**QUESTION**

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1. If $A$ and $B$ are Hermitian operators, prove that (1) the operator $A B$ is only Hermitian if $A$ and $B$ commute, that is, if $A B=B A$, and (2) the operator $(A+B)^{n}$ is Hermitian. 2. Prove that $A+A^{\dagger}$ and $i\left(A-A^{\dagger}\right)$ are Hermitian for any operator, as is $A A^{\dagger}$. 3. Prove that if $H$ is a Hermitian operator, then the Hermitian conjugate operator of $e^{i H}$ (defined to be $\sum_{n=0}^{\infty} i^{n} H^{n} / n$ !) is the operator $e^{-i H}$. 4. An operator is said to be unitary if it has the property that \[ U U^{\dagger}=U^{\dagger} U=\mathbb{1} \] (a) Show that if $\langle\psi \mid \psi\rangle=1$ then $\langle U \psi \mid U \psi\rangle=1$. (b) Show that if $H$ is Hermitian, then $e^{i H}$ is unitary. (c) Show that if the $\left\{u_{a}\right\}$ form an orthonormal complete basis set, with \[ \left\langle u_{a} \mid u_{b}\right\rangle=\delta_{a b} \] then the set \[ \left|v_{a}\right\rangle=U\left|u_{a}\right\rangle \] with $U$ unitary is also orthonormal. (The meaning of the above is a unitary operator acting on a set of basis states yields another set of basis states.)