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27. Alan:We are going to play in the championship final next Saturday.Lisa:______________________(A) It’s my treat! (B) I don’t know.(C) I need a farm. (D) Wow! I wish you good luck in the final.

28. Albert:_____________________Elissa:Oh, I see. I’ll add you later.(A) See you tomorrow. (B) Merry Christmas!(C) I just sent a friend request to you on Facebook. (D) I am full.

29. Anne:When will the football game start?Bill:In five minutes. ___________ We’re late.(A) Hurry up! (B) Time to go home.(C) Where? (D) It rains in summer.

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) $E = -\hbar^2 / 2m\alpha^2$ ; $\psi(x) = \sqrt{\kappa} e^{+\kappa |x|}$

10. In quantum mechanics, which of the following pairs of operators share simultaneous eigenstates? (A) Position along the $x$-axis $(\hat{x})$ and linear momentum along the $x$-axis $(\hat{p}_x)$. (B) The $x$-component of angular momentum $(\hat{L}_x)$ and the $y$-component of angular momentum $(\hat{L}_y)$. (C) Position along the $x$-axis $(\hat{x})$ and the $z$-component of angular momentum $(\hat{L}_z)$. (D) The squared magnitude of orbital angular momentum $(\hat{L}^2)$ and the $z$-component of orbital angular momentum $(\hat{L}_z)$. (E) The $x$-component of spin angular momentum $(\hat{S}_x)$ and the $z$-component of spin angular momentum $(\hat{S}_z)$.