Final answer:
To solve the tight binding Hamiltonian for a charged particle in an electric field with perturbation Δ, one would find first-order eigenstates as slightly delocalized combinations of adjacent site states and second-order eigenvalues as corrected energy levels proportional to Δ².
Step-by-step explanation:
The question pertains to the tight binding Hamiltonian for a charged particle in an electric field, treated within quantum mechanics. The Hamiltonian operator H indicates the total energy of the system and includes a term for the electric potential energy due to the electric field E, the charge of the particle e, and the distance between lattice sites a. The wavefunction ψ represents the quantum state of the charged particle. When introducing a small perturbation Δ, that enables hopping between adjacent sites, we employ perturbation theory.
First, to find the eigenstates of H to first order in Δ, one would begin with the unperturbed Hamiltonian, whose eigenstates are simply the localized states |n⟩. Then, to incorporate Δ as a perturbation, it would introduce a mixture between states |n⟩ and |n+1⟩, which to first order would slightly delocalize the eigenstates.
For the eigenvalues to the second order in Δ, second-order perturbation theory needs to be applied. This requires calculating matrix elements involving the perturbation Δ and summing over all possible intermediate states. The result will be a correction to the eigenvalues that includes terms proportional to Δ².
Both the eigenstates and the eigenvalues are essential for understanding the quantum mechanical behavior of a charged particle in a lattice under the influence of an electric field. The eigenstates provide information about the localization and delocalization of the particle, whereas the eigenvalues give information about the energy levels, which will be shifted due to the perturbation.