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de Broglie–Bohm, delayed-choice double-slit experiment, and density matrix

James S. Bell

Published Jun 3, 2004• 11 citations

Abstract

I will try to interest you in the de Broglie–Bohm version of non-relativistic quantum mechanics. It is, in my opinion, very instructive. It is experimentally equivalent to the usual version insofar as the latter is unambiguous. But it does not require, in its very formulation, a vague division of the world into ‘system’ and ‘apparatus,’ nor of history into ‘measurement’ and ‘nonmeasurement.’ So it applies to the world at large, and not just to idealized laboratory procedures. Indeed the de Broglie–Bohm theory is sharp where the usual one is fuzzy, and general where the usual one is special. This is not a systematic exposition, but only an illustration of the ideas with a particularly nice example, and then some remarks on the role of the density matrix – in tribute to the title of this conference. No one more eloquently than John A. Wheeler has presented the delayed-choice double slit experiment. A pulsed particle source S (see Fig. 1) is so feeble that not more than one particle is emitted per pulse. The associated wave pulse B falls on a screen with slits 1 and 2. The transmitted pulses B′ are focussed by off-centred lenses into intersecting plane wave trains which fall finally on particle counters C 1 and C 2 – unless a photographic plate P is pushed into the region where the two wave trains interpenetrate. The decision, to interpose the plate or not, is made only after the pulse has passed the slits.

DOI: 10.1017/cbo9780511815676.016

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I will try to interest you in the de Broglie–Bohm version of non-relativistic quantum mechanics. It is, in my opinion, very instructive. It is experimentally equivalent to the usual version insofar as the latter is unambiguous. But it does not require, in its very formulation, a vague division of the world into ‘system’ and ‘apparatus,’ nor of history into ‘measurement’ and ‘nonmeasurement.’ So it applies to the world at large, and not just to idealized laboratory procedures. Indeed the de Broglie–Bohm theory is sharp where the usual one is fuzzy, and general where the usual one is special. This is not a systematic exposition, but only an illustration of the ideas with a particularly nice example, and then some remarks on the role of the density matrix – in tribute to the title of this conference. No one more eloquently than John A. Wheeler has presented the delayed-choice double slit experiment. A pulsed particle source S (see Fig. 1) is so feeble that not more than one particle is emitted per pulse. The associated wave pulse B falls on a screen with slits 1 and 2. The transmitted pulses B′ are focussed by off-centred lenses into intersecting plane wave trains which fall finally on particle counters C 1 and C 2 – unless a photographic plate P is pushed into the region where the two wave trains interpenetrate. The decision, to interpose the plate or not, is made only after the pulse has passed the slits.

The Bohm-de Broglie quantum mechanics has made possible to calculate the trajectories of electrons in a typical double-slit experiment[C. Philippidis et al., Il Nuovo Cimento, 52 B, 15-28 (1979)]. The trajectories do not correspond to an uniform movement but to an accelerated one. The acceleration is caused by the quantum potential. From the quantum theoretical point of view, the accelerated electrons should, with a certain probabil-ity, emit photons during its movement from the slits to the screen. According to the Copenhagen interpretation we found that this quantity is strictly zero because the electron moves as a free particle along its path after it leaves the slit and before reach the screen. Then, there is no emission of photons. On the other hand, by using the Bohm-de Broglie (BdB) approach we calculate the emission power, giving a general formula and which results, for a concrete real experiment, in a very tiny but not a vanishing value. We give an idea of the type of spectrum that could be measured. Therefore we have shown that the theoretical predictions of the usual quan-tum mechanics and that of Bohm-de Broglie’s interpretation are diﬀerent. An experiment could determine the correct prediction.

Based on the basic version of the Double-slit experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen — a result that would not be expected if light consisted of classical particles. Our new variant is as below: We shut one of the two parallel slits once the light passed the slits. Or we keep the two slits shutting and opening randomly in high speed. We believe this variant experiment could lead to a farther fundamental understanding of the Quantum Mechanics.

Abstract David Bohm published his “Suggested Interpretation of Quantum Theory in Terms of Hidden Variables” some twenty five years after Louis de Broglie first presented his similar Pilot Wave theory of quantum mechanics. In the following 30 years what became known as the de Broglie–Bohm approach to quantum theory was to a large extent ignored within the physics community. Even David Bohm himself became somewhat disillusioned with the lack of impact of his interpretation of quantum theory and he directed his interest elsewhere. But some 27 years after Bohm had published his interpretation of quantum theory, interest was rekindled in part by new, detailed calculations that demonstrated clearly and graphically, exactly how his interpretation explained quantum phenomena in terms of well defined individual particle trajectories. These computations encompassed two-slit interference, quantum tunnelling, neutron interferometry, Wheeler’s delayed choice experiment, orbital and intrinsic angular momentum, quantum measurement and Einstein–Podolsky–Rosen nonlocal correlations for orbital angular momentum, intrinsic angular momentum and correlated particle interferometry. Since then, the acceptance of the validity of de Broglie–Bohm theory has steadily grown, as has the interest in the consequences of the approach. For my contribution to the current celebratory volume I was asked to provide a personal review specifically of this novel work within its historical context of the last quarter of the twentieth century.