Abstract
In this chapter we have collected mathematical topics pertaining to quantum theory. The main subject is a systematic discussion of linear operators. It is supported by short articles on topological spaces, the Lebesgue integral, probability theory and on generalized functions, or distributions. Linear operators are mappings of a Hilbert space into itself. Its subspaces are characterized by projection operators. Of central importance are normal operators, such as unitary, self-adjoint, positive and probability operators. We also discuss functions of operators. Further topics are translations, the Fourier transform, position and momentum, ladder operators and transformation groups.
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- 1.
\(A\subset B\) includes the case \(A=B\).
- 2.
Here in the sense of non-negative.
- 3.
Causality usually refers to time, and we write t for the argument of the causal functions and \(\omega \) for the argument of its Fourier transform.
- 4.
In the sense of topology.
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Hertel, P. (2017). Mathematical Aspects. In: Quantum Theory and Statistical Thermodynamics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-58595-6_7
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DOI: https://doi.org/10.1007/978-3-319-58595-6_7
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58594-9
Online ISBN: 978-3-319-58595-6
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