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\begin{document}
\begin{equation} \label{Eq:4.31}
\begin{aligned}
& \textbf{E}=-\frac{1}{c}\frac{\partial \textbf{A}}{\partial t}-\nabla V \\
& \textbf{B}=\nabla \cross \textbf{A}
\end{aligned}
\end{equation}
A guage transformation upon these potential components takes the form:
\begin{equation} \label{Eq:4.32}
\begin{aligned}
& V \longrightarrow V -\frac{1}{c}\frac{\partial \Lambda}{\partial t} \\
& \textbf{A} \longrightarrow \textbf{A} + \nabla \Lambda
\end{aligned}
\end{equation}
for some scalar field $\Lambda$, inserting this transformation into their
defining equations gives:
\begin{equation} \label{Eq:4.33}
\begin{aligned}
& \textbf{E}=-\frac{1}{c}\frac{\partial (\textbf{A}+\nabla \Lambda)}
{\partial
t}-\nabla (V -\frac{1}{c}\frac{\partial \Lambda}{\partial t}) \\
& =-\frac{1}{c}\frac{\partial \textbf{A}}{\partial t}-\cancelto{0}{\frac{1}
{c}\frac{\partial \nabla \Lambda }{\partial t}}-\nabla V-\cancelto{0}{\nabla
\frac{1}{c}\frac{\partial \Lambda}{\partial t}} \\
& \textbf{E}=-\frac{1}{c}\frac{\partial \textbf{A}}{\partial t}-\nabla V \\
& \textbf{B}=\nabla \cross (\textbf{A}+\nabla \Lambda) = \nabla \cross
\textbf{A} + \cancelto{0}{\nabla^2\Lambda} \\
& \textbf{B}=\nabla \cross \textbf{A}
\end{aligned}
\end{equation}
showing invariance under these transformations, though they add a phase-like
term to the solution to the electromagnetic wave equation i.e.:
\begin{equation}
\Psi' \longrightarrow \Psi exp[i\Lambda/\hbar c]
\end{equation}
\begin{equation}
\begin{aligned}
& i\hbar\frac{\partial\Psi_1}{\partial t}=\frac{\textbf{p}^2}
{2m}\Psi_1+eV(t)\Psi_1 \\
& i\hbar\frac{\partial\Psi_2}{\partial t}=\frac{\textbf{p}^2}{2m}\Psi_2
\end{aligned}
\end{equation}
\end{document}
导致此错误：
! LaTeX Error: Bad math environment delimiter.
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l.230 \begin{equation}