1322
1. \( \hat{x} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix} \)
2. \( \hat{D} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \)
3. \( |k=0\rangle = \frac{1}{2}(|0\rangle + |1\rangle +|2\rangle + |3\rangle) \\ |k=\frac{\pi}{2}\rangle = \frac{1}{2}(|0\rangle +i|1\rangle -|2\rangle -i|3\rangle) \\ |k=\pi\rangle = \frac{1}{2}(|0\rangle - |1\rangle +|2\rangle - |3\rangle) \\ |k=\frac{3\pi}{2}\rangle = \frac{1}{2}(|0\rangle -i|1\rangle -|2\rangle +i|3\rangle) \)
\( \lambda = 1,-i,-1,i \)
4. \( \hat{D} = e^{-i\hat{p}} \)
5. \( H = \begin{pmatrix} 0 & c & 0 & c \\ c & 0 & c & 0 \\ 0 & c & 0 & c \\ c & 0 & c & 0 \end{pmatrix} \)
6. \( H = c\hat{D} + c\hat{D}^{-1} \)
7. \( H = 2c \cos(\hat{p}) \\ E = 2c,0,-2c,0 \)
9. \( |+,+\rangle = \frac{1}{2}(|00\rangle + |10\rangle +|11\rangle +|01\rangle) \\ |+,-\rangle = \frac{1}{2}(|00\rangle + |10\rangle -|11\rangle -|01\rangle) \\ |-,-\rangle = \frac{1}{2}(|00\rangle - |10\rangle +|11\rangle -|01\rangle) \\ |-,+\rangle = \frac{1}{2}(|00\rangle -|10\rangle -|11\rangle +|01\rangle) \)
10. \( E = 2c,0,-2c,0 \)