The wavefunction of a Gaussian wavepacket for a particle of mass $m$ moving in one dimension is

$$\psi(x, t)=\frac{1}{\pi^{1 / 4}} \sqrt{\frac{1}{1+i \hbar t / m}} \exp \left(-\frac{x^{2}}{2(1+i \hbar t / m)}\right)$$

Show that $\psi(x, t)$ satisfies the appropriate time-dependent Schrödinger equation.

Show that $\psi(x, t)$ is normalized to unity and calculate the uncertainty in measurement of the particle position, $\Delta x=\sqrt{\left\langle x^{2}\right\rangle-\langle x\rangle^{2}}$.

Is $\psi(x, t)$ a stationary state? Give a reason for your answer.

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