In lattice model, we can define $\hat x|n\rangle=n|n\rangle$, and $\hat p|k\rangle=k|k\rangle$, where $|k\rangle=\sum_ne^{ikn}|n\rangle$, but we cann't get $[\hat x,\hat p]|k\rangle=i|k\rangle$ for any $k$. So I want ask what the correspond commutator of $[x,p]=i$ in lattice model is.
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3$e^{ip}$ is the lattice translation $T$, so the analog of $[x,p]=i$ is $TxT^{-1}=x+1$. – Meng Cheng Sep 02 '22 at 02:54
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That is interesting, it's very helpful, thanks a lot! If there are some reference about this? – photon Sep 02 '22 at 04:34
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Santhanam, T.S., Tekumalla, A.R., Quantum mechanics in finite dimensions, Found Phys 6 (1976) 583–587, eqns (20-21). This is the standard QM around the clock. – Cosmas Zachos Sep 02 '22 at 21:04
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Look at Q10 of this exam. – Cosmas Zachos Sep 02 '22 at 21:09
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Linked. – Cosmas Zachos Sep 02 '22 at 21:15