I calculated the time evolution of the free particle $$\psi(x,0)=A\exp(-x^2/2\sigma^2)\exp(i\alpha x^2),$$ with a positive real parameter $\alpha>0$.
$$\Psi(x,t)=A\sigma\sqrt{\frac{m}{m\sigma^2+i\hbar t(2i\sigma^2 \alpha+1)}}\exp\left[\frac{k^2m(2i\sigma^2\alpha+1)}{2(m\sigma^2+i\hbar t(2i\sigma^2 \alpha+1))}\right]$$
and I am going to study how does $\alpha$ affect the motion of the particle
First of all, for the constant infront of exponential, when $\alpha$ increase, the wave packet should spread out more, and there is a time dependence also in the denominator $\propto t$ which creates a further dispersion effect?
For the exponential term, more neatly, written $i\alpha'=2i\sigma^2\alpha+1$ would gives $$\exp\left[\frac{ik^2m}{2(\frac{m\sigma^2}{i\alpha'}-\hbar t))}\right].$$ I don't know how to track the effect of the imaginary part, where the exponent has no clear imaginary part, that the imaginary unit i couple in a very complicate way
