would suggest that it probably is never appropriate at the level at whieh the question usually is advanced by students in their early or first exnerience with quantum chemi~try.~ But over and ahove this point, we believe that our stationary state treatment may be sounder than Lowe's critique suggests. First of all, one might contend that the original questioner really means to ask (A) "Does the mndition tZ= 0 at a node of a stationary state imply that a particle cannot pass through the node?" This is really what bothers the nonce; he seldom cares "how?" or "by what path?" hut simply "doesn't the node stop it?". If the answer to question (A) is that passage is not prevented, then other questions may follow (although in practice, they ordinarily do not); for example (B) "Does a particle in a stationary state really pass through a node?" (C) "How often does it pass through this node?" Sinee we proved that U f 0 in any element hr containing a node ib2 = 0, it follows that the node does not necessarily prevent passage of the particle. We went on to assume implicitly an affirmative answer to (B). Concerning that asumption, we advance the following support. Since quantum mechanics gives no answerto (B),our assumption does not lead to contradiction. Although our assumption is not necessary, it does permit one to keepapieture that isconsistent with both classical and quantum theories. This may he a matter of personal preference. However, is it not the desire to keep as much of classical theory as quantum theory will allow that leads to questions like (A) in the first place? We might add that this point of view is not uncommon but also hiehlv in aoolied ~. . oraetical . .. auantum theorv. For examnle. scattering and rranrmii\ion corttiricnrs opproprmtt. to thr om-dimmrmnnl square putcntial barrier (as well as mure romplicated hnrrwrs) are generally olrtainrd f n m an unhound slntmnnry stare of the timrindependent wave equation. We do not feel that the counterexamplegiven by Lowe in the semnd paragraph of his Comment applies to the central question under discussion. The student observes 5.2 f 0 on both sides of anodal point, and he wonders if the oartide can cross over. Question (A)above refers to a node in which $ 0 only over a region ofzero volume. Our result is not meant to apply to a boundary at and beyond which $10.' For the more advanced student, we view Lowe's suggested treatment asvery interesting and useful. Wenote that in hisarticle there are several references to the measurement process itself. This is one of the mast controversial areas in quantum mechanics (3).Lowe's approach might also be conveniently used to introduce the advanced student to discussion of this subject of heated debate.
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Literature Cited (1) Rspp, D.. '"QuantumMeehanics.('Holt, Rinehsrt and Winston, Inc., New York, 1971. " p. m
(2) See, for example, Blinder, S. M., "Fwndationsaf Quantum ~amics,"AesdemicPms, New York, 1974. p. 203.
1 One might
add, "whether or not a node intervenes." We call the reader's attention to reference ( 1 ) which provides details and useful figures corresponding to Lawe's discussion. 3 When we were writing our original article, it occurred to us that since quantum electrodynamics requires spontaneous emission, it follows that no excited state can he strictly stationary (2). This clouds the original question, and also leads to ultimate decay of Lowe's asd a t i n g nonstationarystate. Though basic and fascinating, we believe that spontaneous emission is best left unsaid when first responding to a student worried about the initial problem concerning nodes. 4 In order for an infinitely high barrier to be impenetrable, it must have finite thickness, hut this produces a region of finite thickness throughout which $ 0.
Frank 0. Ellison C. A. Hollingsworth
University of Pittsburgh Pittsburgh, Pennsylvania 15260 Periodic Sub-Groups
To the Editor: Fifty years of "Chemical Education" apparently have left unresolved a minor but irritating problem, namely the designation of t h e elements of the periodic chart with letters A or B. Roughly half the textbooks, papers and charts issued in the last decades call 3A to 7A* (or 8A) the transition elements which the other half call 3B t o 7 B (or 8B). Further comments seem unnecessary and discussing the advantages of one or the other choice is pointless. Would i t not he possible to suggest to authors and chart manufacturers that they stick to thesame rule . . . the one recommended by IUPAC for example. ["Nomenclature of Inorganic Chemistry (1970 Rules)," 2nd Ed., Butterworth, London, 1971, p. 11.1 7A 3A 5A 6A 4A 1A 2A K. Ca Sr Ti V Cr Mn -- -~ . .-. Zr Nb Mo Tc Y Sr Rb Re Ta W La' Hf Cs Ba ~
"lnrluding rhc Innthnnoids. %duding thr arrlnoida, hut thorium, pn,lacrinium and uranium may aliu he plnred in gnlups 4A, 5A and 6A. Ch. Mazieres Laborstoire de Phvsieochemie MinCrale UniwrsitC de Parir-Sud 9110.5 Orsay Crdcx. France ~~~~
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Precipitating PbCI2 To the E d i t o r T h e problem raised by Pyriadi and Polis [54,439 (1977)] has a rather simple solution, if one remembers t h a t PhClz is not extremely insoluble (K,, = 2 X 10-5) and on t h e other hand HgClz is a rather stable covalent (coordination) com10-13). When the two ions are pound (instability const. 82-I present together in solution they compete for the chloride ions according t o
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750 1 Journal of Chemical Education
Because of the relative stability constants, the mercuric ions consume, in preference, all the chloride ions before any lead ions can precipitate. Thus in presence of Hg2+ one must cool (to lower t h e solubility of PbClz), add excess Cl-and scratch t h e walls (of t h e test tube) in order t o precipitate PhCIz. The Hebrew University of Jerusalem Jerusalem, Israel
Lina Ben-Dar
