Infinite Square Potential

Animate[Manipulate[Show[Plot[a Re[Sin[n π x] Exp[I (n^2) t]]+b Re[Sin[m π x] Exp[I (m^2) t]]+c Re[Sin[l π x] Exp[I (l^2) t]],{x,0,1},PlotRange->{{0,1},{-(a+b+c),(a+b+c)}},PlotStyle->{Black,Thick},AxesLabel->{"x","ψ"}],
Plot[a Re[Sin[n π x] Exp[I (n^2) t]],{x,0,1},PlotRange->{{0,1},{-3,3}},PlotStyle->{Blue,Dashed}],Plot[b Re[Sin[m π x] Exp[I (m^2) t]],{x,0,1},PlotRange->{{0,1},{-3,3}},PlotStyle->{Green,Dashed}],Plot[c Re[Sin[l π x] Exp[I (l^2) t]],{x,0,1},PlotRange->{{0,1},{-3,3}},PlotStyle->{Gray,Dashed}]],Style[(Subscript["ψ", "1"]),16,Bold],Style["(Blue)",10],{{a,0.5,(Subscript["C", 1])},0,1,0.1},{n,1,4,1},Delimiter,Style[(Subscript["ψ", "2"]),16,Bold],Style["(Green)",10],{{b,0,(Subscript["C", 2])},0,1,0.1},{m,1,4,1},Delimiter,Style[(Subscript["ψ", "3"]),16,Bold],Style["(Gray)",10],{{c,0,(Subscript["C", 3])},0,1,0.1},{l,1,4,1},ContentSize->{1000}],{{t,0,"Time"},0,2 π},ContentSize->{1000},AnimationRunning->False]

Animate[Manipulate[Show[Plot[a Re[Sin[n π x] Exp[I (n^2) t]]+b Re[Sin[m π x] Exp[I (m^2) t]]+c Re[Sin[l π x] Exp[I (l^2) t]],{x,0,1},PlotRange->{{0,1},{-(a+b+c),(a+b+c)}},PlotStyle->{Black,Thick},AxesLabel->{"x","ψ"}],
Plot[a Re[Sin[n π x] Exp[I (n^2) t]],{x,0,1},PlotRange->{{0,1},{-3,3}},PlotStyle->{Blue,Dashed}],Plot[b Re[Sin[m π x] Exp[I (m^2) t]],{x,0,1},PlotRange->{{0,1},{-3,3}},PlotStyle->{Green,Dashed}],Plot[c Re[Sin[l π x] Exp[I (l^2) t]],{x,0,1},PlotRange->{{0,1},{-3,3}},PlotStyle->{Gray,Dashed}]],Style[(Subscript["ψ", "1"]),16,Bold],Style["(Blue)",10],{{a,0.5,(Subscript["C", 1])},0,1,0.1},{n,1,4,1},Delimiter,Style[(Subscript["ψ", "2"]),16,Bold],Style["(Green)",10],{{b,0,(Subscript["C", 2])},0,1,0.1},{m,1,4,1},Delimiter,Style[(Subscript["ψ", "3"]),16,Bold],Style["(Gray)",10],{{c,0,(Subscript["C", 3])},0,1,0.1},{l,1,4,1},ContentSize->{1000}],{{t,0,"Time"},0,2 π},ContentSize->{1000},AnimationRunning->False]

Harmonic Oscillator

Animate[Manipulate[Show[Plot[a Re[HermiteH[n,x]x Exp[-x^2]Exp[-I (n+1/2) t]]+b Re[HermiteH[m,x]x Exp[-x^2]Exp[-I (m+1/2) t]]+c Re[HermiteH[l,x]x Exp[-x^2]Exp[-I (l+1/2) t]],{x,-4,4},
PlotRange->{{-3,3},{-(-1+n+m+l),(-1+n+m+l)}},PlotStyle->{Black,Thick},AxesLabel->{"x","ψ"}],Plot[a Re[HermiteH[n,x]x Exp[-x^2]Exp[-I(n+1/2) t]],{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{Blue,Dashed}],Plot[b Re[HermiteH[m,x]x Exp[-x^2]Exp[-I(m+1/2) t]],{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{Green,Dashed}],Plot[c Re[HermiteH[l,x]x Exp[-x^2]Exp[-I(l+1/2) t]],{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{Gray,Dashed}]],Style[(Subscript["ψ", "1"]),16,Bold],Style["(Blue)",10],{{a,0.5,(Subscript["C", 1])},0,1,0.1},{n,0,3,1},Delimiter,Style[(Subscript["ψ", "2"]),16,Bold],Style["(Green)",10],{{b,0,(Subscript["C", 2])},0,1,0.1},{m,0,3,1},Delimiter,Style[(Subscript["ψ", "3"]),16,Bold],Style["(Gray)",10],{{c,0,(Subscript["C", 3])},0,1,0.1},{l,0,3,1},ContentSize->{1000}],{{t,0,"Time"},0,4π},ContentSize->{1000},AnimationRunning->False]

Animate[Manipulate[Show[Plot[a Re[HermiteH[n,x]x Exp[-x^2]Exp[-I (n+1/2) t]]+b Re[HermiteH[m,x]x Exp[-x^2]Exp[-I (m+1/2) t]]+c Re[HermiteH[l,x]x Exp[-x^2]Exp[-I (l+1/2) t]],{x,-4,4},
PlotRange->{{-3,3},{-(-1+n+m+l),(-1+n+m+l)}},PlotStyle->{Black,Thick},AxesLabel->{"x","ψ"}],Plot[a Re[HermiteH[n,x]x Exp[-x^2]Exp[-I(n+1/2) t]],{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{Blue,Dashed}],Plot[b Re[HermiteH[m,x]x Exp[-x^2]Exp[-I(m+1/2) t]],{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{Green,Dashed}],Plot[c Re[HermiteH[l,x]x Exp[-x^2]Exp[-I(l+1/2) t]],{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{Gray,Dashed}]],Style[(Subscript["ψ", "1"]),16,Bold],Style["(Blue)",10],{{a,0.5,(Subscript["C", 1])},0,1,0.1},{n,0,3,1},Delimiter,Style[(Subscript["ψ", "2"]),16,Bold],Style["(Green)",10],{{b,0,(Subscript["C", 2])},0,1,0.1},{m,0,3,1},Delimiter,Style[(Subscript["ψ", "3"]),16,Bold],Style["(Gray)",10],{{c,0,(Subscript["C", 3])},0,1,0.1},{l,0,3,1},ContentSize->{1000}],{{t,0,"Time"},0,4π},ContentSize->{1000},AnimationRunning->False]