22.214.171.124 Solution hione-a
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 from the normalization requirement that the total probability of finding the particle integrated over all possible positions is one. Then evaluate the energy as , where according to chapter 3.5.3, the Hamiltonian is
Compare the ground state energy with the exact value,
(Hints: 6 and 30)
To satisfy the normalization requirement that the particle must be somewhere, you need 1, or substituting for ,
And by definition, chapter 2.3, the final inner product is just the integral which is given to be 30. So you must have
Now evaluate the expectation energy:
You can substitute in the value of from the normalization requirement above and apply the Hamiltonian on the function to its right:
The inner product is by definition the integral , which was given to be 6. So the final expectation energy is
The error in the approximation is only 1.3%! That is a surprisingly good result, since the parabola and the sine are simply different functions. While they may have superficial resemblance, if you scale each to unit height by taking 4 and 1, then the derivatives at 0 and are 4 respectively , off by as much as 27%.
If you go the next logical step, approximating the ground state with two functions as
where and are related by the normalization requirement 1, you find a ground state energy to a stunning, (for a two term approximation,) accuracy of 0.001 5%! However, the algebra becomes impossibly messy, so it was left out of the questions list. Similarly, a two point, linear interpolation, finite element version was left out, since there is so much baggage, it would distract from the true purpose of this book, to bring across the basic ideas of quantum mechanics to engineers.
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.