The wavefunction of a normalised Gaussian wavepacket for a particle of mass $m$ in one dimension with potential $V(x)=0$ is given by

$$\psi(x, t)=B \sqrt{A(t)} \exp \left(\frac{-x^{2} A(t)}{2}\right)$$

where $A(0)=1$. Given that $\psi(x, t)$ is a solution of the time-dependent Schrödinger equation, find the complex-valued function $A(t)$ and the real constant $B$.

[You may assume that $\int_{-\infty}^{\infty} e^{-\lambda x^{2}} d x=\sqrt{\pi} / \sqrt{\lambda} .$ ]