbitals and to those of the 3d wave function is apparent. The contours for the next highest energy level for which n, = n, = 1, n, = 3 are given in Figure 5. These contours may be related to no, molecular o r h i t a l ~ . ~ Both of these wave functions are triply degenerate and contain two nodal planes. I n contrast to the wave functions for a particle in a one-dimensional box in which the number of nodes in the wave function in%OULSON,C. A,, "Valence," (2nd ed.), Oxford University Press, London, 1952, p. 83. ' COHEN,I R ~ I NJ., CHEM.EDUC.,38.20 (1961). 6 Seep. 92 of reference in footnote 3. WAHL,ARNOLD C., Science, 151,961 (1966).
x-0o
a
Figure 1.
Conmur functions in the xy plone a t r = 0 1 2 for .n = nu = n, = I . he dot represene the position of maximum probability. The conmvrr given represent J.2/#%lr = 0.9, 0.7, 0.5, 0.3, 0.1 moving fmm the position of moximum probability toward the edge of the boi.
Figure 2. Conmur function$ in the xy pione a t r = 012 for n, = n, = 1, n. = 2. The contours given are the some or those of Figure 1.
Figure 3. Conmur functions in the xy plane mt r = a12 for n. = n. = 1, n. = 2. The contours given are the same or those of Figure 1.
Figure 4. Contour functions in the xy plone ot r = 0 1 2 for n, = n. = 2, a = 1. The contours given ore tho same or those for Figure 1.
186 / Journal of Chemical Education
Figure 5. Contour functions in the x y plane o t z = 4 2 for n. = 3, n. = n, = 1. Tho contours ore the same as those of Figure 1.
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Figure 6. Contour functions in the xy plane a t r = 0 1 2 for .n = 3, n, = 2, n, = 1. The contours given w e the some as those of Figure 1.
Figure 7. Contour functions in tho xy plane ot r ;= 0 1 2 for n. = n, = 1 , n, = 2. The y dimension has been altered to 3/2a while h e x and r dimension* remain a t length 0. The contours given ore the some as those of Figure I.
creases for each successive energy level, the wave functions corresponding to t,he energy levels of Figures 4 and 5 have the same number of nodes. In further contrast, to the one-dimensional box problem, the wave function with quantum numbers 3,2,2 which has four nodes is lower in energy than the 4,1,1 state which has three nodes. The number of nodes of Figures 4 and 5 is the same as the total number of nodes of the third principle state of atomic systems, although this quantum state for atomic system contains three sub-levels. There are two quantum levels with wave functions having two nodes for the particle in the 3-D box. This analogy is general in that in all cases except the lowest energy state the number of quantum levels having wave functions containing the same number of nodes for the particle in the 3-D box is one less than the number of sub-levels in the corresponding atomic quantum state. Figure 6 shows the introduction of a third nodal plane in the contour function for the n, = 3 , n, = 2, n, = 1 state. Corresponding to the previous wave functions there are two other quantum states, those with quantum numbers 2,2,2, and 4,1,1 having the same number of nodes. If the linear dimensions of a 3-D box are equal or if their ratio is a ratio of integers degeneracy will exist. If no integral relationship exists between the dimensions of the box, the energy levels will all be nondegenerate. The contour functions for the same quantum numbers as those of Figures 2 and 3 are given in Figures 7 and 8 for a 3-D box jn which they dimension has been altered to 3/2a while maintaining the x and z dimensions a t a. The degeneracy has now been removed with the wave functions corresponding to Figure 7 being associated with the higher energy level.
Figure 8. Contour functions in the xy plone a t z = 012 for n, = n, = 1, n, = 2. The y dimen.ionr has been oitcred to 3/20 while the x and r dimensions remain at length a. The contours given are the same or those of Figure 1.
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