ψ[x_]:=(((β (Sqrt[π])))^(-1/2))Exp[-(((((x-x0))^2) )/(2 (β^2)))]Exp[I k0 x]

ψk[k_]:= (1/(Sqrt[2π]))(∫_(-∞)^∞)Exp[-I k x]ψ[x]\[DifferentialD]x

ψxt[x, t]:= (1/(Sqrt[2 π]))(∫_(-∞)^∞)Exp[I (k x - c (k^2) t)]ψk[k]\[DifferentialD]k (*note: c is not speed, c is constant*)

Assuming[c>0&&β>0&& t∈Reals && k0∈Reals && x∈Reals && ω∈Reals&& x0∈Reals,FullSimplify[ψxt[x, t] Conjugate[ψxt[x, t]]]]

Assuming[c>0&&β>0&& t∈Reals && k0∈Reals && x∈Reals && ω∈Reals&& x0∈Reals,FullSimplify[ψxt[x, t] ]]

s:=(1/(2 (b^2)))(((I (x-x0)+(β^2)k0))^2)-((β^2)/2)(k0^(2 ))+ I k0 x0

Assuming[c>0&&β>0&& t∈Reals && k0∈Reals && x∈Reals && ω∈Reals&& x0∈Reals,FullSimplify[  Exp[s]Conjugate[Exp[s]]       ]]

FullSimplify[ψk[k]]

Assuming[a∈Reals&&b∈Reals&&x∈Reals&&y∈Reals,FullSimplify[Conjugate[((a+b I)/(x+ y I))]]]

0

FullSimplify[D[(1/(Sqrt[2π t]))Exp[-((x^2)/(2t))],{x,2}]]

FullSimplify[D[(1/(Sqrt[2π t]))Exp[-((x^2)/(2t))],{t,1}]]