\begin{equation} \label{Eq:4.31}
& \textbf{E}=-\frac{1}{c}\frac{\partial \textbf{A}}{\partial t}-\nabla V \\
& \textbf{B}=\nabla \cross \textbf{A}
A guage transformation upon these potential components takes the form:
\begin{equation} \label{Eq:4.32}
& V \longrightarrow V -\frac{1}{c}\frac{\partial \Lambda}{\partial t} \\
& \textbf{A} \longrightarrow \textbf{A} + \nabla \Lambda
for some scalar field $\Lambda$, inserting this transformation into their 
defining equations gives:
\begin{equation} \label{Eq:4.33}
& \textbf{E}=-\frac{1}{c}\frac{\partial (\textbf{A}+\nabla \Lambda)}
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}
showing invariance under these transformations, though they add a phase-like 
term to the solution to the electromagnetic wave equation i.e.:
\Psi' \longrightarrow \Psi exp[i\Lambda/\hbar c]

& 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


Results in this error:

! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.

l.230 \begin{equation}

closed as unclear what you're asking by David Carlisle, user31729, egreg, Stefan Pinnow, Schweinebacke Nov 3 '17 at 12:37

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 2
    please always show a complete small document that reproduces the error. here as the error is detected on \begin{equation} the error is not in this equation (which has not been read yet) but a previous math construct was not properly closed. – David Carlisle Nov 2 '17 at 20:40
  • It's a very long document, I'm unsure of how much to include. – I.Auguste Nov 2 '17 at 20:44
  • 1
    @I.Auguste - How about (a) the part of the preamble that's relevant and (b) the five to ten lines of code that immediately precede the code block you've already posted? – Mico Nov 2 '17 at 20:45
  • 1
    you have still only posted unconnected fragments, please edit to be a complete document that people can test – David Carlisle Nov 2 '17 at 21:04
  • 1
    The code you've posted so far does not generate the error you're looking to fix. – Mico Nov 2 '17 at 21:05

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