Quantum Systems in Finite-Dimensional Hilbert Spaces

It is much more common than one might imagine that, for a given problem concerning a system with M degrees of freedom, the number of quantum states to consider is not the entire set \(|q_{1}\ldots q_{M}\rangle \) , but only a finite number of states \(|\varphi _{k}\rangle \) (each of which can obviously be expressed as a linear combination of the kets \(|q_{1}\ldots q_{M}\rangle \) ).