psi[y_,t_]=Exp[-(2 k t -y)^2/(4 (sigma^2 + I t)) + I k y - I k^2 t]/Sqrt[sigma^2 + I t]

I D[psi[y,t],t]+D[D[psi[y,t],y],y]
Simplify[%]
sigma=1; k=2 Pi
Plot[Re[psi[y,0]],{y,-4,16},PlotRange->All]
Plot[Re[psi[y,1]],{y,-4,16},PlotRange->All]
Plot[Re[psi[y,0]],{y,-4,4},PlotRange->All]
Plot[Re[psi[y,.05]],{y,-4,4},PlotRange->All]
Plot[Re[psi[y,.1]],{y,-4,4},PlotRange->All]

group ~ 12.5/1  (4 Pi expected)
phase ~ .7/.1   (2 Pi expected)

