Quadratic excitation of an oscillator ground state ($X^2|0⟩\propto |0⟩+\sqrt{2}|2⟩$)

Multiplying the quantum harmonic oscillator ground state by $x^2$ produces a quantum superposition of only its $n=0$ and $n=2$ energy eigenstates. In the conventional phase choice for the creation and annihilation operators, $X=\sqrt{\hslash /(2m\omega )}(b+b^{†})$ and $b|0⟩=0$. ThereforeAfter wavefunction normalization, the energy measurement probabilities are $1/3$ at $\hslash \omega /2$ and $2/3$ at $5\hslash \omega /2$. Rephasing the second basis state can change the displayed relative coefficient's sign but cannot change these probabilities. For example, defining the unscaled creation operator as $a_{+}=P+im\omega X=i\sqrt{2m\hslash \omega }{b}^{†}$ reverses the sign of the $n=2$ basis function generated from $(a_{+}{)}^2|0⟩$.