Schroedinger Equation for Harmonic Oscillator

Clear["Global`*"]

v[x_] = (1/2) m (ω^2) (x^2)

v[x] ψ[x] -(1/2) m (((ψ'))^(′ ))[ x]==en ψ[x]

eq  =  %

m = 1;

ω = 1 ;

en = 0.50;

en1 = 0.49;

en2 =  0.51;

eq

ulim  =  3;

x0 =  0;

plotpot =  Plot[ v[x],  {x, x0, ulim}];

NDSolve[ {eq, ψ[x0]==1, ψ'[x0]==0}, ψ, {x,x0,ulim} ] ;

plot1 = Plot[ Evaluate[ ψ[x] /. % ], {x,x0,ulim} ] ;

en  =  en1;

NDSolve[ {eq, ψ[x0]==1, ψ'[x0]==0}, ψ, {x,x0,ulim} ] ;

\(\(plot2\  = \ 
      Plot[\ Evaluate[\ ψ[x]\  /. \ %\ ], \ {x, x0, ulim}, 

        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotStyle\  -> \ Dashing[{0.01, 0.01}]\ ]\ ; \)\)

en  =  en2;

NDSolve[ {eq,  ψ[x0] == 1,  ψ'[x0] == 0},  ψ, 
       {x, x0, ulim} ] ;

\(\(plot3\  = \ 
      Plot[\ Evaluate[\ ψ[x]\  /. \ %\ ], \ {x, x0, ulim}, 

        \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
        PlotStyle\  -> \ Dashing[{0.01, 0.01}]\ ]\ ; \)\)

Show[plotpot, plot1, plot2, plot3];