Past exam of the quantum theory course of the University of Oxford / 2023 / 2 / b

Since $X$ and $P$ are self-adjoint operators, the adjoint operator relation requires $c_{+}=c_{-}=c$ and $d_{+}=-d_{-}=-d$. Thus $a_{-}=cX+idP$ and $a_{+}=cX-idP$. The canonical commutation relation $[X,P]=i\hslash I$ givesMatching the $X^2$, $P^2$ and identity operator coefficients in $H=\hslash \omega (a_{+}{a}_{-}+\lambda I)$ yieldsConsequently all real solutions areThe conventional creation and annihilation operators use $\epsilon =1$. The other choice multiplies both by minus one and satisfies all the given equations. This harmless freedom is the oscillator ladder-operator sign convention; the subsequent formulas use the positive choice in both coordinates.