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- #1

\begin{align}

L\phi &= \lambda\phi\\

\phi_t &= M\phi

\end{align}

where \(L\) and \(M\) are operators and \(\lambda\) a constant.

I want to show the compatibily condition is \(L_t + [L,M] = 0\) where \([,]\) is the commutator.

\[

(L\phi)_t = L_t\phi + L\phi_t = \lambda\phi_t = \lambda M\phi

\]

That is, we have \(L_t\phi + L\phi_t - \lambda M\phi = L_t\phi + LM\phi- \lambda M\phi = 0\).

\begin{align}

[L_t + LM - \lambda M]\phi &= 0\\

L_t + LM - \lambda M &= 0

\end{align}

Can I just let \(\lambda = L\) which doesn't make since sense one is an operator and the other a constant? If not, how do I get the commutator \(LM - ML\) part?