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25. Matt:Hi, Alice. How are you?Alice:Not so good. I don’t feel very well.Matt:_____________________________(A) That’s wonderful. (B) You are sorry.(C) That’s too bad. (D) I’m glad to know that.

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.

12. The Pauli-X matrix, denoted by $\hat{\sigma}_x$, is represented as $$ \hat {\sigma} _ {x} = \left( \begin{array}{c c} 0 & 1 \\ 1 & 0 \end{array} \right). $$ Find the expectation value $\langle \hat{\sigma}_x\rangle$ in the spin state: $$ | \psi \rangle = \alpha_ {+} \left( \begin{array}{c} 1 \\ 0 \end{array} \right) + \alpha_ {-} \left( \begin{array}{c} 0 \\ 1 \end{array} \right), $$ where $\alpha_{+}$ and $\alpha_{-}$ are complex numbers. (A) 0 (B) $2\operatorname{Re}(\alpha_{+}^{*}\alpha_{-})$ (C) $2\operatorname{Im}(\alpha_{+}^{*}\alpha_{-})$ (D) $|\alpha_{+}|^{2} - |\alpha_{-}|^{2}$ (E) $|\alpha_{+}|^{2} + |\alpha_{-}|^{2}$

13. Consider a free electron gas in $d$ spatial dimensions. Let $n$ be the electron number density (defined as the number of electrons per unit length, area, or volume, depending on the dimension). Which of the following correctly describes the dependence of the Fermi energy $E_{F}$ on the density $n$? (A) $E_{F}\propto n^{2 / d}$ (B) $E_{F}\propto n^{d / 2}$ (C) $E_{F}\propto n$ (D) $E_{F}\propto n^{2}$ (E) $E_{F}\propto n^{1 / d}$