5.3.2 Solution 2state-b
Show that it does not have an effect on the solution whether or not the basic states and are normalized, like in the previous question, before the state of lowest energy is found.
This requires no detailed analysis; just check that the same solution can be described using the nonorthogonal and orthogonal basis states. It is however an important observation for various numerical solution procedures: your set of basis functions can be cleaned up and simplified without affecting the solution you get.
Using the original basis states, the solution, say the ground state of lowest energy, can be written in the form
for some values of the constants and . Now the expression for the orthogonalized functions,
can for given and be thought of as two equations for and that can be solved. In particular, adding times the second equation to the first gives
Similarly, adding times the first equation to the second gives
If this is plugged into the expression for the solution, , it takes the form
So, while the constants and are different from and , the same solution can be found equally well in terms of and as in terms of and .
In the terms of linear algebra, and
span the same function space as and : any wave function that can be described as a combination of and can also be described in terms of and , although with different constants. This is true as long as the definitions of the new functions can be solved for the old functions as above. The matrix of coefficients, here
must have a nonzero determinant.