Preprints
https://doi.org/10.5194/mr-2024-18
https://doi.org/10.5194/mr-2024-18
04 Nov 2024
 | 04 Nov 2024
Status: this preprint is currently under review for the journal MR.

Analytical expressions for time evolution of spin systems affected by two or more interactions

Günter Hempel

Abstract. Analytical expressions for the description of the time evolution of spin systems beyond the product-operator formalism (POF) can be obtained if a low-dimensional subspace of the Liouville space has been found. The latter can be established by a procedure that consists of a repeated application of the commutator of the Hamiltonian with the density operator. This iteration continues as long as the result of such a commutator operation contains a term that is linearly independent of all the operators appearing in the previous commutator operations. The coefficients of the resulting system of commutator relations can be immediately inserted into the generic propagation formulae given in this article if the system contains two, three or four equations. In cases not satisfying the validity conditions of any of these propagation formulae, the coefficients are used as intermediate steps to obtain both the Liouvillian and propagator matrices of the system. Several application examples are given where an analytical equation can be obtained for the description of the time evolution of small spin systems under the influence of two or more interactions.

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Günter Hempel

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • CC1: 'Comment on mr-2024-18', Tom Barbara, 11 Nov 2024
    • AC1: 'Reply on CC1', Günter Hempel, 13 Nov 2024
      • CC2: 'Reply on AC1', Tom Barbara, 13 Nov 2024
        • AC2: 'Reply on CC2', Günter Hempel, 14 Nov 2024
  • RC1: 'Comment on mr-2024-18', Anonymous Referee #1, 19 Nov 2024
    • AC3: 'Reply on RC1', Günter Hempel, 26 Nov 2024
      • CC3: 'Reply on AC3', Tom Barbara, 27 Nov 2024
        • AC5: 'Reply on CC3', Günter Hempel, 28 Nov 2024
  • RC2: 'Comment on mr-2024-18', Anonymous Referee #2, 25 Nov 2024
    • AC4: 'Reply on RC2', Günter Hempel, 28 Nov 2024
Günter Hempel

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Short summary
The time evolution of a spin system in NMR can be represented in simple cases by an analytical equation, which is often better suited to promote physical understanding for teaching as well as for scientific discussion than numerical results. A method is presented that can be used to find analytical equations even for moderately complex problems. This is based on the fact that the coefficients of suitable commutator equation systems can be inserted directly into templates for propagation rules.