Vanishing Integral Relations and Expectation Values for Bloch Functions in Finite Domains

TL;DR

This paper derives integral identities and expectation value properties for Bloch functions in finite periodic systems, revealing vanishing matrix elements and invariance properties crucial for understanding quantum behaviors in finite domains.

Contribution

It introduces new integral identities and expectation value properties for Bloch functions in finite domains, extending prior infinite-system results.

Findings

Matrix elements of certain Bloch functions vanish.

Expectation values are real, time-independent, and satisfy summation properties.

Derived identities hold for finite periodic systems with integer unit cells.

Abstract

Integral identities for particular Bloch functions in finite periodic systems are derived. All following statements are proven for a finite domain consisting of an integer number of unit cells. It is shown that matrix elements of particular Bloch functions with respect to periodic differential operators vanish identically. The real valuedness, the time-independence and a summation property of the expectation values of periodic differential operators applied to superpositions of specific Bloch functions are derived.