On the Derivation of Quantum Mechanical Expectations in the Differential Form

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  • KONDO Shin-ichiro
    Dept. Materials Science and Engineering, Nagasaki University
  • KONDO Tatsushi
    Sumitomo Corporation, Suzuki Pharmacy
  • KONDO Atsushi
    Graduate School of Medicine, University of Tokyo
  • YOSHIMURA Kazuyoshi
    Dept. Chemistry, Graduate School of Science, Kyoto University Department of Energy & Hydrocarbon Chemistry, Graduate School of Engineering, Kyoto University

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  • 微分形による量子力学的期待値導出について
  • ビブンケイ ニ ヨル リョウシ リキガクテキ キタイチ ドウシュツ ニ ツイテ

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Abstract

<p>Usually, expected values for various physical quantities, such as the number of electrons occupying certain states or the Coulomb interaction between different states of electrons, can be expressed in terms of integrals. In contrast, our method, based on differential forms, shows that expected values can be obtained by averaging over time. To confirm the validity of our method, we prepare the two cases: one is a very simple case with no many-body interaction, and the other is the case where the many-body term is included (the simplest Anderson Hamiltonian). Regarding the simple case without inclusion of many-body term, we prove analytically that the number of electrons occupying any state derived from our method is equivalent to the analytical one evaluated from the Greenʼs function method. When the many-body term is included, our results show good numerical agreement with the analytical ones derived from the Greenʼs function method. By the two cases, the calculation of expected values based on our method is considered valid.</p>

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