9. A particle of mass $m$ is subject to a one-dimensional attractive delta potential given by: $$ V(x) = -\alpha \delta(x), $$ where $\alpha > 0$. This potential supports exactly one bound state. Determine the energy eigenvalue $E$ and the normalized eigenfunction $\psi(x)$ for this state. We define $\kappa = \sqrt{-2mE} / \hbar$.
(A) $E = -m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{-\kappa |x|}$
(B) $E = -m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{\kappa |x|}$
(C) $E = -m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{-\kappa x^2}$
(D) $E = +m\alpha^2 / 2\hbar^2$ ; $\psi(x) = \sqrt{\kappa} e^{-\kappa x^2}$
(E) $E = -\hbar^2 / 2m\alpha^2$ ; $\psi(x) = \sqrt{\kappa} e^{+\kappa |x|}$

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