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Confinement states of quantum wells and superlattices

In the case of the confinement state the operator kz is replaced by the operator -i∂/∂z. When dealing with quantum wells the wave function should tend to zero at large z:
and the Hamiltonian can be written in the symmetrized form that takes into account interface boundary conditions, the conservation of flux:
For superlattices the band edge profile is periodic with periodicity d which imply the following condition:
where q is the wave vector of the superlattice in z direction. The Hamiltonian has the form

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