= Shifted two-dimensional oscillator in a uniform electric field
{title2=$H'=H(x-a,y)-a^2/2,\quad a=e\mathcal E$}
Completing the square in $H'=\tfrac12(p_x^2+p_y^2+x^2+y^2)-e\mathcal E x$ shifts the oscillator centre to $(a,0)$ and lowers every energy by $a^2/2$. Translated eigenfunctions keep their normalization. The <angular momentum> about the new centre is $L'=(x-a)p_y-yp_x$, which commutes with $H'$, whereas the <angular momentum> about the old centre has $[L,H']=-i\hbar ay$. For $a\ne0$, a common normalizable eigenfunction of $L,H'$ would have $y\psi=0$ and hence vanish in $L^2(\mathbb R^2)$. This stronger argument excludes individual common eigenstates; noncommutativity alone generally excludes only a complete common eigenbasis.
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