16. Which of the following mathematical expressions correctly represents Planck's formula for the energy density per unit frequency $\rho(\omega)$ of a blackbody at absolute temperature $T$? ($k_B$: Boltzmann constant, $\omega$: angular frequency of light.) (A) $\frac{\hbar \omega^3}{\pi^2 c^3} \frac{1}{e^{\hbar \omega / (k_B T)} - 1}$ (B) $\frac{\hbar \omega^3}{\pi^2 c^3} k_B T$ (C) $\frac{\hbar \omega^3}{\pi^2 c^3} e^{-\hbar \omega / (k_B T)}$ (D) $\frac{\hbar \omega^3}{\pi^2 c^3} \frac{1}{e^{\hbar \omega / (k_B T)} + 1}$ (E) $\frac{\hbar \omega}{e^{\hbar \omega / (k_B T)} - 1}$

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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