Comments on "The Probability Equals Zero Problem in Quantum Mechanics" T o the Editor: In a recent article in this Journal, F. 0.Ellison and C. A. Hollingsworth [53,767 (1976)] have presented a very clear discussion of a question frequently raised in elementary quantum chemistry courses. The question has to do with how a particle gets from one side of a nodal surface to the other. If we restrict our discussion to a nonrelativistic point-particle in a one-dimensional box with infinite potential steps a t x = 0 and L, and consider the system to be in its first excited stationary (time-independent) state (n = 2), then the question Ellison and Hollingsworth ask is, "How does the particle pass from the lefthalf region to the right-half region if it is never observed between x = (112)Land x = (112)L dx?" As a way of answering this question, they expand upon the fact that no finite differential element dx or du, containing a node, can contain a zeroamount of probability density since the nodal point or surface is of zero thickness. Hence 0 in most of the differential element. Thus. it is false to sav that the particle "is never observed between x = (112)~and x = (112)~ dx," so the problem goes away. I contend that this approach is not the most satisfactory.
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A Closer Look at the Question I t seems to me that thelinkage between the first and second parts of the above question is weak. The fact that one sometimes mill ohserve the oarticle in any finite element dx on either side of z = L/2 does not fbree us to conilude that it can pass from one side of the box to the other. I t simply means that the partiele will occasionally be found very close to the node. In fact, a counterexample is provided bythesamesystem. Ifwe take the regionx = Otoz = 0 dx, he., the left-hand wall of the boa) we again obtain a finite probability for finding the particle if dx is finite. But we don't interpret this to mean that the particle crosses this "node," for that would take the particle outside of the boa. Here, then, we have a case where the particle has a finite probability for being arbitrarily close to a barrier it cannot penetrate. This raises the suspicion that the finiteness of prohability density around a node does not even tell us "whether," let alone "how," the particle crosses the node.
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Does the Particle Cross the Node? Implicit in the question, "How does the partiele pass from the left-half region etc. etc." is an assumption-namely that the particle really does move across the node and that the problem lies in our understanding of how it manages the trick. But I am not aware of anything in our quantum mechanical postulates that enables us to make such an assumption. Maybe, in t h e n = 2 state, the particle remains tamely in whichever half of the box i t finds itself. How could we test this? How could we ascertain whether the partiele does in fact pass from one side of the box to the other? One way might he to measure the position of the particle two times in succession. If we find it on the left side for the first measurement and on the right side for the second, it hascrossed over. If it is found to be on the same side far both measurements, we have proved nothing. We would need to take third, fourth, etc. measurements and, if we neuer found it on the other side, we could build up our confidence level that it does not cross over. But there is a problem here. The measurements of position affect the state of the system. Thus, even if it starts in an n = 2 state, it departs from this when we first measure the particle's position. Even if we return the system to t h e n = 2 state and find, in the second measurement, that the particle has crossed over, we can't be sure this happened while the system was in t h e n = 2 state. I t may have happened during the measurement. T o get around this problem, we might consider a million separate oarticle-in-a-hox svstems in t h e n = 2 state and measure the oosition iur each parr~rlronre. Rut this is clearlv not gome: to nnsuer t h ~ quesrwn. Evrn ~fwe i d the first partirkon the lrfr and t h e * ~ c m u or, rhr right, rhls pmves ntlthing almut whethrrn p a n ~ rcrimes l~ the node. [One might postulate a million separate systems, all in t h e n =
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?stateand all with the paniclra ~n~tinlly in the left-hitndGde Hut t h ~ s is self-cuntradictory. I f U,P knou the particle is initialh utl the lctt. then V'W must vanish on rhc n g h r But such a V ir tnr.t upproprlate for the n = 2 stationary state.] The dilemma is becoming clearer. A stationary state for our particle-in-a-box system is a solution for an undisturbed system. Measuring particle position disturbs the system. So the question has been reduced to, "Does a nonrelativistic point-particle cross a nodal surface when wearen't looking? (i.e., in astationarystate?)."The answer has to be, "I can't tell unless I look." So the question is unanswerable. The crux of the matter is that a stationary state function is incapable of describing any observable as changing with time. Since "path" involves change of position with time, it is a concept which is not applicable to stationary states. The whole problem arises due to a lack of awareness of the full implications of the term "stationary."
Can We Discuss Path? All this does not mean that the concept of path is outside the domain of quantum mechanics. I t means that time-dependent wavefunctions are needed to describe particle motion. For instance, I referred earlier to the case where we knew that a particle was initially on the left-hand side of the box, and indicated that V a t the initial instant must be finite on the left side and zero on the right side. A simple approximation to this situation is produced by summing the n = 1 and n = 2 wavefunctions. In the left half of the box these are both positive, so they augment each other. In the right half they are of opposite sign and partially cancel. The result favors finding the partiele on the left. Such a wavefunction does not describe a stationary state since i t is not an eigenfunetion of the Hamiltonian. However, with appropriate time-dependent coefficients, this wavefunction is a solution for the time-dependent Schrodinger equation. As time passes, the coefficients oscillate a t different rates for t h e n = 1and n = 2 components of the wavefunction. This causes the maximum in the wavefunction to move back and forth in the box. Thus, we have a crude "wave-oacket" descriotion which shows the oarticle moving
What Do We Tell the Students? I have found that this question a b u t particles crossing nodes is very useful for bringing out these somewhat subtle hut basic points about quantum mechanics. That W*W for a stntionory state cannot give &wers to such questions about a particle's path is a fact whichone often sees stated but rarely clearly explained. Therefore, I prefer t o treat the question along the lines given here rather than those developed by Ellison and Hollingsworth. T h e Pennsylvania S t a t e University University Park, 16802
J o h n P. Lowe
T o the E d i t o r First of all, we would like to call special attention to Lowe's paraphrase of our answer to the question as to how a particle gets from one side of a nodal surface to the other: ". ...no finite differential element d r or do [Ax or Au], containinganode, can contain a zeroamount of probability density since the nodal point or surface is of zero thickness. Hence, $ # 0 in most of the differential element. Thus, it is false t o say that the partiele'isnever observed between x = W2)L and x = (112)L dx,' so the problem goes away." We would recommend to teachers this concise and cogent answer to the question so often posed bv students. In the remainder of our orieinal oaoer. " . . we eo on to oropost- nn Extended Born I'n,hability I ' O S ~ I I I ~ I P.and . wp show spec ~ f i d l y hou to lakulatt. thr actual non-mro prohnhhry 1P frlr A w i n g a particle in an element l x at somr pmnt wrhm w h i ~h $ = U. Now Lowe has taken a much closer look a t our original paper and concluded that we do not really succeed in answering the central question. That question, "How does a partiele pass from one region to another,"' requires knowledge of path; path involves change of poaition with time, a concept not applicable to stationary states. Lowe thus sueeests ,.~. that the orieinal .. ouestion be utilized as a oedaeoeical , .... oprnrr tor dwussmp. time-depmdcnt motitx of bound port~rle.~ In nun-statimary ~ t ? t e % . ~ Although Lowe'irriticism iscertainly correct and illurnmatma, we
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Volume 55. Number 11, November 1978 I 749
