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**DiscreteW** 1.) Derive the expression of $\displaystyle \hat{p}$ in $\displaystyle x$-space. Prove $\displaystyle \hat{p}$ is linear. Then, prove it is hermitian.

2.) Find the following commutators, showing two different ways for finding them:

$\displaystyle [x,\hat{H}]$

$\displaystyle [\hat{p}, \hat{H} + x]$

(For first way use $\displaystyle [x,p] = i\bar{h}$, and second way using properties of commutators).

3.) Show $\displaystyle Ae^{-ikx}\ \ \ A, k \in \mathbb{R}$ is an eigenfunction of $\displaystyle \hat{p}$. Determine the eigenvalue.