Dice Throwing as an Analogy for Teaching Quantum Mechanics Benicio d e Barros Neto Universidade Federal de Pernamhuco, 50000 Recife, PE, Brasil Those who teach physical chemistry are well aware of the difficulties encountered when trying to get across to beginning students the abstract concents of auantum mechanics. In many cases one has to resort to some easily visualizable physical system and extract from it conclusions that, through analogy, are extended to quantum systems. One of the most .nouular . of such models is. without doubt. the vihratine" strine.". which is employed by nearly all texts treating quantum mechanics. It is my experience, however, that quantum concepts, as derived from the vibrating string model, are not easily grasped by chemistry students. Making students understand the v i h r a t i i string peculiarities seems to be almost as difficult as teaching them right from the start the quantum concepts that are supposed to be clarified through the analogy. This paper describes an alternative approach, which we have been using with considerable success. We employ another model-dice throwing-that serves to illustrate as many important concepts as does the vibrating string (perhaps more). Students find this model much easier to understand, perhaps hecause i t requires only the sort of mental ability necessary to grasp arguments involving discrete notions, as in aleebra. - . whereas the vibratine" strine reauires an understanding of the continuum, as in calculus. We beein with the idea of auantization. The onlv exuerimental aspect we are interested in (the only observable) is the same that interests a olaver: the number of uoints on the unward face. ~ h v i o u s l y t h & number can onlybe 1,2,3,4,5, br 6, that is, the result of a throw is quantized. We say that the die can stop on the table in only one of six differentstates, and to represent one of these states we use the symbol I >, Dirac's Ket. Thus, when the numher of points is n, we say that the die is in state In>, where n is a quantum number. Defining a value for n is enough to completely characterize one state. Now we associate to the observable "numher of points" a certain operator@ and, making no fuss about its explicit form, we write the eigenvalue equation n l n > = nln> which we interpret as follows. The only ~ossibleoutcomes of a .'measureme$" of the number of poi%sare thr ei~enualurc n of operaw! N. The stare In>, to whirh the number of points n cor;esponds, is an eigehstate of operator @. I t m i s t he stressed that these remarks apply only to stationary states, i.e., states whose number of points does not vary with time. Since we know all the possible eigenstates and eigenvalue, equation NI n> = n in> can be viewed as a definition of N. This is, of course, a fortunate instance. For most quantum systems only the operator is precisely known a priori. The throwing of a die is an experiment whose result we cannot predict. All we can do is calculate the probability of occurrence of a certain number of points. In an honest die this prohability is the same for all possible values, being equal to 116, as everyone knows. Now, why 1/6? The answer is simple and also enlightening. There are six possible outcomes of a throw, and all of them have the same probability, p , of occurring. For example, the prohahility of ohtaining 1in a given throw is p. The prohahility of ohtaining 1 or 2 is p p = 2p. The probability of
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obtaininglor2or3or40r5or6isp+p+p+p+p+p= 6p. Since one of these values will he necessarily obtained, 6p corresponds to certainty, which by convention receives the value 1.Then 6p = 1,a n d p = 116. This is an example of normalization. It is entirely analogous to normalization of wavefunctions, which comes from interpreting lil.12d~as a urobabilitv. In dice throwing each of the possible results is completely independent of the others. This leads us to the conceut of orthogonality. Ket Il>, for instance, which corresponds to obtaining 1on the upward face, represents a state that cannot be produced from combination of Kets 12> to 16>. Kets I I>, 12>, . . . 16> are then said to be ortho~onal.Furthermore, they ionstit& a complete set, in the sense that any state la> can he represented by a linear combination of 11>, 12>, . . . 16>:
At this point we emphasize that the state itself is not an observable. The only observable, as we declared at the beginning, is the numher of uoints on the uuward face. We are now in ;position to i ~ u i t r a t ethe so-called reduction of the state hv the measurine nrocess. Let us consider the die still in the ai;, before fallingupon the table. Let us denote &s state hv I a > , which need not be an eigenstate of onerator N. that is,it'do& not necessarily correspo~dto a definite number of points. It does corresnond, . . however, to a linear combination of ~ e t I1> s to 16>, as we have just seen. When the die at last comes to rest on the tabletop and we see n points facing upward, we conclude that its state is now the eigenstate in>, corresponding to the number of points we see. State la>, which was not necessarily an eigenstate, was therefore reduced to eigenstate In> by the act of measuring. It should be stressed that this reduction is probabilistic, since we are not able to predict which will be the final eigenstate. When treating the particle in a one-dimensional box, a frequent demonstration is that for any state the expected
EigenStateJand eigenvalues far the throwing of two dice.
value for position is the center of the box, although the center is a node for half of the states. This asDect, which is a touah one for beginning students, can also be clarified with the help of our analow. The expected ualue or average value, ii. of the number of points is simply the average of all possihle numhers of points, weighted by their probahlitws of occurrence: ii = 116 X 1
+ 116 X 2 + 116 X 3 + 1W X 4
+ 116 X 5 + 116 X 6 = 3.5
We see that ii is not one of the wssible numben of noints. and we conclude that an expectedvalue need not be equal to an eieenvalue. Such is the case of the oarticle-in-a-box. where for h z f of the states the expected valie for position cdrresponds to a forbidden position. Let us consider the throwing of two dice. The (total) number of ooints now varies from 2 to 12, and two auantum numbers are required for its characterization. Accordingly, a generic eieenstate will be denoted by nm>. There are :{ti eicenstates, shbwn in the table together with their eigenvalues. state 113>, for instance, corres~ondsto scoring 1with the first die and 3 with the second. 1t; eigenvalue is and we notice that there are two other states, 122> and 131>, that also have this eigenvalue. Whenever there are eigenvalues corresponding to more than one eieenstate we speak of degeneracy. In our example, we say that4 is triply degenerate &, alternatively, that its degree of degeneracy is 3. The only nondegenerate eigenvalues are 2 and 12. corresooudine to states h1> and 166>. , . respectively. We mav use the table to illustrate still another interestine aspect. ~ G t enm> (scoring n points with the first die and k with thesecond) and statr , m n > (scoringvi points with the first die and n with the second) both ha\.e eiyenvalue n m. They are therefore indistinguishahk with respect to the total
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number of points and, since this is the only observable, state Inm> cannot in practice he distinguished from state imn> (nor from anv state with eieenvalue n m. as a matter of fact). These two states are an example of permutational symmetry, since one of them can be obtained from the other simnlv hv " interchanging the dice. So far we have emoloved what mav be considered the orthodox interpretation ofquantum mechanics. I t is curious to observe that we may also use the dice throwing analogy to illustrate the famous argument concerning the existence of hidden variables governing quantum behavior. If we consider die motion as descrihed (or at least describable) by classical mechanics, we mav conclude that. with a thoroueh knowledge of all the "ariablei involved (airresistance, elasticity of t i e die and the table, forces applied when throwing, etc.), we would be able to predict with security the outcome of agiven throw. In other words, our inability to predict a result stems not from an intrinsic indeterminacy associated to the physical phenomenon, but rather from our ignorance of the variables governing it. Substitute "electron" for "die" and we have an example of the hidden variable argument. An advocate for the hidden variables argument would be able to predict the trajectory of the die within any desired accuracy, simply hy choosing Newtonian mechanics to describe its motion. An adherent to the orthodox interpretation, on the other hand, would prefer a die considered from the point of view of probability theory, i.e., intrinsically indeterministic.
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Acknowledgment I would like to thank 0. M. Malta. J. G. R. Tostes. and R. E. Rruns for carefully reading the manuscript and for'making several important cwnmrnts. Financial s u ~ ~ ofrom r t CNPu ( ~ o u s e l h o ~ ~ a c i ode n aDesenvolvimento l ~ieutificoe ~ e c h : nol6gico) is also gratefully acknowledged.
Volume 61
Number 12 December 1984
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