Floquet eigenfunctions

This shows that the time evolution is exactly like that of a stationary state of a fime-independenf Hamiltonian, provided the probing is limited to T, or any multiple of T. Since, (0)) is equal fo 4> (0)), Eq. (27) also shows fhaf exp ( - iEiT/h) is an eigenvalue of fhe evolution operator over one period of the field. Suppose now thaf we wish to follow fhe developmenf in fime of an arbitrary initial wavepacket rj 0). We can expand it over the complete set of Floquet eigenfunctions of a given Brillouin zone af fime f = 0 ... 

This relation emphasizes again the similarity between the properties of Floquet eigenfunctions and the ordinary stationary eigenfunctions (see Ref. [35] for more details). Although it has been proven here only for stroboscopic times, Eq. (29) may be written more generally by changing NT into an arbitrary t. 

We notice that we have obtained the Floquet eigenfunctions... 

By using this operator, one can in principle know the time development of an arbitrary initial wave function. However, we are rather interested in obtaining the Floquet quasi-energies and eigenfunctions for our Hamiltonian, i.e., we are looking for functions f) that satisfy equation (46). Bearing this in mind, we propose the following initial function ... 

The universal structural feature of perfect crystals is their periodicity. An immediate consequence of this periodicity is the Bloch-Floquet theorem stating that the one-electron eigenfunctions are of the form... 

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