Wave-Particle Duality || Unsharp Particle—Wave Duality in Double-Slit Experiments

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  • CHAPTER 10




    The quantum mechanical double-slit experiment was discussed from an epistemo-logical point of view in the Bohr-Einstein debate. (I) According to Bohr the particle-wave duality comes from the fact that the measuring instruments for the measurement of a particle (the path) and for the measurement of the wave (the interference pattern) are mutually exclusive. This was demonstrated by Bohr for many Gedanken experiments by means of the uncertainty relation between momentum p and position x. The same uncertainty relation can, however, also be used to show that in spite of the strict particle-wave duality, the path (momentum p) and the interference pattern (position x) can at least be measured simultaneously in an approximate sense. This surprising result was first demonstrated by Wootters and Zurek(2) within a quantum mechanical reformulation of one of the Gedanken experiments discussed by Bohr and Einstein. (I)

    A more rigorous treatment of unsharp quantum mechanical measurements must make use of the general theory of unsharp observables in quantum me-chanics, which replaces the sharp observables, which can be described by projector-valued measures, by effect-valued measures which correspond to un-sharp observables (see Busch(3. If in the double-slit experiment the observables of the path and the interference pattern are replaced by unsharp observables of this kind, unsharp joint measurements of the path (momentum p) and the interference pattern (position x) turn out to be possible. Unsharp measurements of this kind must not be confused with inaccurate ones since the unsharpness expresses the

    PETER MITTELSTAEDT Institut fur Theoretische Physik, Universitiit zu KOln, 0-5000 KOin 41, Germany.

    Wave-Particle Duality, edited by Franco Selleri. Plenum Press, New York, 1992.


    F. Selleri (ed.), Wave-Particle Duality Plenum Press, New York 1992


    objective indeterminacy of the path and of the interference pattern, but not the observer's ignorance about these properties.

    Several attempts have been made to realize unsharp joint measurements experimentally. A photon split-beam experiment has been performed using a modified Mach-Zehnder interferometer. (4) This experiment provides simul-taneous unsharp wave and particle knowledge in full agreement with the theoreti-cal predictions. More sophisticated experiments of this kind were proposed, but have not yet been realized. (5) Other kinds of experiments which were interpreted in the sense of unsharp joint measurements are neutron interference experiments, which were performed by several authors. (6--8) However, these experiments use a slightly different experimental setup which does not allow interpreting them as unsharp measurements, but rather as experiments with "unsharp preparation." This conceptual distinction can also be realized experimentally by extended experiments which are briefly discussed here.


    As an illustration of the Bohr-Einstein debate we consider the photon split-beam experiment shown in Figure 1. In the experimental setup, a Mach-Zehnder interferometer, the incoming photon-state 1'1') is split by the first beam splitter (BS\

    ,-/ y L

    I cBI 1 CA 2 L_~

    (85)2 ~ I I I C~f-- /' L ~


    (85)1 hp>


    FIGURE 1. Photon split-beam experiment. Beam splitters (BS)\ and (BS)2 with transparencies J3 = 112, 'Y = 112, two fully reflecting mirrors M\ and M2, a phase shifter, and two photon counters C\ and C2


    into two components described by the orthonormal states IB) and I-,B), respec-tively. The two parts of the split beam are reflected at the two mirrors M, and M2 and recombined with a phase difference 8 at the second beam splitter (BS)2' In this experiment there are two mutually exclusive measuring arrangements: If the photon counters C" C2 are in the position (q, q), one observes which way (B or -,B) the photon came, and if the counters are in the position (q, q), one observes the interference pattern, i.e., the intensities which depend on the phase difference 8.

    This experiment can be described completely within the framework of the two-dimensional Hilbert space 'JC2 The observables are then given by projection operators P(B) with eigenstates IB) and I-,B) (for the path) and P(A) with eigenstates IA) and I-,A) (for the interference pattern). The state I'P) of the incoming photon can be decomposed in the B-basis as I'P) = 1I'\I2(IB) + exp (i8)1-,B. Hence, the probabilities for the waysB and -,B are given by p('P, B) = 1('PIB)12 = 112 andp('P,-,B) = 1('PI-,B)12 = 112, res~ctively. The interference observable is given by P(A) with eigenstates IA) = ttV2(IB) + I-,B and I-,A) = ttV2(IB) - I-.B. Hence, the probabilities for A and -,A are given by p('P, A) = 112(1 + cos 8) and p('P, -,A) = 112(1 - cos 8), respectively. Obviously the observables P(B) and P(A) have a nonvanishing commutator [P(B), P(AL = 1I2{IB)( -,BI - I-,B)(BI} *' 0 and can thus not be measured simultaneously.

    Pure and mixed state operators of 'JC2 can be represented by means of a three-dimensional unit sphere, the Poincare sphere r;p (Figure 2). By using the Pauli operators {TI ,


    P( B) (particle) --..,-.--..,.-.~

    FIGURE 2. States and operators of the two-dimensional Hilbert space represented by the Poincare sphere. Initial preparation P['P], phase shift 8, path observable P(B), interference observable P(A). Mixed states W(Z) and WL('P, A) lie in the interior of the sphere.

    Mixed states are described by (positive trace one) operators W(Z) and are given by W(Z) = 1I2(Zi


    observable P(A) = 1I2(Aj U"j + TI) in the sense of von Neumann and Liiders consists of two steps (I) and (II)

    Step (I), the objectification, transforms the preparation P['P] into the Liiders mixture

    WL('P, A) = p('P, A)P(A) + p('P, -,A)P(-,A)

    whereas step (II) merely reduces the observer's subjective ignorance by reading the final result P(A), say, of the measuring process. For the above-mentioned preparation I'P) = 1I2(1B) + exp(i8)I-,B) one obtains either the mixture

    W L ( 'P, B) = 1I2P(B) + 1/2P( -,B)

    by measuring the path or the mixture

    WL('P, A) = 112(1 + cos'P)P(A) + 112(1 - cos'P)P(-,A)

    by measuring the interference pattern. In the Poincare sphere the mixed states WL('P, B) and WL('P, A.) are obtained

    geometrically by orthogonal projections of the vector q, onto the radius vectors ii and A, respectively. For a given angle 8 S.t. 8 = arccos.ffA the probabilities for P(A) and P( -,A) are p('P, A) = 112(1 + cos 8) and p('P, -,A) = 112(1 - cos 8), respectively (Figure 2).


    In contrast to the sharp first kind measurement described above, the final result of an unsharp measurement is not a projection operator P E C!P('/1e) but an effect operator E E ~('/1e). Effects are selfadjoint operators E, the spectrum Ell. of which lies within the interval [0,1]. In the two-dimensional Hilbert space '/1e2 effects are given by operators of the kind

    For the sake of simplicity we will restrict the discussion here to the special case a = 1. These special effect operators

    correspond to points in the interior of the Poincare sphere, and are formally


    equivalent to mixture operators W. To any effect there exists a counter effect E which is given by

    E = 11 - E = 112(11 - Xp)

    The spectral decomposition of an effect E(i..) of this kind reads

    E = 1I2(~(j + 11) = 112(1 + 1~I)p(A) + 112(1 - 1~1)p( - A)

    with projection operators

    The eigenvalues have the interpretation

    E+ = 112(1 + I~I) > 1!2-reality degree of property P(~) E- = 112(1 - I~I) < 1I2-unsharpness of property P(~)

    where we have assumed that E+ > 112 and E- < 112. By means of these considerations we are now prepared to describe the

    unsharp measuring process. An unsharp measurement of the observable P(A) = 112(1 + Aio) leads to the effect

    E(A,AA) = 1I2(AAAi



    FIGURE 3. Unsharp measuring process P(A) wiIh prepaIation P['Pj. The vector C lies in the plane spanned by Ihe vector


    and with eigenvalues E(A,AA) and E(B,AB), respectively. 1\vo arbitrary observ-abIes Ml and M2 are called coexistent, if the ranges on the M; are contained in the range of one joint observable M. For the unsharp observables E(A,AA) and E(B ,AB) this means that these two effects are coexistent iff

    If in particular Xjj = 0, this condition reads Ai + A~ ,.,; 1. The special observables P(A)-(of the interference pattern) and P(B)-(of

    the path) fulfill the relationXjj = O. Moreover, we consider here the limiting case of maximal coexistence Ai + A~ = 1. The corresponding effects E(A, AA) and E(B, AB) can then be estimated jointly by the ideal Liiders measurements of an observable

    P(C) = 1I2(C;0'; + ~) with Iq = 1 which fulfills the two conditions CiA; = AA and C;B; = AB. The observables P(C) with CX = AA lie on a circle CA and those with Cjj = AB on a circle CB' From the geometry of the Poincare sphere (see Figure 4) it is obvious that for values AA and AB with Ai + Ah = 1 there is exactly oneobservableP(C) for which both conditions are fulfilled. However, since C is neither in the plane spanned by qi and X nor in the

    P( 8) (particle) -~~--

    FIGURE 4. Poincare sphere descrip-tion of an unsharp joint measurement of the observables P(A) and P(A) by means of an i


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