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No! ##\hat{D}(a)## is the crucial point and not a mere pedagogical tool. It selects the convenient Bloch energy eigenstates. As I showed in my previous posting the important point is the discrete translation symmetry of the (here simplified case of 1D) crystal lattice. Due to this symmetry there's a common eigenbasis of ##\hat{H}## and ##\hat{D}(a)##.You still emphasize the wrong point. D(a) is a pedagogical tool to get the intuition right. For a completely periodic potential, you will get a whole family of operators corresponding to D(na), where n is an integer. Any of them represents a symmetry of the system. Any complex [itex]\lambda[/itex] for D(a) will be a real [itex]\lambda[/itex] for D(na) for some n. The reason why you get wavefunctions that do not share the periodicity of the lattice is that all Bloch wave functions in a fully periodic potential need to be simultaneous eigenstates of the Hamiltonian and D(na) for some n, not necessarily of the Hamiltonian and D(a).

The reason, why here common eigenstates of ##\hat{H}## with a unitary operator, representing a symmetry transformation, are considered and not common eigenvectors of ##\hat{H}## and some self-adjoint operator(s) representing observable(s) is that here we deal with a discrete symmetry group and not a continuous Lie group. In the latter case the generators of the corresponding Lie algebra define conserved observables.

For our example the continuous case is that all ##\hat{D}(a)## (i.e., for all ##a \in \mathbb{R}##) are symmetry operators. After some (not too simple) analysis for a single particle in a Galilei-invariant theory it turns out that this is only fulfilled for a free particle, with the Hamiltonian fixed to ##\hat{H}=\hat{p}^2/(2m)##.

Of course to have only a discrete translation symmetry a periodic potential for the particle (electron) is also allowed, i.e., there's a much larger class of Hamiltonian fulfilling the symmetry. This is almost trivial: The less operations are symmetry operations the less restricted is the Hamiltonian by these symmetries.