Abstract
The Heisenberg microscope and its analysis by Weizsäcker are used by Grete Hermann in her 1935 essay on the foundations of quantum mechanics to argue her claims about causality in quantum mechanics. In this chapter, I wish to draw a comparison between Hermann’s use of the Heisenberg microscope and another famous use of a very similar thought experiment : Bohr’s analysis of the suspended single slit in his reply to EPR . I shall argue that Hermann’s use of different aspects of the classical pictures in the treatment of the Heisenberg microscope makes her treatment closer to Bohr’s discussion of the suspended slit than to Heisenberg’s own treatment of the microscope. This suggests that Hermann, who equally gave an extensive analysis of complementarity in her 1935 essay, may be an especially acute interpreter of Bohr’s views. I conclude by looking at Hermann’s (and Bohr’s) approach in the context of more general examples of measurement and at possible limitations of the approach.
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Notes
- 1.
Cf. the classic textbook argument claiming to show that if ‘particles’ really went through the upper or lower slit, there would be no interference pattern in a two-slit experiment.
- 2.
In particular, it goes unmentioned in the correspondence between Bohr and Heisenberg in September 1935 on the latter’s reply to EPR . Hermann did visit Bohr’s institute in May 1935, however, and her work was discussed in a colloquium. In a letter of July 1936, Heisenberg replies to Hermann about an objection by Bohr relating to the movability of the cut, which Heisenberg, however, thinks should not affect Hermann’s considerations. For Heisenberg’s reply and his correspondence about it with Bohr, see Bacciagaluppi and Crull (2011), for his July 1936 letter to Hermann, see [Herrmann 2017, Part III, Letter 16].
- 3.
- 4.
Only near-maximal, because the slit has a finite width.
- 5.
This point is emphasised also by Pauli in a letter to Schrödinger of 9 July 1935, in which he describes Bohr’s (as yet unpublished) reply (Pauli 1985, pp. 419–422).
- 6.
In the Como lecture, [Bohr 1928, p. 580] writes: ‘the quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected. Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation’.
- 7.
Of course, insofar as we can see Hermann as explicating the concept of complementarity, the highlighting of analogies between Hermann and Bohr lends indirect support to Howard’s analysis in the first place (cf. also Chap. 10).
- 8.
Note that Pauli (unrigorously) treats also continuous observables as if they were discrete, but the treatment is meant to cover both.
- 9.
Since traditional quantum observables are self-adjoint operators, one can associate them one-to-one with their spectral measures, which are projection-valued measures (PVM) over the real line. Nowadays, observables are identified with more general positive-operator-valued measures (POVMs) . ‘Unsharp’ measurements of a traditional observable are represented by commutative POVMs (all of whose elements share the same spectral measure), while non-commutative POVMs can be thought of as a form of joint realisation of unsharp measurements of more than one traditional observable. For general and thorough treatments of modern measurement theory, see Busch et al. (1991) and Busch et al. (1997); for the last point about interpreting non-commutative POVMs, see Cattaneo et al. (1997).
- 10.
In modern measurement theory, the first stage corresponds to the most general measurements described using projection-valued measures, while the combination of the two stages corresponds already to a very special case of a POVM .
- 11.
More precisely, the case in which the condition is not satisfied with respect to any basis \(|u_n\rangle \) (otherwise we are trivially back to the preceding case).
- 12.
Note that precisely such a general case of preparation is actually described in the EPR paper, in their Eqs. (7) and (8), just before the special example of the EPR state (Einstein et al. 1935, p. 779). (This is somewhat ironic, given the conclusions we shall eventually arrive at below.).
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Bacciagaluppi, G. (2016). Bohr’s Slit and Hermann’s Microscope. In: Crull, E., Bacciagaluppi, G. (eds) Grete Hermann - Between Physics and Philosophy. Studies in History and Philosophy of Science, vol 42. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-0970-3_9
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