发表机构
Clarkson University（克拉克森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究为复弱动量的虚部建立严格泛函分析框架，证明其期望值恒为零，揭示玻姆量子势源于该虚部、布洛赫带群速度仅由实部决定等性质，且结果具规范不变性并可推广至弯曲黎曼流形。
AI 中文摘要
我们为复弱动量 $p_Q = p_R + ip_I = \frac{\hbar}{i}\nabla\ln\psi$ 的虚部 $p_I$ 建立了严格的泛函分析框架。对于索伯列夫空间 $H^1(\Omega)$ 中的每一个波函数，包括那些具有节点集的波函数，我们证明 $p_I$ 的期望值恒为零，这是 $\rho = |\psi|^2$ 的正则性导致的结果，且无需借助波函数的极形式 $\psi = Re^{iS/\hbar}$。我们进一步证明：(i) 玻姆量子势完全源于 $p_I$，且满足 $\langle V_{\mathrm{qu}}\rangle = \langle p_I^2\rangle/2m$，由此可推导出已知的费舍尔信息关系作为推论；(ii) 布洛赫带的群速度是 $p_R/m$ 的密度加权平均，无 $p_I$ 的贡献；(iii) 该分解在磁最小耦合下具有规范不变性，矢量势仅作用于 $p_R$；(iv) 上述结果可推广至弯曲黎曼流形。
英文摘要
We establish a rigorous functional-analytic framework for the imaginary component $p_I$ of the complex weak momentum $p_Q = p_R + ip_I = \frac{\hbar}{i}\nabla\lnψ$. For every wavefunction in the Sobolev space $H^1(Ω)$, including those with nodal sets, we prove that the expectation value of $p_I$ vanishes identically, as a consequence of the regularity of $ρ= |ψ|^2$ and without appeal to the polar form $ψ= Re^{iS/\hbar}$. We further show that \textnormal{(i)} the Bohm quantum potential arises entirely from $p_I$ and satisfies $\langle V_{\mathrm{qu}}\rangle = \langle p_I^2\rangle/2m$, from which the known Fisher information relation follows as a corollary; \textnormal{(ii)} the group velocity of a Bloch band is the density-weighted average of $p_R/m$, with no contribution from $p_I$; \textnormal{(iii)} the decomposition is gauge-invariant under magnetic minimal coupling, the vector potential acting only on $p_R$; and \textnormal{(iv)} the results extend to curved Riemannian manifolds.
