Abstract
This chapter offers a reassessment, from the perspective of this study, of Schrödinger’s work, especially of his concept idea of quantum waves, as it extended from Max Born’s 1926 interpretation of Schrödinger’s wave or ψ-function in terms of probability to Bohr’s complementarity and then to the viewpoint of modern-day quantum information theory, especially in its Bayesian version. Section 5.1 gives a general introduction to the subject of quantum waves. Section 5.2 discusses those aspects of the old quantum theory and Heisenberg’s matrix mechanics that help place Schrödinger’s work in its proper historical context. Sections 5.3, 5.4, and 5.5 consider Schrödinger’s wave mechanics as a wave theory of subatomic processes. I close with a discussion of the concepts of quantum state, quantum entanglement, and quantum information, via the cat-paradox paper. These themes will be developed in Chapters 6, 7, and 8.
This is an extremely funny thing. The contrast between light-quanta and wave radiation can be traced even in the atom as: (a) single orbiting electrons; or (b) a standing vibration of the whole atomic region. In the interpretation (a) [we have]: the Hamiltonian partial differential equation; the separation of variables; and the quantization in the well-known manner. In the interpretation (b) [we have]: the wave equation; the separation of variables in the old, well-known sense that the unknown function is assumed to be, e.g., the product of a function of [radial coordinate] r, of a function of [the angle] θ, and a function of [the angle] φ; searching for the “normal vibrations.” The frequencies emitted [by atoms] appear as frequency differences, i.e., as beat frequencies of the normal frequencies. Something must be hidden behind that. I hope to formulate the results soon in an organized way.
—(Erwin Schrödinger, A Letter to Wilhelm Wien, January 8, 1926)
[The ψ-function] is now the means for predicting the probability of measurement results. In it is embodied the momentarily-attained sum of theoretically based future expectation, somewhat as laid down in a catalogue.
—Erwin Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik” [The Present Situation in Quantum Mechanics] (Schrödinger 1935a, p. 158)
If two separate bodies, each by itself known maximally, enter the situation in which they influence each other, and separate again, then there occurs regularly that which I [call] entanglement [Verschränkung] of our knowledge of the two bodies.
—Erwin Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik” [The Present Situation in Quantum Mechanics] (Schrödinger 1935a, p. 161)
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Notes
- 1.
For Schrödinger’s collected papers on wave mechanics, see Schrödinger (1928), and for his other important papers on quantum mechanics, see Schrödinger (1995). This chapter cannot pretend and does not aim to offer a comprehensive treatment of Schrödinger’s thought, which, it may be observed, was given rather less attention in literature than that of Bohr and Heisenberg, or Einstein (even as concerns his engagement with quantum theory). Arguably the best available study is Bitbol (1996). On the other hand, in Mehra and Rechenberg’s treatise Schrödinger is given more space than any other founding figure (MR 5), and my analysis in this chapter is indebted to their discussion of Schrödinger. As in earlier chapters, however, I am less in accord with and, at several key junctures, shall contest their philosophical argumentation.
- 2.
As explained earlier, such traces are dot-like only at a low resolution, which “masks” a very complex physical object, composed of millions of atoms, and particle-like only in the sense that a classical object idealized as a particle would leave a similar trace. See, again (Ulfbeck and Bohr 2001) and (Bohr et al. 2004).
- 3.
The investigation of these rules and their interpretations would require a separate discussion, which is beyond my scope and is not germane for my argument in this study.
- 4.
On Weyl’s assistance to Schrödinger, see Mehra and Rechenberg (MR5, pp. 484–485).
- 5.
The idea is by no means dead. Thus, J. S. Bell was an ardent supporter of quantum “beables,” mostly along the lines of de Broglie–Bohm’s thinking (e.g., Bell 1987, p. 173).
- 6.
It is not clear to what degree Einstein realized that Bohr’s approach was not grounded in wave-particle complementarity, especially in Bohr’s post-EPR writings. Einstein does not appear to have been at ease with the concept of complementarity (in the narrow sense) and says at one point that he was “unable to attain …the sharp formulation … [of] Bohr’s principle of complementarity” (Einstein 1949b, p. 674).
- 7.
Once can hardly miss yet another parallel here, that between his formula for S and his fellow-Viennese Ludwig Boltzmann’s famous formula for entropy (inscribed on Boltzmann’s gravestone), which could have suggested at least certain formal connections between waves and entropy and hence between information and probability.
- 8.
Archive for the History of Quantum Physics Microfilm No. 40, Section 6, p. 1 (cited as Notebook II, p. 1 in MR 5, p. 543).
- 9.
See Mehra and Rechenberg’s discussion (MR 5, pp. 570–571).
- 10.
As is clear from the last section of his cat-paradox paper, Schrödinger was as suspicious of quantum field theory as of quantum mechanics (Schrödinger 1935a, QTM, p. 167).
- 11.
As emphasized throughout this study, these considerations do not mean that an alternative formalism for such predictions is not possible, which, however, does not bear on my argument here, since Schrödinger’s formalism is that of the standard quantum mechanics.
- 12.
It may be noted that, in this view, the problem of time-reversibility of physical processes—or, rather, their description (there is no actual physical evidence thus far that any actual physical processes are time-reversible)—found in classical physics or relativity does not arise in quantum mechanics. One might see (and many do) the time-dependent Schrödinger’s equation as, mathematically, time-reversible, although this reversal involves a complex conjugation of variables. This fact already poses difficulties (since in principle such a procedure implies that the resulting equation refers to a different “time-reversed” quantum system than the original one), customarily disregarded by those who argue the case. In any event, in the present view, Schrödinger’s equation does not offer any physical description but only, in each case, a probabilistically predictive algorithm concerning the outcome of future experiments.
- 13.
The paper appeared in German in three parts in 1935 in Die Naturwissenschaften and was accompanied by two related articles in English (Schrödinger 1935b, 1936).
- 14.
See, again, Fuchs (2001), (2003); Caves et al. (2007). The program these works pursue is based on the so-called POVMs (Positive Operator Value Measures), rather than on the type of measurements considered in this study, which correspond to ideal cases defined by wave functions or “pure states.” In von Neumann’s version of the formalism, the latter measurements are associated with “projection operators” or “projectors” in the corresponding Hilbert spaces. For those who are unfamiliar with Hilbert spaces, this operation is analogous to a projection of vectors in a two-dimensional plane onto a coordinate axis defined by a given vector. POVMs are analogous to density operators or matrices corresponding to the so-called “mixed states,” as opposed to “pure states,” which are vectors rather than matrices. Unlike a mixed state, a pure state cannot be represented by a mixture (convex combination) of other states. Pure states correspond to ideal experiments (which presuppose ideally functioning instruments and the absence of outside interferences in the course of such experiments). In such cases, one could always assign a definite probability to a given outcome, but, again, only a probability, rather than have it strictly determined as in classical mechanics. Actual quantum experiments require using mixed states and density operators, or POVMs, for predicting their outcomes. These qualifications do not affect my argument here or elsewhere in this study. In the present view, all quantum states are mathematical objects that, supplemented by Born’s or equivalent rules, enable our expectation catalogues concerning the outcomes of quantum experiments. Indeed, it is crucial that this view is applied here even to the (ideal) case of pure states, since it could be more expected for mixed states and in fact appears less “distressing” (more “realist” and “objective”). This understanding of pure states corresponds to the view that idealized causal models of the type used in classical mechanics are impossible in considering quantum phenomena.
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Plotnitsky, A. (2010). Schrödinger’s Waves: Propagation and Probability. In: Epistemology and Probability. Fundamental Theories of Physics, vol 161. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85334-5_5
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