Issue |
EPL
Volume 149, Number 2, January 2025
|
|
---|---|---|
Article Number | 26001 | |
Number of page(s) | 7 | |
Section | Condensed matter and materials physics | |
DOI | https://doi.org/10.1209/0295-5075/ada4d4 | |
Published online | 28 January 2025 |
Revisiting the matrix elements of the position operator in the crystal momentum representation
1 School of Materials and Energy, Lanzhou University - Lanzhou 730000, China
2 Department of Physics, Indiana State University - Terre Haute, IN 47809, USA
Received: 22 October 2024
Accepted: 2 January 2025
Fewer operators are more fundamental than the position operator in a crystal. But since it is not translationally invariant in crystal momentum representation (CMR), how to properly represent it is nontrivial. Over half a century, various methods have been proposed, but they often lead to either highly singular derivatives or extremely arcane expressions. Here we propose a resolution to this problem by directly computing their matrix elements between two Bloch states. We show that the position operator is a full matrix in CMR, where the off-diagonal elements in crystal momentum only appear along the direction of the position vector. Our formalism, free of singular derivative and degeneracy difficulties, can describe an array of physical properties, from intraband transitions, polarization with or without spin-orbit coupling, orbital angular momentum, to susceptibilities.
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