Abstract
We quickly review in this chapter the most relevant common features of quantum systems. Readers interested in a concise introduction to the physics of Quantum Mechanics (QM) will profit from [SaTu94]: putting aside the mathematical rigour, it discusses Dirac’s formulation of QM from a modern and smart perspective. Here the intention is to formalize in a simple way the ideas that will be developed in full in the subsequent chapters, after introducing the appropriate tools.
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- 1.
\(f : \mathbb R \to \mathbb C\), defined up to zero-measure sets, is the weak derivative of \(g \in L^2(\mathbb R, dx)\) if \(\int _{\mathbb R} g \frac {dh}{dx} dx = -\int _{\mathbb R} f h dx\) for every \(h \in C_c^\infty (\mathbb R)\). If g is differentiable, its standard derivative coincides with the weak one.
- 2.
The restriction should be defined so that it admits a unique selfadjoint extension. A sufficient requirement on \({\mathcal {S}}\) is that every Q(f) is essentially selfadjoint on it, see the next chapter.
- 3.
The factor ħ has to be added in the left-hand side of (1.14) if our unit system has ħ ≠ 1.
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Moretti, V. (2019). General Phenomenology of the Quantum World and Elementary Formalism. In: Fundamental Mathematical Structures of Quantum Theory. Springer, Cham. https://doi.org/10.1007/978-3-030-18346-2_1
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DOI: https://doi.org/10.1007/978-3-030-18346-2_1
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