Hempel Günter
Martin-Luther-Universität Halle-Wittenberg, Institut für Physik - NMR, Betty-Heimann-Str. 7, 06120 Halle, Germany.
Magn Reson (Gott). 2025 Feb 27;6(1):77-92. doi: 10.5194/mr-6-77-2025. eCollection 2025.
Analytical expressions for the description of the time evolution of spin systems beyond product-operator formalism (POF) can be obtained if a low-dimensional subspace of the Liouville space has been found in which the time evolution of the spin system takes place completely. This can be achieved using a procedure that consists of 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 where the validity conditions of any of these propagation formulae are not met, 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. This procedure for finding the Liouvillian matrix is not limited to time-independent interactions. Some examples illustrate the treatment of time-dependent problems using this method.
如果在刘维尔空间中找到了一个低维子空间，其中自旋系统的时间演化完全发生，那么就可以得到超越乘积算符形式主义（POF）的自旋系统时间演化描述的解析表达式。这可以通过一种程序来实现，该程序包括重复应用哈密顿量与密度算符的对易子。只要这种对易子操作的结果包含一个与先前对易子操作中出现的所有算符线性无关的项，这种迭代就会继续。如果系统包含两个、三个或四个方程，那么所得对易子关系系统的系数可以立即插入到本文给出的通用传播公式中。在任何这些传播公式的有效性条件不满足的情况下，这些系数用作中间步骤来获得系统的刘维尔算符和传播子矩阵。给出了几个应用示例，其中可以得到一个解析方程来描述在两个或更多相互作用影响下小自旋系统的时间演化。这种寻找刘维尔算符矩阵的程序不限于与时间无关的相互作用。一些示例说明了使用此方法处理与时间有关的问题。
