12. The normalized common eigenstates of Hamiltonian \( \hat{H} \), angular momentum squared \( \widehat{L}^{2} \) and angular momentum along the \( z \) axis \( \hat{L}_{z} \) of a hydrogen atom are denoted as \( |n, l, m\rangle \), with corresponding eigenvalues \( E_{n} \) for \( n \in\{1,2, \ldots\}, l(l+1) \hbar^{2} \) for \( l \in\{0,1, \ldots, n-1\} \), and \( m \hbar \) for \( m \in\{l, l-1, \ldots,-l\} \).
We define an operator \( \hat{A} \) as \( \hat{A} \psi(x, y, z)=\psi(y, x, z) \), for any arbitrary wavefunction \( \psi(x, y, z) \) in the Cartesian coordinates \( (x, y, z) \).
(a) What is the physical meaning of the operator \( \hat{A} \) ? (3 marks)
(b) Prove that \( \hat{A} \) is an Hermitian operator (i.e., \( \hat{A}=\hat{A}^{\dagger} \) ). (3 marks)
(c) Show that \( \hat{A}=\hat{A}^{-1} \) (i.e., \( \hat{A}^{2}=1 \) ). What possible eigenvalues can \( \hat{A} \) have? (4 marks)
(d) Show that \( \left[\hat{L}_{z}, \hat{A}\right] \neq 0 \) (Hint: Check how \( \hat{A} \) acts on a function in the spherical coordinates). (4 marks)
(e) Show that \( [\hat{H}, \hat{A}]=0 \) and \( \left[\hat{A}, \hat{L}^{2}\right]=0 \). What symmetry of the hydrogen atom have as indicated by the commutator \( [\hat{H}, \hat{A}]=0 \) ? (5 marks)
(f) Find all the common eigenstates \( \left|\psi_{2,1, a}\right\rangle \) of \( \widehat{H}, \hat{L}^{2} \), and \( \hat{A} \) with \( \widehat{H}\left|\psi_{2,1, a}\right\rangle=E_{2}\left|\psi_{2,1, a}\right\rangle \), \( \hat{L}^{2}\left|\psi_{2,1, a}\right\rangle=2 \hbar^{2}\left|\psi_{2,1, a}\right\rangle \), and \( \hat{A}\left|\psi_{2,1, a}\right\rangle=a\left|\psi_{2,1, a}\right\rangle \). Find the corresponding eigenvalues \( a \) of the operator \( \hat{A} \). (No need to write down the explicit wavefunction \( \psi_{2,1, a}(x, y, z) \). Instead, you need to find \( \left|\psi_{2,1, a}\right\rangle \) as superposition \( |n, l, m\rangle \). ) (7 marks)
