# Canonically conjugate pairs and phase operators

Source: [Canonically conjugate pairs and phase operators](<https://www.sota2.com/research/paper/68fa31e71e209131ce8837c7>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T19:11:33.600Z

Authors: K. Schonhammer

Published: 2002-04-24T11:30:54.000Z

## Abstract

For quantum mechanics on a lattice the position (\`\`particle number'') operator and the quasi-momentum (\`\`phase'') operator obey canonical commutation relations (CCR) only on a dense set of the Hilbert space. We compare exact numerical results for a particle in simple potentials on the lattice with the expectations, when the CCR are assumed to be strictly obeyed. Only for sufficiently smooth eigenfunctions this leads to reasonable results. In the long time limit the use of the CCR can lead to a qualitativel wrong dynamics even if the initial state is in the dense set.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/quant-ph/0204139>)

[PDF](<https://arxiv.org/pdf/quant-ph/0204139.pdf>)