Normalizing Trial Function with 2 Normalization Factors

[timman_24]

Homework Statement

I am given a trial function and before I use the variational method, I need to normalize the trial function. This is easy usually, but I don't know what to do in this specific case:

The trial function is:
X[x]=N1(1-x^2)+N2(x-x^3)
Domain: -1<x<1

N1 and N2 are the normalization factors that I need to solve for.

My problem is that I do not know how to normalize a sum of linear equations each with their own normalization factor. The problem says nothing about weighing one trial term more over another. I thought about normalizing each separately setting them to 50% weight, but I thought that was a cop out.

2. The attempt at a solution

Another thing I tried was writing two equations and solving for the 2 unknown N1 and N2. I split the domain and integrated X[x]^2 from -1 to 0 in one equation and then from 0 to 1 in the other. I then solved the equations simultaneously. However, one N always goes to zero which I do not want.

Does anyone know the proper way to do this?

Thanks

 

[gabbagabbahey]

The normalization condition is still just

$$1=\int_{-\infty}^{\infty}|\psi(x)|^2dx=\int_{-1}^{1}|X(x)|^2dx$$

perform the integration and you will get a relationship between $N_1$ and $N_2$, leaving you with one free parameter to vary.

 

[timman_24]

Okay, so one of the N's will be the free parameter I solve for to get the ground state energy?

Thanks for the help.