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Some Mathematical Foundations of Quantum Mechanics

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Abstract

Here we recall some mathematical basics of quantum mechanics about which there is no dispute. These fundamentals are equally relevant for all quantum mechanical theories (which are unfortunately—or rather, mistakenly—often referred to as interpretations). Our selection is also determined by our needs in later chapters.

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Notes

  1. 1.

    G. Galilei, Il Saggiatore, Capitolo VI. 1623. [Translation by authors.]

  2. 2.

    From ancient Greek, meaning the study of “that which is”.

  3. 3.

    Compare with the quote from Einstein at the beginning of Chap. 8.

  4. 4.

    Born had first thought of |ψ| as a candidate for a probability density.

  5. 5.

    In the sense of mathematical measure theory.

  6. 6.

    Strictly speaking, we should use Lebesgue’s theorem of dominated convergence here, because we exchange integration with taking a limit, but rigor does not bring new insights, so let’s ignore that here.

  7. 7.

    In an ideal situation, we might imagine sending only one particle through the slits each day …

  8. 8.

    …and that could take years.

  9. 9.

    D. Dürr and S. Teufel, Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, 2009.

  10. 10.

    The rather beautiful German word “unanschaulich” is often used to describe configuration space, but it has also been abused in the context of quantum mechanics to suggest that one cannot develop any coherent picture of what is going on in the microcosm.

  11. 11.

    It is often stated in quantum theory that it is of great philosophical importance that in Hilbert space , the space of wave functions, one can select an arbitrary basis in which to express the wave function in coordinates. So there is a position basis, a momentum basis, an energy basis, and whatever we want, and none of those bases is preferred. This is true mathematically. But every human being, even someone working in the field of quantum physics, lives and dies in position space, at least physically. So it is natural enough to find wave functions in the position representation particularly informative.

  12. 12.

    Mathematically trained students will soon realise that if an initial wave function has compact support the time evolved wave function (solving Schrödinger’s equation) will immediately have support equal to the whole of configuration space. But the function will be almost zero in most of configuration space. Hence all the following statements about disjoint supports and the associated picture can be taken with a grain of salt, i.e., they should be taken in the sense of physics and not pedantic mathematics.

  13. 13.

    Starting with a “product wave function” Ψ(q) = Ψ(x, y) = ψ i(x)Φ 0(y) expresses the idea of the physical independence of system and apparatus before they interact. In the case of no interaction V (x, y) ≈ 0, the Schrödinger equation (1.2) “factorises” into two independent equations, one for each factor of the product wave function. In that situation, the system and apparatus remain “physically independent”. However, it is important to note that V (x, y) ≈ 0 by itself does not imply physical independence because the wave function need not have product structure.

  14. 14.

    Note that a superposition of wave functions does not necessarily mean that interference takes place. By the same token, if a superposition does not lead to interference, i.e., decoherence takes place, that does not mean that there is no superposition.

  15. 15.

    The symbol 〈 ⋅ 〉 is a common way to denote a mean or average.

  16. 16.

    A warning within a warning: the interaction with the environment actually produces a measurement-like situation, as discussed in Chap. 2, i.e., we end up with a superposition of wave packets which are more or less spatially separated. So Chap. 2 is relevant for this too.

  17. 17.

    D. Drr and S. Teufel, Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, 2009.

  18. 18.

    Actually one would like to do the experiment with electrons, but silver atoms have the advantage of being electrically neutral while possessing a net magnetic moment, due to a single outer electron. Therefore, it is as good as an electron as far as spin is concerned. The fact that it is electrically neutral is also good, since the Lorentz force on charged particles does not then contribute.

  19. 19.

    We need to spell out carefully what is meant by this, but superficially we may consider that in a particular physical situation the speed of the electron is much less than the speed of light.

  20. 20.

    See, e.g., M. Nowakowski, The quantum mechanical current of the Pauli equation. Am. J. Phys. 67, 916–919 (1999), and for an overview, W.B. Hodge, S.V. Migirditch, W.C. Kerr, Electron spin and probability current density in quantum mechanics. Am. J. Phys. 82, 681–690 (2014).

  21. 21.

    For more on this, see S. Das and D. Dürr, Arrival time distributions of spin-1/2 particles. Sci. Rep. 9, 2242 (2019), and arXiv:1802.07141.

  22. 22.

    We talked above about spin-1/2. For our purposes, this particular value is merely an unimportant convention and has been absorbed in the gyromagnetic factor μ.

  23. 23.

    We shall often simply write L 2 for \(L^2(\mathbb {R}^n, \mathrm{d} ^n q)\).

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Dürr, D., Lazarovici, D. (2020). Some Mathematical Foundations of Quantum Mechanics. In: Understanding Quantum Mechanics . Springer, Cham. https://doi.org/10.1007/978-3-030-40068-2_1

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