Nonlinear Variation The Hydrogen Atom

We have already seen (Chapter 4) that the lowest-energy eigenfunction for the hydrogen atom is (in atomic units) 

Suppose we did not know this and used the variation method to optimize the normalized trial function 

In this example, when = becomes identical to but in more complicated systems the trial function never becomes identical to an eigenfunction of the hamiltonian. Nevertheless, this is a good example to start with since there are few mathematical complexities to obscure the philosophy of the approach. 

Incorporating this into Eq. (7-8) gives (after integrating 9 and cpiadv to give 47t) -3x 2 

Now we have a simple expression for i as a function of f. To obtain the minimum, we set the derivative of E to zero  

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