To analyze the many-body problem based on differential forms, we apply the Heisenberg equation of motion to the Hamiltonian including the many-body term, focusing on the simplest electron–phonon interaction. Simultaneous differential equations are obtained after applying the Heisenberg equation of motion. To confirm our proposed method, we compare the numerical solutions to the simultaneous differential equations with the analytical results based on the Green’s function method. Good agreement is observed between the results, indicating that the analysis based on the Heisenberg equation of motion is suitable for analyzing the steady state. To examine the non-steady state, we investigate weakly pulsed phonon excitation. The numerical calculations suggest that phonons may be conserved if the excitation period is very short. Furthermore, based on the Pauli exclusion principle, the number of simultaneous differential equations that exactly describe electron–electron interactions appears to be finite but becomes extremely large as the number of electrons increases.

中文翻译：

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海森堡运动方程在最简单的电子-声子相互作用中的多体问题中的应用

为了分析基于微分形式的多体问题，我们将海森堡运动方程应用于包含多体项的哈密顿量，重点是最简单的电子-声子相互作用。应用海森堡运动方程后，可获得联立微分方程。为了确认我们提出的方法，我们将数值解与联立微分方程进行了比较，并基于格林函数方法得出了分析结果。结果之间观察到良好的一致性，表明基于海森堡运动方程的分析适合于分析稳态。为了检查非稳态，我们研究了弱脉冲声子激发。数值计算表明，如果激发周期很短，声子可以守恒。此外，