The Harmonic Oscillator

ClearAll["Global`*"]
xmax0 = (Sqrt[2 0 + 1]); 
xmax1 = (Sqrt[2 1 + 1]); 
xmax2 = (Sqrt[2 2 + 1]); 
xmax3 = (Sqrt[2 3 + 1]); 
xmax4 = (Sqrt[2 4 + 1]); 
potential = 
      Plot[((x^2)/2), {x, -5, 5}, PlotStyle -> {RGBColor[0, 0, 1]}, 
        Epilog -> {Line[{{-xmax0, 0.5}, {xmax0, 0.5}}], 
            Text["n=0", {xmax0 + 1, 0.5}], 
            Line[{{-xmax1, 1.5}, {xmax1, 1.5}}], 
            Text["n=1", {xmax1 + 1, 1.5}], 
            Line[{{-xmax2, 2.5}, {xmax2, 2.5}}], 
            Text["n=2", {xmax2 + 1, 2.5}], 
            Line[{{-xmax3, 3.5}, {xmax3, 3.5}}], 
            Text["n=3", {xmax3 + 1, 3.5}], 
            Line[{{-xmax4, 4.5}, {xmax4, 4.5}}], 
            Text["n=4", {xmax4 + 1, 4.5}]}];

n = 2; 
xmax = (Sqrt[2 n + 1]); 
hermite2 = 
      Plot[((2 (HermiteH[n, x] (E^(-((x^2)/2)))))/(Sqrt[(2^n) n!])) + n + 0.5, {x, -5, 5}, PlotRange -> All, 
        DisplayFunction -> Identity, 
        PlotStyle -> {RGBColor[1, 0, 1]}]; 
prob2 = 
      Plot[((4 (((HermiteH[n, x] (E^(-((x^2)/2)))))^2))/((2^n) n!)) + n + 0.5 + 
          4, {x, -5, 5}, PlotRange -> All, 
        DisplayFunction -> Identity, 
        PlotStyle -> {RGBColor[0, 1, 0]}]; 
potential = 
      Plot[((x x)/2), {x, -5, 5}, 
        PlotRange -> {{-5, 5}, {0, 12}}, 
        DisplayFunction -> Identity, 
        PlotStyle -> {RGBColor[0, 0, 1]}, 
        Epilog -> {Line[{{-xmax, n + 0.5}, {xmax, n + 0.5}}], 
            Line[{{-xmax, 0}, {-xmax, 15}}], 
            Line[{{xmax, 0}, {xmax, 15}}], Text["n=2", {1, 11}], 
            Text["ψ(x)", {4, n + 0.5}], 
            Text["ψ*(x)ψ(x)", {4, n + 0.5 + 4}]}]; 
Show[potential, hermite2, prob2, 
      DisplayFunction -> $DisplayFunction]; 
n = 3; 
xmax = (Sqrt[2 n + 1]); 
hermite3 = 
      Plot[((2 (HermiteH[n, x] (E^(-((x^2)/2)))))/(Sqrt[(2^n) n!])) + n + 0.5, {x, -5, 5}, PlotRange -> All, 
        DisplayFunction -> Identity, 
        PlotStyle -> {RGBColor[1, 0, 1]}]; 
prob3 = 
      Plot[((4 (((HermiteH[n, x] (E^(-((x^2)/2)))))^2))/((2^n) n!)) + n + 0.5 + 
          4, {x, -5, 5}, PlotRange -> All, 
        DisplayFunction -> Identity, 
        PlotStyle -> {RGBColor[0, 1, 0]}]; 
potential = 
      Plot[((x x)/2), {x, -5, 5}, 
        PlotRange -> {{-5, 5}, {0, 12}}, 
        DisplayFunction -> Identity, 
        PlotStyle -> {RGBColor[0, 0, 1]}, 
        Epilog -> {Line[{{-xmax, n + 0.5}, {xmax, n + 0.5}}], 
            Line[{{-xmax, 0}, {-xmax, 15}}], 
            Line[{{xmax, 0}, {xmax, 15}}], Text["n=3", {1, 11}], 
            Text["ψ(x)", {4, n + 0.5}], 
            Text["ψ*(x)ψ(x)", {4, n + 0.5 + 4}]}]; 
Show[potential, hermite3, prob3, 
      DisplayFunction -> $DisplayFunction];

n = 70; 
xmax = (Sqrt[2 n + 1]); 
hermiten = 
      Plot[((HermiteH[n, x] (E^(-((x^2)/2))))/(Sqrt[(2^n) n!])), {x, -14, 14}, PlotRange -> All, 
        PlotStyle -> {RGBColor[1, 0, 1]}, 
        Epilog -> {Line[{{-xmax, -1}, {-xmax, 1}}], 
            Line[{{xmax, -1}, {xmax, 1}}]}]; 
prob = 
      Plot[((((HermiteH[n, x] (E^(-((x^2)/2)))))^2)/((2^n) n!)), {x, -14, 14}, PlotRange -> All, 
        DisplayFunction -> Identity, 
        PlotStyle -> {RGBColor[1, 0, 1]}, 
        Epilog -> {Line[{{-xmax, -1}, {-xmax, 1}}], 
            Line[{{xmax, -1}, {xmax, 1}}]}]; 
classical = 
      Plot[(1/(Sqrt[π ((2 n + 1) - x x)])), {x, -xmax + 0.1, 
          xmax - 0.1}, PlotStyle -> {RGBColor[0, 0, 1]}, 
        DisplayFunction -> Identity]; 
Show[prob, classical, 
      DisplayFunction -> $DisplayFunction];