4.6.6.1 Solution hione-a

**Question:**

The solution for the hydrogen molecular ion requires elaborate evaluations of inner product integrals and a computer evaluation of the state of lowest energy. As a much simpler example, you can try out the variational method on the one-dimensional case of a particle stuck inside a pipe, as discussed in chapter 3.5. Take the approximate wave function to be:

Find

Compare the ground state energy with the exact value,

(Hints:

**Answer:**

To satisfy the normalization requirement that the particle must be somewhere, you need

And by definition, chapter 2.3, the final inner product is just the integral

Now evaluate the expectation energy:

You can substitute in the value of

The inner product is by definition the integral

The error in the approximation is only 1.3%! That is a surprisingly good result, since the parabola

If you go the next logical step, approximating the ground state with two functions as

where

Now, if you read the next subsection, you will see that in real-life, multi-dimensional, problems, getting results this accurate is difficult. Still, if you are desperate for a good solution of these very complex problems by a simple and reliable means, variational methods are hard to beat.