 Solution

The number operator commutators established above imply that ${\hat{a}}_x^{†}|n⟩$ is proportional to $|n+1⟩$ and ${\hat{a}}_x|n⟩$ to $|n-1⟩$. Their norms determine the magnitudes of the proportionality constants:Choose the phase of the normalized ground eigenstate once, then define recursivelyThis removes the otherwise arbitrary relative phases of the eigenstates. The Hermitian adjoint relation then fixes the lowering coefficient with the same positive sign, givingAt $n=0$, the second expression means the zero vector, ${\hat{a}}_x|0⟩=0$; it does not introduce a physical state $|-1⟩$.