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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.

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 $(\hat{S}_z)$. 

11. Consider a quantum particle in a one-dimensional infinite square well with potential $V(x) = 0$ for $-L < x < L$ and $V(x) = \infty$ for $|x| \geq L$. The particle has an initial wave function $\psi(x, t = 0) = \frac{1}{\sqrt{3}} \psi_1(x) + \sqrt{\frac{2}{3}} \psi_2(x)$, where $\psi_1$ and $\psi_2$ are the ground state and the first excited state of the system with eigenenergies $E_1$ and $E_2$, respectively. $\psi_{1,2}$ are real functions. What are the probability density $|\psi(x,t)|^2$ and the expectation value of energy $\langle H \rangle$ at later times $t > 0$? (A) $\langle H\rangle = \frac{1}{3} E_1 + \frac{2}{3} E_2$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2 +\frac{2\sqrt{2}}{3}\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$ (B) $\langle H\rangle \neq \frac{1}{2} (E_1 + E_2)$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2 +\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$ (C) $\langle H\rangle = \frac{1}{3} E_1 + \frac{2}{3} E_2$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2$ (D) $\langle H\rangle = \frac{1}{3} E_1 + \frac{2}{3} E_2\cos (t)$ ; $|\Psi (x,t)|^2 = \frac{1}{3}\psi_1^2 +\frac{2}{3}\psi_2^2 +\frac{\sqrt{2}}{3}\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$ (E) $\langle H\rangle = E_1 + \sqrt{2} E_2$ ; $|\Psi (x,t)|^2 = \psi_1^2 +2\psi_2^2 +2\sqrt{2}\psi_1\psi_2\cos \left(\frac{(E_2 - E_1)t}{\hbar}\right)$