泡利方程 / Pauli equation

Woit · §34.1 The Pauli-Schr¨odinger equation and free spin 1/2 particles in d=3

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Using this Dirac operator (often called in this context the “Pauli operator”) we can write a two-component version of the Schr¨odinger equation (often called the “Pauli equation” or “Pauli-Schr¨odinger equation”)

$$\begin{array}{rl}i\frac{\partial }{\partial t}\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)& =-\frac{1}{2m}{\left({\sigma }_1\frac{\partial }{\partial q_1}+{\sigma }_2\frac{\partial }{\partial q_2}+{\sigma }_3\frac{\partial }{\partial q_3}\right)}^2\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)\\ & =-\frac{1}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)\end{array}\text{\hspace{1em}(34.3)}$$

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正文：英文 OCR 草稿

来源版本：2025-10-20

原文位置：§34.1；PDF 页 377；印刷页 360

处理记录：MinerU OCR

核对状态：英文 OCR 草稿 · 待校对

中文机译草稿

Author-named definition; machine draft source identity and human-review status are preserved.

机器草稿中的定义提及

使用这个狄拉克算符（在此语境中常称为“泡利算符”），我们可以写出薛定谔方程的两分量版本（常称为“泡利方程”或“泡利-薛定谔方程”）

$$\begin{array}{rl}i\frac{\partial }{\partial t}\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)& =-\frac{1}{2m}{\left({\sigma }_1\frac{\partial }{\partial q_1}+{\sigma }_2\frac{\partial }{\partial q_2}+{\sigma }_3\frac{\partial }{\partial q_3}\right)}^2\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)\\ & =-\frac{1}{2m}\left(\frac{{\partial }^2}{\partial q_1^2}+\frac{{\partial }^2}{\partial q_2^2}+\frac{{\partial }^2}{\partial q_3^2}\right)\left(\genfrac{}{}{0ex}{}{{\psi }_1(\mathbf{q})}{{\psi }_2(\mathbf{q})}\right)\end{array}\text{\hspace{1em}(34.3)}$$

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来源与版本

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正文：中文机译草稿

来源版本：2025-10-20

原文位置：§34.1；PDF 页 377；印刷页 360

处理记录：MinerU OCR；machine-translation/deepseek-v4.1-flash

核对状态：中文机译草稿 · 待校对