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Testing wave function collapse and the complementarity principle using neutron self-interference and tunneling
Physica B 174 (1991) 403-405 North-Holland
Testing wave function collapse and the complementarity principle using neutron self-interference and tunneling Partha G h o s e S.N. Bose National Centre for Basic Sciences, DB-17, Sector-I, Salt Lake, Calcutta 700 064, India
Dipankar Home Physics Department, Bose Institute, Calcutta 700 009, India
We propose two different experiments (A and B) to test wave function collapse and the complementarity principle using neutron self-interference and tunneling, respectively.
1. Introduction The concept of wave function collapse and the complementarity principle are two of the most fundamental ingredients of quantum mechanics. There have been recent experiments [1] using neutron interferometry and other methods which give partial 'wave knowledge' and partial 'particle knowledge' (i.e. unsharp particle and wavelike properties) in terms of the 'which path' information and the corresponding contrast of the interference pattern. This already goes beyond the usual discussions of the complementarity principle implying mutual exclusiveness between complete 'particle-knowledge' and complete 'wave-knowledge'. In this paper we propose an experiment (B) which confronts the complementarity principle in a sharper way. We also propose another experiment (A) to test wave function collapse in the following sense: if one treats the system-apparatus combination quantum mechanically, then because of the assumed orthogonality of the macroscopically distinct states of the measuring apparatus, the reduced density operator of the system becomes diagonal. This quenching of the off-diagonal elements of the density operator of the system is
one necessary component of wave function collapse, the other component being the reduction of the system-apparatus pure state to a mixed state. It is only the first component of wave function collapse that our proposed experiment (A) can clearly test.
2. Experiment (A) An experiment that can test this type of effect on the wave function of a single neutron is a variant of the neutron interferometry experiment of Summhammer et al. [2] who introduced first a static partial absorber and then a time-dependent one with the same mean transmission efficiency into one of the interfering beams. The amplitudes of the resulting interference patterns were found to be different although the mean intensities were the same. Instead of replacing the static absorber by a time-dependent one, we suggest replacing it by a perfect neutron absorber with a hole of appropriate dimension such that the mean transmission efficiency remains the same but the neutrons that go through it do not suffer any scattering or diffraction.
0921-4526/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
P. Ghose, D. Home / Wave function collapse and the complementarity principle
404
If the static absorber of mean transmission efficiency b is inserted into, say, the left beam, the expected final intensity will be 11 = IV UL + U R eiX'12 = b[ULI 2 +
IURI 2 Jr 2x/-bULUR COSX',
(1)
where X' is the relative phase shift between the two coherent beams UL and U R. When the static partial absorber is replaced by a perfect absorber with a hole, then U L = UL(a) + UL(t ), where UL(a) is the absorbed part of the wave function and UL(t ) is the transmitted part; the transverse spread of UL(t ) is smaller than that of U L, being determined by the hole size which is so adjfisted that the mean transmission probability is again b. The expected intensity in the forward direction (•2) can then be obtained from the final total wave function Uf given by Uf = x/b(UL(t) + U R e'X)~o(a) + x/1 - b(UL(a) + UR)~e(a),
(2)
where qte(a) and ~0(a) are, respectively, the wave functions of the apparatus corresponding to neutron absorption and no absorption. Hence, I2 = blUL(t) + Ur~ e~XI2+ (1 - b)IURI z = blUL(t)l 2 + l u l l 2 + 2 b U L ( t ) U a cos X,
(3)
provided we assume that ~e(a) and ~0(a) are macroscopically distinguishable and orthogonal so that the neutron wave function can be said to collapse from the coherent sum (U L + U R e ix) to an incoherent mixture of (UL(t) + U r e ix) and U a with weigh factors b and (1 - b), respectively (remembering that UL(a) is the absorbed part which is not available subsequently). A test of eq. (3) can therefore be interpreted as evidence of collapse in the sense of this loss of coherence of the neutron wave function. It is curious that a time-dependent absorber (e.g., a periodic or random chopper) is found to produce a pattern similar to 12. We must emphasize, however, that the physics of the two situations are different. With a time-dependent chopper one is actually
performing two different experiments alternately. In one, where the chopper blocks the left beam, there is no question of any interference. In the other, both beams are completely unobstructed and one should see the full interference effect. The resultant pattern observed must therefore be viewed as the time averaged effect of these two independent patterns with appropriate weightages [3]. In contrast, the experiment with the hole discussed here is a single arrangement. It should be noted that the experiment with a Cd lattice [2] may appear, in principle, to be similar to the experiment with a hole in a perfect absorber. However, the hole arrangement is simpler than the lattice and more convenient for the purpose of clarifying the conceptual significance of such experiments. (In fact, there are departures from /2, i.e., eq. (3) in the observed interference pattern when the Cd lattice is rotated around the beam axis. The lattice experiment was conceived primarily to study the intermediate situations between I 1 and I2. )
3. Experiment (B) This is a variant of the microwave experiment carried out by J.C. Bose in 1897 [4] with two prisms placed opposite each other. With a large gap between them he detected only total internal reflection in the first prism. As the gap was reduced and made smaller than the wavelength, the total internal reflection was 'frustrated' and Bose found that the radiation could partially tunnel through the gap. This was a striking demonstration of the wave nature of the radiation he had discovered. We propose an analogous experiment using neutron optics with a source producing one neutron at a time. The neutrons internally reflected by the first 'prism' can be detected by a counter-1 and those tunneling through the gap between the 'prisms' by a counter-2. With the gap larger than the neutron wavelength, only counter-1 should click. As the gap is reduced to less than the wavelength, the two counters should click in perfect anticoincidence implying both particle-like (anticoinci-
P. Ghose, D. Home / Wave function collapse and the complementarity principle
dence) and wave-like (tunneling) propagation of neutrons. What conceptually distinguishes this experiment from 'which path' ('welcher Weg') experiments is that tunneling (wave-like propagation) rather than interference in conjunction with perfect anticoincidence (particle-like propagation) rather than 'which path' information implies simultaneously sharp particle and wave-like properties. In fact, in the proposed experiment one can label each neutron registered in one of the two counters as coming after transmission through the gap or after internal reflection (analogous to 'which path' information) and at the same time the ratio of the numbers of transmitted and internally reflected neutrons displays a wave-like property. This is irreconcilable with the usual formulation of the complementarity principle but is consistent with both the Einstein-de Broglie version of wave-particle dualism [5] and the view-point advocated by Heisenberg [6] who wrote in 1959: " . . . the concept of complementarity introduced by Bohr into the interpretation of quantum theory has encouraged the physicists to use an ambiguous rather than an unambiguous language, to use the classical concepts in a somewhat vague manner in conformity with the principle of uncertainty . . . . When this vague and unsystematic use of the language leads into difficulties, the physicist has to withdraw into the mathematical scheme and its unambiguous correlation with the experimental facts."
405
4. Concluding remarks It is a tribute to modern technology that we are able in a conference on neutron scattering to propose feasible experiments which can put some of the long-standing foundational issues of quantum mechanics to critical test.
Acknowledgement One of us (D.H.) acknowledges the financial support provided by the Department of Science and Technology (Government of India).
References [1] H. Rauch, in: Proc. 3rd Int. Syrup. on Foundations of Quantum Mechanics in the Light of New Technology, eds. S. Kobayashi et al. (The Physical Society of Japan, Tokyo, 1990). P. Mittelstaedt et al., Found. Phys. 17 (1987) 891. [2] J. Summhammer, H. Rauch and D. Tuppinger, Phys. Rev. A 36 (1987) 4447. [3] D. Home and P.N. Kaloyerou, J. Phys. A 22 (1989) 3253. [4] J.C. Bose, in: Collected Physical Papers (Longmans, Green and Co., London, 1927) pp. 44-49. [5] F. Selleri, Quantum Paradoxes and Physical Reality (Kluwer, Dordrecht, 1990) Ch. 3. [6] W. Heisenberg, Physics and Philosophy (Harper and Row, New York, 1959) p. 179.
Testing wave function collapse and the complementarity principle using neutron self-interference and tunneling Partha G h o s e S.N. Bose National Centre for Basic Sciences, DB-17, Sector-I, Salt Lake, Calcutta 700 064, India
Dipankar Home Physics Department, Bose Institute, Calcutta 700 009, India
We propose two different experiments (A and B) to test wave function collapse and the complementarity principle using neutron self-interference and tunneling, respectively.
1. Introduction The concept of wave function collapse and the complementarity principle are two of the most fundamental ingredients of quantum mechanics. There have been recent experiments [1] using neutron interferometry and other methods which give partial 'wave knowledge' and partial 'particle knowledge' (i.e. unsharp particle and wavelike properties) in terms of the 'which path' information and the corresponding contrast of the interference pattern. This already goes beyond the usual discussions of the complementarity principle implying mutual exclusiveness between complete 'particle-knowledge' and complete 'wave-knowledge'. In this paper we propose an experiment (B) which confronts the complementarity principle in a sharper way. We also propose another experiment (A) to test wave function collapse in the following sense: if one treats the system-apparatus combination quantum mechanically, then because of the assumed orthogonality of the macroscopically distinct states of the measuring apparatus, the reduced density operator of the system becomes diagonal. This quenching of the off-diagonal elements of the density operator of the system is
one necessary component of wave function collapse, the other component being the reduction of the system-apparatus pure state to a mixed state. It is only the first component of wave function collapse that our proposed experiment (A) can clearly test.
2. Experiment (A) An experiment that can test this type of effect on the wave function of a single neutron is a variant of the neutron interferometry experiment of Summhammer et al. [2] who introduced first a static partial absorber and then a time-dependent one with the same mean transmission efficiency into one of the interfering beams. The amplitudes of the resulting interference patterns were found to be different although the mean intensities were the same. Instead of replacing the static absorber by a time-dependent one, we suggest replacing it by a perfect neutron absorber with a hole of appropriate dimension such that the mean transmission efficiency remains the same but the neutrons that go through it do not suffer any scattering or diffraction.
0921-4526/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
P. Ghose, D. Home / Wave function collapse and the complementarity principle
404
If the static absorber of mean transmission efficiency b is inserted into, say, the left beam, the expected final intensity will be 11 = IV UL + U R eiX'12 = b[ULI 2 +
IURI 2 Jr 2x/-bULUR COSX',
(1)
where X' is the relative phase shift between the two coherent beams UL and U R. When the static partial absorber is replaced by a perfect absorber with a hole, then U L = UL(a) + UL(t ), where UL(a) is the absorbed part of the wave function and UL(t ) is the transmitted part; the transverse spread of UL(t ) is smaller than that of U L, being determined by the hole size which is so adjfisted that the mean transmission probability is again b. The expected intensity in the forward direction (•2) can then be obtained from the final total wave function Uf given by Uf = x/b(UL(t) + U R e'X)~o(a) + x/1 - b(UL(a) + UR)~e(a),
(2)
where qte(a) and ~0(a) are, respectively, the wave functions of the apparatus corresponding to neutron absorption and no absorption. Hence, I2 = blUL(t) + Ur~ e~XI2+ (1 - b)IURI z = blUL(t)l 2 + l u l l 2 + 2 b U L ( t ) U a cos X,
(3)
provided we assume that ~e(a) and ~0(a) are macroscopically distinguishable and orthogonal so that the neutron wave function can be said to collapse from the coherent sum (U L + U R e ix) to an incoherent mixture of (UL(t) + U r e ix) and U a with weigh factors b and (1 - b), respectively (remembering that UL(a) is the absorbed part which is not available subsequently). A test of eq. (3) can therefore be interpreted as evidence of collapse in the sense of this loss of coherence of the neutron wave function. It is curious that a time-dependent absorber (e.g., a periodic or random chopper) is found to produce a pattern similar to 12. We must emphasize, however, that the physics of the two situations are different. With a time-dependent chopper one is actually
performing two different experiments alternately. In one, where the chopper blocks the left beam, there is no question of any interference. In the other, both beams are completely unobstructed and one should see the full interference effect. The resultant pattern observed must therefore be viewed as the time averaged effect of these two independent patterns with appropriate weightages [3]. In contrast, the experiment with the hole discussed here is a single arrangement. It should be noted that the experiment with a Cd lattice [2] may appear, in principle, to be similar to the experiment with a hole in a perfect absorber. However, the hole arrangement is simpler than the lattice and more convenient for the purpose of clarifying the conceptual significance of such experiments. (In fact, there are departures from /2, i.e., eq. (3) in the observed interference pattern when the Cd lattice is rotated around the beam axis. The lattice experiment was conceived primarily to study the intermediate situations between I 1 and I2. )
3. Experiment (B) This is a variant of the microwave experiment carried out by J.C. Bose in 1897 [4] with two prisms placed opposite each other. With a large gap between them he detected only total internal reflection in the first prism. As the gap was reduced and made smaller than the wavelength, the total internal reflection was 'frustrated' and Bose found that the radiation could partially tunnel through the gap. This was a striking demonstration of the wave nature of the radiation he had discovered. We propose an analogous experiment using neutron optics with a source producing one neutron at a time. The neutrons internally reflected by the first 'prism' can be detected by a counter-1 and those tunneling through the gap between the 'prisms' by a counter-2. With the gap larger than the neutron wavelength, only counter-1 should click. As the gap is reduced to less than the wavelength, the two counters should click in perfect anticoincidence implying both particle-like (anticoinci-
P. Ghose, D. Home / Wave function collapse and the complementarity principle
dence) and wave-like (tunneling) propagation of neutrons. What conceptually distinguishes this experiment from 'which path' ('welcher Weg') experiments is that tunneling (wave-like propagation) rather than interference in conjunction with perfect anticoincidence (particle-like propagation) rather than 'which path' information implies simultaneously sharp particle and wave-like properties. In fact, in the proposed experiment one can label each neutron registered in one of the two counters as coming after transmission through the gap or after internal reflection (analogous to 'which path' information) and at the same time the ratio of the numbers of transmitted and internally reflected neutrons displays a wave-like property. This is irreconcilable with the usual formulation of the complementarity principle but is consistent with both the Einstein-de Broglie version of wave-particle dualism [5] and the view-point advocated by Heisenberg [6] who wrote in 1959: " . . . the concept of complementarity introduced by Bohr into the interpretation of quantum theory has encouraged the physicists to use an ambiguous rather than an unambiguous language, to use the classical concepts in a somewhat vague manner in conformity with the principle of uncertainty . . . . When this vague and unsystematic use of the language leads into difficulties, the physicist has to withdraw into the mathematical scheme and its unambiguous correlation with the experimental facts."
405
4. Concluding remarks It is a tribute to modern technology that we are able in a conference on neutron scattering to propose feasible experiments which can put some of the long-standing foundational issues of quantum mechanics to critical test.
Acknowledgement One of us (D.H.) acknowledges the financial support provided by the Department of Science and Technology (Government of India).
References [1] H. Rauch, in: Proc. 3rd Int. Syrup. on Foundations of Quantum Mechanics in the Light of New Technology, eds. S. Kobayashi et al. (The Physical Society of Japan, Tokyo, 1990). P. Mittelstaedt et al., Found. Phys. 17 (1987) 891. [2] J. Summhammer, H. Rauch and D. Tuppinger, Phys. Rev. A 36 (1987) 4447. [3] D. Home and P.N. Kaloyerou, J. Phys. A 22 (1989) 3253. [4] J.C. Bose, in: Collected Physical Papers (Longmans, Green and Co., London, 1927) pp. 44-49. [5] F. Selleri, Quantum Paradoxes and Physical Reality (Kluwer, Dordrecht, 1990) Ch. 3. [6] W. Heisenberg, Physics and Philosophy (Harper and Row, New York, 1959) p. 179.