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I'm not sure I can ask this question, if it causes some problem, then I would immediately delete the post.

I'm looking at "Spinsim" Package, and at the guide pdf, it shows some math technique: enter image description here

I'm not so good in mathematics, and I have just found out about Lie-Trotter Decomposition in the pdf.

I just don't get how the above $T$ becomes the below $T$ (by leapfrog splitting method, they say)

I tried to read the article in appendix [22]. I think it is related to this: enter image description here

I just don't get how the arctan terms, and $w_x$ squared terms appear just by that. It says about commute relations, but I have no idea what it means since the spin operators do not commute.

Thanks for the help.

Ricky
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user21091084
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  • What are $J_x,J_y,J_z$ ? Pauli matrices ? – Jean Marie Sep 05 '23 at 14:25
  • Well, it's dealing with high spin systems, so it is not actually Pauli matrices. But it is OK to think as Pauli matrices. Actually I have figured out why it works, now I am confused about commutation relations in matrix exponentials. – user21091084 Sep 05 '23 at 15:15
  • Suppose I can write my Hamiltonian as above. J_x, J_y, J_z and Q, etc. are operators.

    It is obvious that I can add the terms as I want: I can add J_x + J_y + J_z + ... or J_y + J_x + J_z ... or etc.

    But from my Hamiltonian, if I get unitary operator and use the Lie-Trotter Product Formula above, now the operators are in exponential and the added up order matters. Why?

     – user21091084 Sep 05 '23 at 15:22

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