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Is the Relation at the Root of all Mutual Exclusiveness?

D. Sen, A. N. Basu, and S. Sengupta Condensed Matter Research Centre, Department of Physics, Jadavpur University, Jadavpur, Calcutta-700032, India

Z. Naturforsch. 52a, 398-402 (1997); received October 10, 1994

It is argued that two distinct types of complementarity are implied in Bohr's complementarity . While in the case of complementary variables it is the mechanical uncertainty relation which is at work, the collapse hypothesis ensures this exclusiveness in the so-called - complementarity experiments. In particular it is shown that the conventional analysis of the double slit experiment which invokes the to explain the absence of the simul- taneous of the which-slit information and the interference pattern is incorrect and implies consequences that are quantum mechanically inconsistent.

Keywords: Bohr's Complementarity Principle, Wave-Particle , Uncertainty Relation, Mu- tual Exclusiveness, Quantum .

Introduction Class II: A property such as interference which de- pends on the superposition of a number of states, and Bohr [1] formulated his complementarity principle the property associated with the "which state" infor- to "harmonise the different views" on wave-particle mation, form a pair of complementary properties. dualism without addressing the of quantum Here ME follows from the principle of collapse of mechanics (QM). Surprisingly, no reference to the ac- wavefunction. tual physical principle which enforce the mutual ex- An important point to note is that here we refer to clusiveness (ME) of sharp knowledge of complemen- complementarity between a pair of properties, rather tary pair of variables or properties was made in this than a pair of variables as in class I. Usually it is diffi- pleonastic formulation. Bohr's complementarity prin- cult to associate specific operators corresponding to ciple (BCP), which is often superficially identified with these properties [3]. The possibility of assigning suit- the wave-particle dualism for historical reasons, how- able pair of operators was explored in [3] and we have ever, is a more general concept. It is natural that the noted that the commutation relation of such suggested actual mechanism which enforces ME varies from one pair of operators does not lead to any uncertainty type experimental situation to another. relation. However, in a recent letter Storey et al. [2] have concluded that the principle of complementarity is a Complementary Pairs in consequence of the Heisenberg uncertainty principle. We disagree [3] with this sweeping generalisation and Various gedanken experiments have been analysed emphasize that two intuitive formulations of the no- in subsequent years, which emphasize complementar- tion of complementarity by Bohr may be classified ity in QM. Early examples include Heisenberg's y-ray into two distinct classes on the of the origin of microscope [4], atomic de-excitation experiment [5] mutual exclusiveness. These are: and Einstein's recoiling (double) slit arrangement [6]. Class I: In this class belong pairs of variables ob- In the first two of the examples the complementarity tained from a of the state vector is between a pair of variables, and hence the origin of and any pair of non-commuting dynamical variables. ME can be traced to some quantum mechanical inde- These include pairs of canonically terminacy relation. Heisenberg's original treatment [4] for which the corresponding operators satisfy the usual of this gedanken experiment deals with the scattering uncertainty relation. For other non-commuting pairs of of a single by an individual . The no- variables uncertainty relations are also true but here tion of uncertainty in a single is not the uncertainty product depends on the state vector. amenable to any interpretation implied by the mathe- matical formalism of QM. A reformulation [7] in Reprint requests to Prof. A. N. Basu. terms of the statistical interpretation of Heisenberg's

0932-0784 / 97 / 0500-0386 $ 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72072 Tübingen D. Sen et al. • Is the Uncertainty Relation at the Root of all Mutual Exclusiveness? 399 may be regarded as a meaningful wave and particle properties [3], It is in this sense that illustration of the quantum mechanical indeterminacy some authors have discussed violation of BCP [8, 9]. relation and complementarity between the canoni- cally conjugate variables x and px. Here, for the com- plementary pairs, it is impossible to have a quantum Wave-Particle Complementarity mechanical state where both the variables have sharp values. The most discussed wave-particle complementarity The atomic de-excitation experiment, on the other experiment is the double slit interference experiment. hand, highlights a different type of uncertainty rela- Quite frequently people have used Heisenberg's un- tion in QM. Even though is not a quantum me- certainty relation to explain the ME aspect. But we chanical , the complementarity between the believe that this approach does not give a correct Fourier pair (£) and time (t), expressed in the description. time-energy uncertainty relation, may be traced to the Before examining how far these explanations are non-commutation of the Hamiltonian and a quantum consistent with the formalism of QM, let is consider mechanical operator, say A corresponding to some the standard arguments. In the famous example of the dynamical variable. double slit interference experiment with [10], From the quantum mechanical definition of the un- when Feynman intends to "watch" the electron, using certainty product a scattered photon as a spy, the interference pattern gets washed out, and Feynman concludes: "If an appa- AA AE > ratus is capable of determining which hole the electron 2i goes through, it cannot be so delicate that it does not and using the quantum mechanical equation of mo- disturb the pattern in an essential way. No one has tion, we have ever found (or even thought of) a way around the uncertainty principle". Hence, AA >ti/(2AE). dt /.p of the illuminating photon. Therefore, which-slit

Now for a particular quantum mechanical state (\j/), information is obtained when /p < a (= width of each the r.h.s. is fixed. So, for any arbitrary A, the l.h.s. has slit). But such a light quantum will deliver to the elec- a lower bound r, characteristic of \j/ having the dimen- tron a which is uncertain by Ap > h/a. So sion of time. Then, the angular dispersion of the scattered electron is: AO ~ Ap/p > h/(ap) = x /a; which is obviously greater T A£ > h/2 . el than the angular difference between the successive

Also, we have complementarity between compo- minima of the pattern, i.e. /.el/d, where d is the separa-

nents of , jx and jy or jy and jz, etc. tion between the midpoints of the two slits (d > a). which are not canonically conjugate variables. The Thus the which-slit information destroys the interfer- non-commutation relations in these cases lead to un- ence pattern. certainty products which depend on the state function Our first criticism is that the above type of analysis - not the usual type of uncertainty relation. Our point is not strictly based on the principles of QM. The is that for complementary variables ME follows quite uncertainty relation that follows from non-commuta- clearly from the general quantum mechanical uncer- tion of canonically conjugate variables [12] nowhere tainty principles. uses the concept of simultaneous approximate mea- The crux of the problem is with the class II type of surement of non-commuting . But this is pairs appearing in . Does the uncer- the essence of Heisenberg's formulation [4] used in the tainty relation have any role in ME in these cases? We standard explanations. In particular, what happens to want to emphasize here, in the first instance, that all a state vector after an approximate measurement of so-called wave-particle complementarity experiments say , is not specified by QM. Interference is belong to class II, and QM does not guarantee the basically a property of the characteristic wavefunction claim of complementarity between arbitrarily defined in a double slit experiment. Any explanation of the 400 D. Sen et al. • Is the Uncertainty Relation at the Root of all Mutual Exclusiveness? 400 disappearance of the interference due to a measure- accuracy of the order of the slit width and the corre- ment without clarifying how the statevector is being sponding uncertainty in the slit momentum is always altered by the measurement, seems incomplete. In par- much greater than the momentum transfer in the scat- ticular, Heisenberg's uncertainty relation uses a pecu- tering process. Consequently the "welcher Weg" infor- liar mixture of classical and quantum concepts and in mation is not available when the do produce most cases offers a semi-ciassical explanation for some an interference pattern. On this point we also recall quantum phenomena. that Bacry [8] and Hauschildt [13] have criticised the The entire analysis here envisages classical Bohr-Feynman explanation invoking Heisenberg's like passage (see the next paragraph) of the electrons uncertainty principle in a similar vein. (even when they do interfere!) through the double slit, Incidentally, we also recall Englert et al.'s [14] anal- while a correct quantum mechanical explanation ysis of the double slit experiment in terms of the causal should be obtained in terms of the quantum mechan- interpretation of quantum mechanics as advanced by ical wavefunction (consistent with the boundary con- Böhm. Some interesting points and counterpoints re- ditions on the double slit surface) of the interfering garding this interpretation, in particular the "notion" beam plus the detector states - the concept of momen- of the slit through which the particle passes in the tum transfer being out of place. sense of causal interpretation and formal interpreta- The paths of (micro) particles in the Wilson cloud tion have, subsequently, been debated and discussed chamber, in the cathode ray oscillograph and in many by Dürr et al. [15] and again by Englert et al. [16]. other instruments can be excellently precalculated ac- Although the above investigations provide some clar- cording to the laws of and visually ification about the nature of Böhm trajectories vis-a- observed. A moving electron in a Wilson cloud cham- vis the double slit experiment they, however, do not ber actually leaves a very real cloud track. From an provide any clue to the understanding of the actual elementary analysis of quantum mechanical uncer- mechanism responsible for the mutual exclusiveness tainty relations one can show that an approximate in the so-called wave-particle complementarity. track with a finite transverse extension may be as- Invoking Heisenberg's uncertainty relation, alter- cribed to each microparticle of an ensemble. The con- natively, the electron beam is also considered as a cept of a microparticle trajectory is, therefore, only classical wave with a definite wavelength A = h/p. By meaningful when the lateral extension can be ne- an approximate measurement of the position, we alter glected compared to the dimension of the target. If a the situation only in that the wavelength of the beam slit of the same order of width(s) is placed across, the is being randomly altered, and hence the interference beam of microparticles will produce or in- pattern disappears as happens in the case of a classical terference pattern instead of uniform distribution. It is wave. It appears that the details of Storey et al.'s [2] erroneous to think that some actual path of the analysis (to be published elsewhere) can be replaced by microparticle does exist somewhere in a more or less this simpler argument. In double slit like experiments narrow spatial domain and limited range of momenta, "which path"-information is obtained by a detector, and Schrödinger (E. Schrödinger, Nobel Lecture, which interacts with the interfering particle and there Dec. 12, 1933) writes: "Admittedly the individual path is an energy exchange, i.e. a momentum exchange. It of a mass point loses its proper physical significance may also be argued that such experiments basically and becomes as fictitious as the individual isolated ray imply a position measurement (albeit approximate, of light". Accordingly, quantum mechanics does not say to the extent dx) of the particle and hence induces provide a framework in which we can talk about the an uncertainty in the momentum of the particle path of a particle in a meaningful way. The path prior dpx ~ h/dx. If dx d, the separation between the slits, to the detection of any physical entity can be retrod- one can show that the uncertainty in momentum and icted but it is a matter of personal belief whether such hence in /. is large enough to destroy the interference. an extrapolation of the past history can be ascribed to We think this picture is again not consistent with the any physical reality or not. quantum mechanical description of the phenomenon. Similarly in the archetypal example of a recoiling- The mental picture emerging from the analysis in double slit arrangement, following the Bohr-Feynman terms of Heisenberg's uncertainty relation is that each arguments [6, 10] we note that in order to observe the individual microparticle produces its own pattern, interference pattern, the slits must be localised with an and if we have the which-slit information "shifted in- D. Sen et al. • Is the Uncertainty Relation at the Root of all Mutual Exclusiveness? 401 terference patterns add together, washing out the the very outset assumes the particle-detector corre- fringes" [2]. This implies a thoroughly unacceptable lated state which together with a particular "outcome conclusion that superposition is not lost even after the of the detector measurement" [2] may well be regarded which-slit detection. In actual experiment we see that as the operational expression for the collapse hypoth- each registration of the micro-entity on the screen con- esis. In the remaining part of the description of arbi- forms to one of the patterns (either a continuous distri- trary welcher-Weg detectors Storey et al. have shown, bution or interference fringes) depending on whether using , that the maximum transferred the experimental set-up is providing the which-slit in- momentum for a perfectly efficient detector cannot be formation or not. Talking about the interference pat- less than that required by the appropriate quantum tern of individual particle/entity is meaningless. mechanical uncertainty relation. In other words, this From a detail and involved analysis of three types analysis provides a sort of inner consistency of the of interferometry experiments Hauschildt [13] has quantum mechanical formalism. However, we note concluded that the extinction of interference by a that, since the precise physical mechanism depends on "path measurement" is a consequence of a very ele- the choice of the detector basis, a Fourier analysis mentary assumption about the quantum formalism leading to the "momentum kick amplitude distribu- which is consistent with the projection postulate or tion" [2] is not always at work. The entire analysis has collapse hypothesis. From a purely quantum mechan- some significance only for some special detectors for ical analysis, Böhm [11] has also identified the collapse approximate . But BCP is defined for of wavefunction as the physical mechanism which en- unambiguous detection only. forces ME in the so-called wave-particle complemen- The analysis of Scully et al. [17] clearly shows that tarity experiments. If both the uncertainty principle the quantum mechanical formalism guarantees the and the collapse hypothesis are accepted as alternative validity of ME in experiments involving superposed mechanisms enforcing ME in these experiments, then states (class II types of complementarity). Whenever one must be implied in the other. But either of these one has which-slit information, the interference pat- conclusions is wrong. tern gets washed out. From an elaboration of this Our second criticism is that for the disappearance of analysis, we finally conclude [3] that only the quantum interference it is not always necessary that an ex- mechanical formulation in terms of the wavefunction change of energy will take place between the detector of the entire system including the detector states can and the particle. For example, if we put a 100% effi- present an unambiguous description of the which-slit cient detector close to slit 1 and record on the screen detection scheme, and the observed complementarity only those events for which the detector does not click, in the so-called wave-particle duality experiments is the result does not show any interference. In this case enforced by the quantum mechanical collapse hypoth- collapse of wavefunction offers the only explanation. esis. For a fixed double slit with an arbitrary welcher- Weg detector, Storey et al.'s preliminary analysis at

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