Harmonic oscillator

For a one-dimensional harmonic oscillator, the creation operator $a^\dagger$ acting on the $n$-th state $|\psi_n\rangle$ gives $a^\dagger |\psi_n\rangle = \sqrt{n+1} |\psi_{n+1}\rangle$. The position operator $x$ in the basis of the first three states is proportional to

Options:

• (A) $\sqrt{\frac{\hbar}{2m\omega}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{3} \end{pmatrix}$
• (B) $\sqrt{\frac{\hbar}{2m\omega}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$
• (C) $\sqrt{\frac{\hbar}{2m\omega}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & \sqrt{2} \\ 0 & \sqrt{2} & 0 \end{pmatrix}$
• (D) $\sqrt{\frac{\hbar}{2m\omega}} \begin{pmatrix} 1 & 0 & \sqrt{3} \\ 0 & 0 & 0 \\ \sqrt{3} & 0 & 1 \end{pmatrix}$