How to Teach the Postulates of Quantum Mechanics without Enigma Jose J. C. Teixeira-Dias Universidade de Coimbra, 3000-Coirnbra, Portugal Despite recent and reliable experimental evidence ( I ) and enormous practical success, quantum theory is so contrary to common sense that even our language is inadequate to express microohvsical ohenomena. Frequently, one has to resort to confiitiig words of everyday lifeio name microphysical ideas, concepts, and models. Moreover, a t the fundamental and philosophical level, the conceptual interpretation is still the subject of different viewpoints, heated debate, and controversy (2,3), as can be easily inferred from the many articles that auestion the ability of quantum theory to mirror reality. Therefore, it is not su;prisfng that various articles in THIS JOURNAL have considered several difficulties in the teaching of basic concepts and principles of quantum mechanics, for example, the wave-particle dualism ( 4 , 51, the uncertainty principle (41, and the superposition and correspondence principles (5).These articles also stress that the commonly encountered teaching "philosophy," just go on, and faith will soon return, is both misleading and dangerous when dealing with onantum theorv. A general inspecti& of quantum mechanics and quantum chemistry textbooks reveals three distinct ways of introducing the h a m concepts of quantum mechanics. The first follows the historical path, starting with classical Hamiltonian mechanics and going through the old quantum theory until the heuristic derivation of the Schroedineer eauation. The second is the axiomatic approach which'briefly introduces the mathematical laneuaee and formalism of auantum mechanics and then presents a ;et of postulates expressing the physical interpretation of the mathematical operations in a straightforward manner. A much less common approach, henceforward named statistical, considers a number of real and ideal experiments with waves and particles to present the waveparticle dualism and the uncertainty and superposition principles, thus leading the reader to the orthodox statistical interpretation of quantum mechanics (the Copenhagen interpretation) and its axiomatic statements. The primary objective of this article is to show that the statistical approach can help students accept the postulates of quantum mechanics in a natural and less enigmatic way and not see auantum theorv as the intellectual outbreak of eenial " minds. This approach will also make the student aware of the ohilosoohical and humanistic imolications of auantnm mechanics: This will prevent the student from thinking that the ereat success of auantum mechanics is due to the fact that
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approach asserts the pr&acy of experiment'and of facts over theorv and intellectual constructs, it is less enigmatic in the features they often display and become statements which simply absorb and summarize previous concepts and principles supported by a clear and strong experimental background (6). With better understanding, the quantum mechanics postulates will also provide a basis for further generalization enabling the student to progress rapidly in the matter. In manv. resoects the statistical aooroach is a hvbrid ohe. nomenological/historical approa% similar to the r e c h y advocated for the teaching of chemistry (7).
The Conceptual Basis of Quantum Mechanics and the Teaching of its Postulates In order to build the necessary background for making the postulates plausible while preserving the primacy of experiment over theory, the following sequence for general orientation is sumested (6):(1) imoortant ex~erimentsin auantum m e c h a n i c s ~ ~ 2conventionai ) statistick interpretation; (3) mathematical language . . of auantum mechanics; and (4) oostulates. The striking similarity of diffraction patterns obtained with x-rays and electrons of comparable De Broglie wavelength has been widely adopted in the prediction of microphysical results using a wave model. Moreover, a lot of recent experiments performed a t the subatomic level, most of them concerned with measurements of spin components of protons and the polarization of photons on correlated systems (1,3), clearly support quantum mechanics. Interpretation of particle diffraction experiments both with ideal ex~erimentsand sinele-nhoton interference ex~eriments behavior of quantum systems while showing the inability of t orecise outcome of measureauantum theorv to ~ r e d i cthe ments on individuai microsystem. This will enable the student to understand the statistical interpretation of quantum mechanics: quantum mechanics predictions can only be verified with experiments carried out on a large number of similarly prepared systems (henceforward designated as ensembles). At this instance, the student should realize that probability in quantum mechanics does not have the same meaning as in classical mechanics. In fact, while classical statistics uses probability descriptions for practical reasons, quantum mechanics resorts to it by virtue of essential restrictions imposed by the uncertainty principle which simply forbids predictive laws based on coordinates and momenta. We can now proceed by considering the principle of statistical determinism, the probability distributions of oboeriments one can easilvrecoenize the heo ore tical necessitv ior complex quantities, the so~calledprobability amplitude;, and lav down the concent of suoeroositiou of orohahilitv . . amplitudes in quantum mechanics (9).The typical interference ohenomena usuallv- disolaved . " bv" microohvsical . " svstems can be nicely confirmed, from the experimental point of view, hv neutron scatterine- from a crvstal(8). Followine" this. the superposition principle will naturally prompt a discussion on the meaning of superposition in quantum mechanics. We thus reach topic (3) where the language and concepts described in touics (1) and (2) are laid down in a formal mathematical
aid absorb the basic concepts and of quantum mechanics. Since there is no unique set of postulates, we proceed with a typical set taken from an excellent quantum chemistry book (10) to illustrate the above mentioned considerations. Volume 60
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The first postulate says that all the experimentally accessible information about the state of a system is described hv a function of the coordinates and the time, the so-called statefunction or wavefunction. In order to understand the exnerimental foundations of this postulate one should consider the following concept of statkfunction. The probability distributions of ohservables are objective characteristics of ensembles. Two ensembles are in the same state when they have the same probability distributions for all of their ohservables. A rule of correspondence between the
the measurement was carried out and the state of the ensemhle. The possibility of performing subsequent measurements on different statistical samples of the ensemble (suhensembles) guarantees that the probability distrihutions of ohservables are objective characteristics of the states of an ensemble. In narticular.,the urohabilitv distributions can he sharp or dispersion-free thus corresponding to eigenstates as in the case of nostulate 3. It is important to point out that this postulate has the most freauent and far reachine in auantum chemistrv . an~lication .. WII(:II 1; cu~nridt+ with the m e r s oprrtltok, the ~:imilton:at;. Tlnc 1s what en.thles the determination ofthe eneruv levels of atoms and molecules. An obvious problem raised by postulate 3 occurs when state $ of our ensemble is not an eigenstate of G and so does not coincide with an eigenfunction of c. This problem is addressed by the following postulates. The fourth postulate asserts that the eigenfunctions of the linear Hermitean operator G form a complete set of functions, and the fifth postulate indicates how to evaluate the average value of the physical observable G at a given time.
.
~
~~
--
state is represented by the function
*(XI
=
(XI*)
(1)
By gathering all this experimental information about the system function (1) fully describes 1 $) thus confirming postulate 1. This postulate follows naturally from the experimentally based quantum mechanical concept of state and expresses an act of faith in quanthm mechanics by asserting that the statefunction contains all the information that can be determined about the system. It should be mentioned that Einstein, Poldolsky, and Rosen contradicted this statement in a paper presented in 1935 by concluding that the description of reality as given by a wavefunction is not complete ( I 11. However, all the recent experimental attempts to find hidden variables (12) have failed so far (I ). By restricting, in the second postulate, the operators representing observables to linear Hermitean operators, one simply recognizes that the results of a measurement as well as the average values of Hermitean operators are real numbers. The requirement of linearity is closely related with the superposition of states previously considered. The third postulate asserts that the only possible values that can result from measurements of a physical observable G are the eigenvalues g; of the equation This postulate translates the concept of measurement into the quantum mechanical formal language through the application of the corresponding operator to the statefunction. In this case the statefunction is an eigenfunction and the nostulate also contains a definition of eieenstate bv relating it with the measurement of an observable. In fact, the eigenfunction of an operator correspondina to the observable G is the function which describe; the state 4; of an ensemble previously prepared with the requirement of providing a unique and certain valueg, for the measurement of G. In this way postulate 3 can be seen as relating eigenstate with experiment. Equation (2) assumes that all the microsystems in the ensrrnhl( w ( w prepared in the ,tat* ( 8 chmicterized 11). the itate property y Followi~~l: thi, rrawnlng 11 is natural 10 ask w h ~ t < ~ ~ u ~ ~ I of I I1 ; . Mathernariis tbe -~ ~-~~result ~ ~ ~ ' ~ J J niemlrement~ cally speaking, this isan easy question to answer as it suffices to repeatedly apply G to both sides of eqn. (2), i.e., ~
~
~
~
~
~
(G) = J+"G$d7
(4)
However, before making practical use of postulate 4 in the evaluation of can. (4) one should dwell hrieflv on it in order to underdmd i ~ experimental a l~ackyround.We start with the fdl,,wing relsrim,hip applicable r i r prubabilit\ i~mplitudei
d C where lo : reprewntia set of states where rne:~.iur~ment \.ield.; uniuur and well defined -. c , value*. Then IC,Irepresents . a complete set of mutually exclusive state properties defining state observable G. Exnression (5) can he seen as an exnerimentally derived relationship in so far as it is successfullyked to interoret interference nhenomena in microphvsical ex. . prrilnents Iror instance neutrrm scatterin:: frcm il crystal tbl) nnd tu r\,aluute the prohahiiities u i thv v.irltms t,utcmnej uf a measurement in state I$). To the same extent expression (5) is valid for any property xk. It can be cast in a more ahstract form known as the superposition principle
where ci = (&I $). It is immediately apparent that the completeness of eigenfunctions I&) (postulate 4) as well as their orthoaonalitv are in consonance with the completeness and the n&ually exclusive character of the state-properties g; comprising the state observable G. In this way postulate 4 appears not so much as an act of faith in a mathematical peculiarity of the eigenfunctions of a particular operator, but rather as a result which emerges from the probability amplitudes of relation (5). Along these lines it is now possible to use postulate 4 in the evaluation of eqn. (4) to conclude that this expression (postulate 6) makes sense as it aerees with the intuitive notion of arrives a t
thus concluding that the ensemble is kept in the same eigenstate 4;. This result can be simply illustrated with a beam of light of a particular color being obtained by the spectral decomnosition of sunlieht (Newton exneriment in 1666). . . and so it is supported evenuat the macrosco& level. However, one could areue that this result contradids the exnerimental observation wkch shows that the measurement oi a property may drastically alter the state of individual microsystems. In order to solve this apparent contradiction the student should realize that there are two distinct features: the state of individual microsystems in the statistical sample of the ensemble where
.~~~
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Journal of Chemical Education
The sense of eqn. (7) can he seen, in a particular case, by applying eqn. (4) to an eigenstate 4 k of G as ( G ) coincides then with gk:
(G)= SdjC$id~= gk
(8)
While eqn. (7) is more general than eqn. (a), their comparison immediately suggests that the prohability of obtaining g;, when a measurement of G is carried out, is P#, = c:c;
(9)
In order to prove this result and confirm eqn. (7), the student should be invited to conclude the results of eqn. (9) by considering a particular case of eqn. (5) when I x k ) is made to coincide with I$). Once again, expression (5) is of crucial importance in the understanding of the postulates and should be seen as an experimentally derived relationship emerging from the analysis of interference phenomena in microphysical systems (8,9). By substituting eqn. (9) hack into ( I ) ,one easily confirms the physical meaning of the integral in eqn. (4), i.e., the average of all possible outcomes of measuring G,
co = ZPP,&'L
iur the i - d n t v i i ~ I I , P I I I I I I Z 11101e that t h i i C Y ~ I ~ C Y I HI S 1111 l(,n:t.r \..III,I d u r ~ ~nu l r rnvcrimc~i~;~l .,l,arrv,~ti$m~, tllr m n t l ~ e ~ ~ ~ n r ~ c a l limit
should exist. By combining eqns. (14) and (15) one then arrives at where
(10)
The sixth postulate is concerned with the time development of a state of an undisturbed system. It shows that By considering the concept of physical continuity as it applies to an undisturbed system and using postulate 1and the superposition principle, one can show how to arrive without difficulty a t the general form of eqn. (11). Detailed and complete information on an ensemble not only presupposes the knowledge of the probability distributions of its ohservables a t a given time hut also their evolution with time. As the statefunction provides all this information a t a given time (postulate I), its time development should also enable one to establish the time evolution of the probability distributions. Then 'P should be related to its time derivative a*/&, and this relationship should be linear in obedience to the superposition principle. Let us then consider an isolated ensemble during a certain time interval and let us represent by ($i(q,t,)) the complete set of all eigenfunctions of a state observable a t time t,. The statefunction a t to can then be expanded in the following way,
The linear relationshiv (12) that defines 'P a t t,, according
=
lim {(i? - 1)iit - to)l
(17)
t-to
and Li is a linear anti-Hermitean operator. However, by multiplying it by the imaginary unit i the student easily verifies that a linear Hermitean operator is then obtained thus arriving a t an equation of type (16) in the form as postulate 6 asserts. At this stage it is quite appropriate for the student to consider the limiting case of classical mechanics along the lines suggested by Landau and Lifshitz (13). Conclusion The above mentioned sketch of quantum mechanics topics as they are fully developed in (6) is intended to carry the student through a simple and easy route from the basic concepts and principles of quantum mechanics to a set of plausible statements, the so-called postulates. These include the basic quantum mechanical doctrine and summarize the actual interpretation of microphysical experiments. An important advantage of the suggested approach should he found in the fact that it establishes the primacy of facts and experiment over the theory, thus being more natural in the way it leads the student to scientific conclusions. Literature Cited
t of the same interval one has Wq,t) = Z c,ji,(q,t)
(13)
The operator that brings 'P(q,t,) into 'P(q,t) should also yield $i(q,t) when applied to +;(q,t,), i.e.,
This equation shows that I? is a linear operator as the superposition principle requires. If one assumes physical continuity
11) (2) 13) (4) 15) (6)
Pipkin, F. M.,Adu.Al. M o k i P h y ~ .14,281 , (1918). DeWitt,R. S.. Phyr~c.7Today.23. 80 (1970). D'Espagnsl. R.. Scient. Arner., 241,128 (1979). Geor~e,D.V., J. CHEM EDUC.,46,663 (1969). Crupper, W. H., J. C H F M EUUC..46,889 (1969). Teireira-Dias. J. J. C., "Quimica Quintica," Funda~eoCalouste Gulbenkian, Lishor,
19R2.
17) Schwartz.A. T., J. CHEM EDUC.,58,314 11981). 18) Feynman. R. P., Leighton. K. B., and Sands, M.. "The Feynman Lectures on Physics: Vol. 111, Addiwx-Wesley. Reading, MA, 1965. 19) Feynman, R. P., RIU.Mod. Phyr.,20,567 (1948). I.N.. "Quantum Chemirtry,"2nd Ed., Allyn and Racon.Inr.. Boston, 1974. (10) Pliys. R e " , 47,777 11955). (11) Einstein. A..Podoisky, &,and Roien, N.. (121 Rell. J. S..f'hyrtrs. 1. 195 (1964). (1%) Landau.L.D..and Lifshitz.E. M ,"quantum Mechanics-Non-relatiiiitiiThiii~," 2nd Ed., Pergamon Presi, Oxford, 1965.
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