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A one-electron or mean-field Hamiltonian with lattice-periodic coefficients used to organize electron states in a crystal. Electron correlation, disorder, finite boundaries, and lattice motion are omitted or encoded only effectively.
wp-physics-bloch-theoremwp-physics-band-structuretong-solid-state-physicsmit-theory-of-solidsEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
A periodic one-electron or mean-field Hamiltonian models electrons in an ideal crystal after the ionic structure and many-electron effects are frozen or encoded effectively.
The target is a model, idealization, restricted ansatz, approximation scheme, phenomenological fit, or historical representation used for the source system or theory. It need not have a universal small error parameter and is not thereby a controlled limit or effective theory. The edge label and detail must identify the precise status.
wp-physics-bloch-theoremwp-physics-band-structuretong-solid-state-physicsmit-theory-of-solidsThis is an authored directed relation from the source endpoint to the target endpoint.
A periodic-electron model uses the nonrelativistic Schrödinger eigenvalue problem with a lattice-periodic one-particle or mean-field Hamiltonian. Correlation, relativistic, disorder, and lattice-motion effects require extensions or different effective descriptions.
The source supplies a substantive formal ingredient, dynamical sector, law, field content, or mechanism of the target theoretical framework.
wp-physics-schrodinger_equationwp-physics-bloch-theoremtong-solid-state-physicsmit-theory-of-solidsEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
For a single-particle Hamiltonian invariant under Bravais-lattice translations, simultaneous diagonalization of the commuting translation operators allows energy eigenstates to be chosen in Bloch form. Degeneracies do not invalidate the existence statement.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-physics-bloch-theoremtong-solid-state-physicsmit-theory-of-solidsThis is an authored directed relation from the source endpoint to the target endpoint.
Solving the periodic Hamiltonian in each crystal-momentum sector yields energy bands; the result belongs to the selected one-particle or mean-field model.
The target represents a state, property, observable, or state-dependent description associated with the source system or theory.
wp-physics-band-structurewp-physics-bloch-theoremtong-solid-state-physicsThis is an authored directed relation from the source endpoint to the target endpoint.
When the periodic potential is weak relative to the free-electron kinetic scale, perturbation theory recovers nearly free dispersion away from Bragg degeneracies and controlled gap opening after degenerate perturbation theory.
The target is recovered from the source either in a stated mathematical or asymptotic limit, or through a quantitatively controlled approximation with identified small parameters or omitted terms. The edge label and detail must state which case applies and its regime of validity.
wp-physics-nearly-free-electronwp-physics-band-structuretong-solid-state-physicsThis is an authored directed relation from the source endpoint to the target endpoint.
Tight binding represents a periodic electronic problem in a truncated localized-orbital basis with fitted or derived hopping amplitudes; its errors depend on band isolation and basis quality rather than one universal small parameter.
The target is a model, idealization, restricted ansatz, approximation scheme, phenomenological fit, or historical representation used for the source system or theory. It need not have a universal small error parameter and is not thereby a controlled limit or effective theory. The edge label and detail must identify the precise status.
wp-physics-tight-bindingtong-solid-state-physicsmit-theory-of-solids