A particle of mass $m$ is in the state
$$
\Psi(x, t)=A e^{-a\left[\left(m x^{2} / \hbar\right)+i t\right]}
$$
where $A$ and $a$ are positive real constants.
(a) Find $A$.
(b) For what potential energy function $V(x)$ does $\Psi$ satisfy the Schrödinger equation?
(c) Calculate the expectation values of $x, x^{2}, p$, and $p^{2}$.
(d) Find $\sigma_{x}$ and $\sigma_{p}$. Is their product consistent with the uncertainty principle?
