When the number of unknowns in a quantum mechanical problem has been reduced to a finite number, the problem can be reduced to a linear algebra one. This allows the problem to be solved using standard analytical or numerical techniques. This section describes how the linear algebra problem can be obtained.
Typically, quantum mechanical problems can be reduced to a finite number of unknowns using some finite set of chosen wave functions, as in the previous section. There are other ways to make the problems finite, it does not really make a difference here. But in general some simplification will still be needed afterwards. A multiple sum like equation (5.30) for distinguishable particles is awkward to work with, and when various coefficients drop out for identical particles, its gets even messier. So as a first step, it is best to order the terms involved in some way; any ordering will in principle do. Ordering allows each term to be indexed by a single counter , being the place of the term in the ordering.
Using an ordering, the wave function for a total of particles can
be written more simply as
(5.32) |
Under those conditions, the energy eigenvalue problem
takes the form:
This can again be written more compactly in index notation:
(5.33) |
Since the functions are known, chosen, functions, and the Hamiltonian is also known, the matrix coefficients can be determined. The eigenvalues and corresponding eigenvectors can then be found using linear algebra procedures. Each eigenvector produces a corresponding approximate eigenfunction with an energy equal to the eigenvalue .
Key Points
- Operator eigenvalue problems can be approximated by the matrix eigenvalue problems of linear algebra.
- That allows standard analytical or numerical techniques to be used in their solution.
As a relatively simple example, work out the above ideas for the 2 hydrogen molecule spatial states and . Write the matrix eigenvalue problem and identify the two eigenvalues and eigenvectors. Compare with the results of section 5.3.
Assume that and have been slightly adjusted to be orthonormal. Then so are and orthonormal, since the various six-dimensional inner product integrals, like
Also, do not try to find actual values for , , , and . As section 5.2 noted, that can only be done numerically. Instead just refer to as and to as :
Find the eigenstates for the same problem, but now including spin.
As section 5.7 showed, the antisymmetric wave function with spin consists of a sum of six Slater determinants. Ignoring the highly excited first and sixth determinants that have the electrons around the same nucleus, the remaining 4 Slater determinants can be written out explicitly to give the two-particle states
Note that the Hamiltonian does not involve spin, to the approximation used in most of this book, so that, following the techniques of section 5.5, an inner product like can be written out like
If you do not have experience with linear algebra, you may want to skip this question, or better, just read the solution. However, the four eigenvectors are not that hard to guess; maybe easier to guess than correctly derive.