In solid-state physics of metals, are the valence-electron wave functions strongly perturbed by neighboring atoms, leading to energy-band formation rather than isolated atomic levels?

Introduction / Context:
Band theory explains why metals conduct electricity so well. In a crystal, each atom is surrounded by many neighbors. The overlap of their valence-electron wave functions means electrons can no longer be described as belonging to single atoms. Instead, their allowed energies spread into bands. This question checks your conceptual grasp of how interatomic interactions in a lattice transform discrete atomic levels into continuous energy bands that govern metallic behavior.

Given Data / Assumptions:

• Periodic crystalline metal with closely spaced atoms (typical metallic bond distances).
• Focus on valence electrons (s, p, or d depending on metal) that participate in bonding and conduction.
• Single-electron picture with a periodic potential is sufficient for first-order discussion.

Concept / Approach:
When many identical atoms come together, their discrete atomic energy levels split due to quantum mechanical overlap and Pauli exclusion. With N atoms, each level splits into N closely spaced levels that merge into bands as N becomes macroscopic. The perturbation from neighboring atoms is therefore strong for valence electrons, producing a conduction band that is partially filled in metals, enabling high electrical and thermal conductivity.

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