= Oscillator ladder-operator sign convention
{title2=$a_\pm\mapsto-a_\pm$}
For real coefficients in $a_\pm=c_\pm X+id_\pm P$, imposing $a_-^\dagger=a_+$, $[a_-,a_+]=I$ and $H=\hbar\omega(a_+a_-+\lambda I)$ gives
$$
c_+=c_-=\varepsilon\sqrt{\frac{m\omega}{2\hbar}},\qquad
d_-=-d_+=\frac{\varepsilon}{\sqrt{2m\hbar\omega}},\qquad
\lambda=\frac12,\qquad\varepsilon=\pm1.
$$
Indeed the <adjoint operator> condition makes the two $c$ coefficients equal and the two $d$ coefficients opposite. The <canonical commutation relation> gives $2\hbar c_-d_-=1$, and matching the coefficients of $X^2$ and $P^2$ fixes their magnitudes. Both signs change together. The conventional <creation and annihilation operators> choose $\varepsilon=1$; the other sign is an equivalent phase convention for the <number states>.
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