---
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"当前已知",
"数学与物理任务",
"可交付成果",
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"deliverable": "均匀背景带限初值的二阶解与包络解、参数单位、误差范数、时间有效域和收敛报告，再扩展一个明确变系数或边界问题。",
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---
## 当前已知
物质波有效方程已经完成从选定有隙延展通道到慢变包络的具体计算。第五章式（5.1.22）给出本征截止频率与二阶传播张量；式（5.1.30）—（5.1.33）将快速载波代入原二阶方程，本征载波项相消，留下包含二阶慢时间导数的精确包络关系。
在式（5.1.34）的能量—载波匹配、长波、低速、弱背景和慢变条件下，式（5.1.35）给出领先有效方程。精确代入、所舍弃的时间导数以及常系数色散的低波数展开都已明确。后续任务是从完整核心—场系统计算该支及系数，并为指定初边值问题给出误差界；不是从零重新寻找一个形式相似的方程。
## 数学与物理任务
先区分三个层次：完整离散连接的色散、长波连续方程和连续方程的慢变包络。第一层到第二层需要 $ka_0\ll1$；第二层到第三层还需 $|\Omega|/\omega_0\ll1$、$ck/\omega_0\ll1$ 和弱背景调制。二阶有效色散在选定连续模型中成立，并不包含全布里渊区的所有晶格修正。
从具体作用量和稳定背景计算延展有隙支、传播张量 $B^{ij}$、归一化系数与 $\omega_0$，核对内禀能量匹配。对带限初值和明确边界控制快支激发，估计保留的精确余项、累计时间误差与失效尺度。变系数时保留散度形式中的系数导数；界面、转折区和突然切换边界分别处理，最早因果前沿仍由原二阶方程决定。误差条件采用频谱或适当范数，避免在包络节点除以其幅度。
## 可交付成果
第一份成果可限定为均匀背景的一族带限初值：提供原二阶解与包络解、相同归一化下的误差范数、时间区间、参数尺度及收敛图。随后再加入一个明确的缓变系数或边界问题。报告须把已知载波代入恒等关系与近似误差估计分开，并列出系数哪些已由核心计算得到、哪些作为输入采用。
## 不能当作完成的捷径
得到 Schrödinger 形式不独立证明微观谱支、质量匹配或 Born 事件统计；反过来，微观闭合未完成也不撤销已给出的载波代入与有效极限。把完整晶格色散、二阶连续色散和慢变包络当成同一个精确方程，会掩盖需要控制的不同误差。
## 研究依赖
- [有效波动方程及其精确余项](/theory/quantum/effective-wave-equation)
- [连续动力学的变量、源与边界](/theory/space-dynamics/continuum-dynamics)
[量子现象总览](/theory/quantum) · [现象学要求](/tests/quantum)