Abstract
We showed in the previous chapter that the Bohmian velocity of a particle cannot be measured according to (7.24). But by now there are many measurement experiments which report on successful measurements of Bohmian velocities. So what is going on? The answer is that these measurements are not of the form so far discussed. The new way of measuring is called weak measurement. It is a measurement in which the wave function of the measured particle (system) is only weakly disturbed. The theory of weak measurements has been developed quite generally for all observables, but for didactic reasons, we shall stick to weak measurements of position, which are relevant for experiments to measure trajectories. And since the measured trajectories are Bohmian, we shall remain for the moment in the realm of Bohmian mechanics.
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Notes
- 1.
W. Heisenberg, Der Teil und das Ganze: Gesprche im Umkreis der Atomphysik. Piper, 1996, S. 80. Translation by the authors.
- 2.
Note that the replacement [see (8.7)] comes with an error 1∕σ and the reader might wonder whether a more careful argument on the order of τ is needed. Indeed it is, but for ease of presentation we leave it as is and refer to D. Dürr, S. Goldstein, and N. Zanghì, On the Weak Measurement of Velocity in Bohmian Mechanics. In: Quantum Physics Without Quantum Philosophy, Springer, 2013, Sect. 7.3, for a more detailed analysis.
- 3.
S. Kocsis, B. Braverman, S. Ravets, M.J. Stevens, R.P. Mirin, L.K. Shalm, and A.M. Steinberg, Observing the average trajectories of single photons in a two-slit interferometer. Science 332, 1170–1173 (2011).
- 4.
H.M. Wiseman: Grounding Bohmian mechanics in weak values and Bayesianism. New Journal of Physics 9, 165 (2007). The experiment referred to in footnote 3 is actually based on Wiseman’s suggestion.
- 5.
Unfortunately, there is no agreement about what the wave function of a photon is. We ignore here this “little” detail for the sake of argument.
- 6.
See, for instance, L. Vaidmann, Past of a quantum particle. Physical Review A 87, 052104 (2013).
- 7.
B.-G. Englert, M.O. Scully, G. Süssmann, and H. Walther, Surrealistic Bohm trajectories. Zeitschrift für Naturforschung 47a, 1175 (1992).
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Dürr, D., Lazarovici, D. (2020). Weak Measurements of Trajectories. In: Understanding Quantum Mechanics . Springer, Cham. https://doi.org/10.1007/978-3-030-40068-2_8
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