Wave function
Common Hilbert spaces

While the space of solutions as a whole is a Hilbert space there are many other Hilbert spaces that commonly occur as ingredients.

More generally, one may consider a unified treatment of all second order polynomial solutions to the Sturm–Liouville equations in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well as Chebyshev polynomials, Jacobi polynomials and Hermite polynomials. All of these actually appear in physical problems, the latter ones in the harmonic oscillator, and what is otherwise a bewildering maze of properties of special functions becomes an organized body of facts. For this, see Byron & Fuller (1992, Chapter 5).

There occurs also finite-dimensional Hilbert spaces. The space C is a Hilbert space of dimension n. The inner product is the standard inner product on these spaces. In it, the "spin part" of a single particle wave function resides.

With more particles, the situations is more complicated. One has to employ tensor products and use representation theory of the symmetry groups involved (the rotation group and the Lorentz group respectively) to extract from the tensor product the spaces in which the (total) spin wave functions reside. (Further problems arise in the relativistic case unless the particles are free. See the Bethe–Salpeter equation.) Corresponding remarks apply to the concept of isospin, for which the symmetry group is SU(2). The models of the nuclear forces of the sixties (still useful today, see nuclear force) used the symmetry group SU(3). In this case, as well, the part of the wave functions corresponding to the inner symmetries reside in some C or subspaces of tensor products of such spaces.

Due to the infinite-dimensional nature of the system, the appropriate mathematical tools are objects of study in functional analysis.
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