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Clear["`*"]; f = 1/Sqrt[Cosh[r]]*((m ω)/(π h))^(1/4)* Exp[-((m ω)/(2 h))*x^2 - Conjugate[α]/ 2 (α + Conjugate[α] E^(I ϕ)*Tanh[r]) - I*ω*t - (m ω)/h E^(I ϕ) Tanh[r]* E^(-2*I*ω*t)* x^2 + (α + Conjugate[α] E^(I ϕ) Tanh[r])* E^(-I*ω*t)*Sqrt[(2 m ω)/h] x - 1/2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[r] x)^2 - Log[1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)]/ 2 - ((E^(I ϕ) Tanh[r]*E^(-2*I*ω*t))/(1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)))*1/ 2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[ r] x)^2] /. {r -> -3, ϕ -> 0, α -> 0, m -> 1, h -> 1, ω -> 1} // Re animation3 = Table[Plot[f, {x, -100, 100}, PlotRange -> {-0.3, 0.3}, PlotStyle -> {Thickness[0.0042], Blue}, Axes -> {False, False}, Frame -> True, PlotPoints -> 100, PlotLegends -> Placed[Framed[LineLegend[{Automatic}, {"r=-3,α=0"}]], ImageScaled[{0.8, 0.9}]]], {t, 0, 8, 0.1}] 

Clear["`*"]; f[x_, t_] =  1/Sqrt[Cosh[r]]*((m ω)/(π h))^(1/4)* Exp[-((m ω)/(2 h))*x^2 - Conjugate[α]/ 2 (α + Conjugate[α] E^(I ϕ)*Tanh[r]) - I*ω*t - (m ω)/h E^(I ϕ) Tanh[r]* E^(-2*I*ω*t)* x^2 + (α + Conjugate[α] E^(I ϕ) Tanh[r])* E^(-I*ω*t)*Sqrt[(2 m ω)/h] x - 1/2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[r] x)^2 - Log[1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)]/ 2 - ((E^(I ϕ) Tanh[r]*E^(-2*I*ω*t))/(1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)))*1/ 2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[ r] x)^2] /. {r -> -3, ϕ -> 0, α -> 0, m -> 1, h -> 1, ω -> 1} // Re animation1 = Animate[Plot[f[x, t], {x, -100, 100}, PlotRange -> {-0.3, 0.3}, PlotStyle -> {Thickness[0.0042], Blue}, Axes -> {False, False}, FrameStyle -> Directive[Black, Thick], FrameTicksStyle -> Directive[Thick, FontSize -> 20], Frame -> True, PlotPoints -> 100, PlotLegends ->   Placed[Framed[LineLegend[{Automatic}, {"r=0,α=0"}]], ImageScaled[{0.8, 0.9}]]], {t, 0, 6, 0.01}, AnimationRate -> 0.15, AnimationRunning -> False] (* animation3 = Table[Plot[f[x,t], {x, -100, 100}, PlotRange -> {-0.3, 0.3}, PlotStyle -> {Thickness[0.0042], Blue}, Axes -> {False, False}, Frame -> True, PlotPoints -> 100, PlotLegends -> Placed[Framed[LineLegend[{Automatic}, {"r=-3,α=0"}]], ImageScaled[{0.8, 0.9}]]], {t, 0, 8, 0.1}] *) 

But what are the values of m and ω and h?

Clear["`*"]; f = ψTimeDep[x_, r_, ϕ_, α_, t_] = 1/Sqrt[Cosh[r]]*((m ω)/(π h))^(1/4)*   Exp[-((m ω)/(2 h))*x^2 -   Conjugate[α]/   2 (α + Conjugate[α] E^(I ϕ)*Tanh[r]) -   I*ω*t - (m ω)/h E^(I ϕ) Tanh[r]*   E^(-2*I*ω*t)*   x^2 + (α + Conjugate[α] E^(I ϕ) Tanh[r])*   E^(-I*ω*t)*Sqrt[(2 m ω)/h] x -   1/2*(E^(-I*ω*t)*(α +   Conjugate[α] E^(I ϕ) Tanh[r]) -   E^(-2*I*ω*t)*   Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[r] x)^2 -   Log[1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)]/   2 - ((E^(I ϕ) Tanh[r]*E^(-2*I*ω*t))/(1 -   E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)))*1/   2*(E^(-I*ω*t)*(α +   Conjugate[α] E^(I ϕ) Tanh[r]) -   E^(-2*I*ω*t)*   Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[   r] x)^2] /. {r -> -3, ϕ -> 0, α -> 0} // Re animation3 = Table[Plot[f, {x, -100, 100}, PlotRange -> {-0.3, 0.3},   PlotStyle -> {Thickness[0.0042], Blue}, Axes -> {False, False},   Frame -> True, PlotPoints -> 100,   PlotLegends ->   Placed[Framed[LineLegend[{Automatic}, {"r=-3,α=0"}]],   ImageScaled[{0.8, 0.9}]]], {t, 0, 8, 0.1}]  

Clear["`*"];  f = 1/Sqrt[Cosh[r]]*((m ω)/(π h))^(1/4)* Exp[-((m ω)/(2 h))*x^2 - Conjugate[α]/ 2 (α + Conjugate[α] E^(I ϕ)*Tanh[r]) - I*ω*t - (m ω)/h E^(I ϕ) Tanh[r]* E^(-2*I*ω*t)* x^2 + (α + Conjugate[α] E^(I ϕ) Tanh[r])* E^(-I*ω*t)*Sqrt[(2 m ω)/h] x - 1/2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[r] x)^2 - Log[1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)]/ 2 - ((E^(I ϕ) Tanh[r]*E^(-2*I*ω*t))/(1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)))*1/ 2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[ r] x)^2] /. {r -> -3, ϕ -> 0, α -> 0, m -> 1, h -> 1, ω -> 1} // Re animation3 = Table[Plot[f, {x, -100, 100}, PlotRange -> {-0.3, 0.3}, PlotStyle -> {Thickness[0.0042], Blue}, Axes -> {False, False}, Frame -> True, PlotPoints -> 100, PlotLegends -> Placed[Framed[LineLegend[{Automatic}, {"r=-3,α=0"}]], ImageScaled[{0.8, 0.9}]]], {t, 0, 8, 0.1}] 

Replace := by = and define a new f

Clear["`*"]; f = ψTimeDep[x_, r_, ϕ_, α_, t_] = 1/Sqrt[Cosh[r]]*((m ω)/(π h))^(1/4)* Exp[-((m ω)/(2 h))*x^2 - Conjugate[α]/ 2 (α + Conjugate[α] E^(I ϕ)*Tanh[r]) - I*ω*t - (m ω)/h E^(I ϕ) Tanh[r]* E^(-2*I*ω*t)* x^2 + (α + Conjugate[α] E^(I ϕ) Tanh[r])* E^(-I*ω*t)*Sqrt[(2 m ω)/h] x - 1/2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[r] x)^2 - Log[1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)]/ 2 - ((E^(I ϕ) Tanh[r]*E^(-2*I*ω*t))/(1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)))*1/ 2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[ r] x)^2] /. {r -> -3, ϕ -> 0, α -> 0} // Re animation3 = Table[Plot[f, {x, -100, 100}, PlotRange -> {-0.3, 0.3}, PlotStyle -> {Thickness[0.0042], Blue}, Axes -> {False, False}, Frame -> True, PlotPoints -> 100, PlotLegends -> Placed[Framed[LineLegend[{Automatic}, {"r=-3,α=0"}]], ImageScaled[{0.8, 0.9}]]], {t, 0, 8, 0.1}] 

Replace := by = and define a new f.

But what are the values of m and ω and h?

Clear["`*"]; f = ψTimeDep[x_, r_, ϕ_, α_, t_] = 1/Sqrt[Cosh[r]]*((m ω)/(π h))^(1/4)* Exp[-((m ω)/(2 h))*x^2 - Conjugate[α]/ 2 (α + Conjugate[α] E^(I ϕ)*Tanh[r]) - I*ω*t - (m ω)/h E^(I ϕ) Tanh[r]* E^(-2*I*ω*t)* x^2 + (α + Conjugate[α] E^(I ϕ) Tanh[r])* E^(-I*ω*t)*Sqrt[(2 m ω)/h] x - 1/2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[r] x)^2 - Log[1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)]/ 2 - ((E^(I ϕ) Tanh[r]*E^(-2*I*ω*t))/(1 - E^(I ϕ) Tanh[r]*E^(-2*I*ω*t)))*1/ 2*(E^(-I*ω*t)*(α + Conjugate[α] E^(I ϕ) Tanh[r]) - E^(-2*I*ω*t)* Sqrt[(2 m ω)/h] E^(I ϕ) Tanh[ r] x)^2] /. {r -> -3, ϕ -> 0, α -> 0} // Re animation3 = Table[Plot[f, {x, -100, 100}, PlotRange -> {-0.3, 0.3}, PlotStyle -> {Thickness[0.0042], Blue}, Axes -> {False, False}, Frame -> True, PlotPoints -> 100, PlotLegends -> Placed[Framed[LineLegend[{Automatic}, {"r=-3,α=0"}]], ImageScaled[{0.8, 0.9}]]], {t, 0, 8, 0.1}]