11.6 For further reading
Concept links · terms present in this machine draft; source roles are unverified: spectral theorem
The use of periodic boundary conditions, or “putting the system in a , thus reducing the problem to that of Fourier series, is a conventional topic in quantum mechanics textbooks. Two good sources for the mathematics of Fourier series are [84] and [27]. The use of the Fourier transform to solve the free particle Schr¨odinger equation is a standard topic in physics textbooks, although the function space used is often not specified and distributions are not explicitly defined (although some discussion of the -function is always present). Standard mathematics textbooks discussing the Fourier transform and the theory of distributions are [89] and [27]. Lecture 6 in the notes on physics by Dolgachev [18] contains a more mathematically careful discussion of the sort of calculations with the -function described in this chapter and common in the physics literature.
For some insight into, and examples of, the problems that can appear when one ignores (as we generally do) the question of domains of operators such as the momentum operator, see [34]. Section 2.1 of [91] or Chapter 10 of [41] provide a formalism that includes a spectral theorem for unbounded self-adjoint operators, generalizing appropriately the spectral theorem of the finite dimensional state space case.
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