how to normalize results from ode45

I solved using the shooting method the first 3 level of the anharmonic quantum potential well

I get a plot like this

since I've found the three energys with fzero I simply run this code

Any suugest in how to normalize?

thank you

x=linspace(-0.5,+0.5);
V=alpha*x.^2+beta*x.^4;
xspan = [-0.5,0.5];
options = odeset('RelTol',1e-8,'absTol',[1e-8,1e-8]);
for i=1:length(ENERGIE)
    Etrial=ENERGIE(i);
    [T,Y] = ode45(@Anharmonic,xspan,init_cond,options);
    plot(T,Y(:,1))
end 

[darova]

What normalazing means?

[Simo]

Integrations of square modulus between -inf and +inf equal to 1

Answer 1

[darova]

What if just to scale the data?

clc,clear
x = -1:.1:1;
y = 2 - x.^2;
plot(x,y)           % plot original data
s = trapz(x,y)      % find area ander the curve
y1 = y/s;           % scale data
line(x,y1)          % plot scaled data
trapz(x,y1)         % calculate area under the scaled curve