letters About Electrons Crossing Nodes To the Editor:
Nelson (1)claims to have shown that the mathematics is wrong in our explanation (2) of why an electron has no uroblem crossine a node. This letter is to uoint out that his Hrgument is flawed. Our opening thesis, acknowledged by Nelson in his eq 7, is obtained from basic probability theory: the cumulative distribution function is given by
where p(t) = v v i s the probability density function. From this, we derived the probability AP that the random variable t fall in the interval x< t < x + Ax: *A%
AP= l p ( t ) d t z
(2)
Assuming that p(t) is continuous and has continuous derivatives, we expanded p(t) in a Taylor Series about the point t = x, substituted into eq 2 above, and then integrated. The result may be written (see ref. (2) for details)
we note that a general expression for dPldx is obtained by differentiating eq 1 dPl& = p ( 4
Note that the differential dP = p(x) dx is the conventional Born probability postulate in quantum mechanics, and is identical to the first term in eq 3 (2). From the fact that the differential probability dP = 0 for x = %, Nelson (I)concludes that the cumulatiue, or actual, probability for observing an electron betweenx = M L and x = IhL + dx is zero. It is on this basis that he claims we are wronz. But we demonstrated and emuhasized in our original paperr2, that the differential probabihy dP is onlv an a~~roximatron Lo theaclual urobabilitv P. This is evidencedby eq 3. To review and illustrate this principle, consider the simple function y = x2, for which dy = 22 dx AY = ( ~ + k ) ~ - z ~ = 2 x & + ( k ) ~ = d y +2 ( k )
Suppose we were asked to find Ay ifx t 0. For small Ax, Ay 22 Ax = dy; that is, dy is an approximation of the actual increment Ay (2,3). Suppose we were asked to find Ay if x = 0, which happens to be a node. Then
-
Ay = (Ad2
Our conclusion was that eq 3 expresses the true probability AP that the random variable; will fall between z and x Ar.If* is not sufficiently small, terms higher than firstorder are required for accuracy. If & is allowed to approach zero and if p(x) + 0, then only the first term AP = p(x)Ax is required. However, if p(x) = 0, then higher order nonvanishing terms in eq 3 are requisite. This is, indeed, the case for regions containing nodes. We applied eq 3 above to the particle in a box of length L in state n = 2. At the midpoint, p(lhL) and p(ll(l/zL)are both equal to zero. For the interval x=I/zLtox=I/zL+AX
-
retaining the first nonvanishing term, we found AP = ( 8 2 1 3 ~ ~+)higher k ~ order terms in AT (4) Once a nonzero Ax is specified, no matter how small, a residual AP results. Therefore, there exists a nonzero probability for observing an electron within the given interval. In the calculus, the differential of the independent variable x is defined equal to its increment: dx =Ax ( 2 4 ) .There are no limits on the size of dx =Ax, although often they are considered infiiitesimal (3, 4). Consequently, we may replace Ax by dx in all of the above equations and text. Also,
(6)
(7)
In this case, dy = 0, the condition for an extreme. Even for infinitesimal Ax, the actual increment Ay is infinitesimally small but definitely nonzero. This is exactly the case for AP in eq 4 for infinitesimal Ax. Rather than deriving eq 4, the actual probability for an electron crossing the node in the specific particle in a box problem, we can showgenemlly that there is nonzero probability for electrons crossing nodes. At a node the probability density p(x) = 0. But at all points adjacent to the node, p(x) > 0. The only way to obtain AP = 0 using the general eq 2 would be to set Ax = 0. But that is clearly contrary to basic probability theory as applied to continuous distributions. Setting Ax = 0 is equivalent to calculating AP over a region of zero volume. If one admitted zero volume, there would be zero urobabilitv for all nonzero distributions even without nodes present r21. \?'e do call attention to one imoortant reference that Nelson has overlooked: Lowe (5) kas given an excellent resume of the more advanced time-dependent quantum mechanical description of this problem; see also ref. 6. Literature Cited 1. Nelson. P G. il C h m . Edue 1880,67,64347. 2. Ellison, F. 0.: Hollingsworth, C. A. J C h .Edue IWB,53,767-170. 3. Swokouaki. E. W Cdeulus;Rindle, W e k r & Schmidt Boaton, 1986:p 112.
Volume 70 Number 4 April 1993
345
Frank 0. Ellison C. A. Hollingsworth
University of Pittsburgh Pittsburgh. PA I 5260
that contain no hydrogen, including a large number of halocarbons, and many metal carbides. Harold Goldwhite California State University, Los Angeles 5151 State University Drive LOS Angeles, CA 90032 Literature Cited 1. Coldwhite, H.JCham. Educ 1964,41,626
To the Editor: The basic difference between Ellison and Hollingsworth's analysis and mine lies in our different understanding and use of infinitesimals. This is a controversial subject, about which there continues to be much discussion (I). Consider their simple function y = x2. For this, the change in y when x changes from x to x + hz is given by ~ y = ( x + h r ) ~ - x ~ = ! ? x h2r + ( k ) (1) The corresponding change in y when x changes from x - hz toxis A ~ = ~ ~ - ( z - L \ x ) ~ = ~ L \ x -2 ( L \ x )
(2)
Now let hz be infinitesimal (&I. I take this to mean that
Equations 1and 2 then give, for x t 0, and for x = 0, Ay = A{y = 0 (compared with 4x)
(4)
The same results can be obtained by taking the limit of AyIhz or A'ylhz
and identifying (with negligible error compared with A&) d with Ax. Ellison and Hollingsworth retain the term in (hzz2 in equation 1 or 2 when Az is infinitesimal, giving or 43' =-(A&)~ a t x = 0. A# = Literature Cited 1. See, for example, Edwards, C. H. Jr Tha Elstoried Lkudopmnt ofthe Cdedus; SpringerVerlag: New York,1979. 2. Compare ref2, pp 255-256,277-218,34t; Xiesler, H.J. Foundations ofinfinitesimnl C&ulus: Rindle. Weber & Schmidt: Boston,Masssehuset&, 1976; & d o n 2A.
P.G. Nelson University of Hull Hull HU6 7RX England
Carbon and HydrogebWhich Has the Most Compounds? To the Editor:
Regarding Ernest R. Birnbaum's letter on What's the Use? [J. Chem. Educ. 1991, 68, 7121 Alton J. Banks shouldn't have surrendered so easily on the question of whether carbon or hydrogen is i'nund in the largest number of compounds. The answer is not obvious, as was pointed out some time ago in this Journal (1).The hydrogen protagonists overlook the large group of carbon compounds 346
Journal of Chemical Education
Suggestions for Truly Evaluating Texts To the Editor: Recently, when several faculty members here were selecting a text for a new general chemistry course, one of them reproduced copies of vour Journal's reviews of most books now in use. &king at the reviews all together, I was struck bv how similar they were. Granted. the books themselves &e quite similar, i s few book companies have the courage to do something different. But the reviews had no "bite." Then I realized that the reviewers have not taught from the books before writing the reviews. Surelv all of us have had thc expcricnce uf flippingthrough book; trymgto find one for adoptim, picking one, and then finding - it to be a disaster when you use it with an actual class! We used Zumdahl's second edition in our general chemistry course. In the second half, which I us&, there were some terrible messes. Especiallv the c h a ~ t e r son kinetics. thermodynamics, and &clear chemistry contained mis: conceptions and outright errors. When I looked a t your reviewer's comments (Haworth, D. T. J. Chem. Educ. 1989, 66, A231), his most serious criticism was ofreflectionofthe light from his desk lamp by the yellow pictures! Virtually all books treat kinetics poorly. Most authors use the same examples, most don't really know how termolecular reactions work, and they fail to realize that probably the major class of unimole&lar reactions in the real world is photolvsis! It's a pity that coverage of kinetics is so poor deipite
