Solution
= Solution
For $\psi=Ae^{-Bx^2}$,
$$
\psi_t=\left(\frac{A'}A-B'x^2\right)\psi,
\qquad
\psi_{xx}=(-2B+4B^2x^2)\psi.
$$
Substitution into the <Schrödinger equation> and comparison of the constant and $x^2$ coefficients gives
$$
\boxed{A'=-i\hbar AB,
\qquad
B'=-\frac{i}{2\hbar}-2i\hbar B^2}.
$$