5.8 Matrix Formulation
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
or in index notation:
where is the total count of the chosen -particle wave
functions and the single counter in replaces a set of
indices in the description used in the previous section. The
-particle functions are allowed to be anything;
individual (Hartree) products of single-particle wave functions for
distinguishable particles as in (5.30), Slater determinants
for identical fermions, permanents for identical bosons, or whatever.
The only thing that will be assumed is that they are mutually
orthonormal. (Which means that any underlying set of single-particle
functions as described in the previous section should
be orthonormal. If they are not, there are procedures like
Gram-Schmidt to make them so. Or you can just put in some correction
Under those conditions, the energy eigenvalue problem
takes the form:
The trick is now to take the inner product of both sides of this
equation with each function in the set of wave
functions in turn. In other words, take an inner product with
to get one equation, then take an inner product with
to get a second equation, and so on. This produces,
using the fact that the functions are orthonormal to clean up the
are the matrix coefficients, or Hamiltonian coefficients.
This can again be written more compactly in index notation:
which is just a finite-size matrix eigenvalue problem.
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 .
- 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.
5.8 Review Questions
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
can according to the rules of calculus be factored into three-dimensional integrals as
which is zero if and are orthonormal.
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 :
Next note that you also have
because they are the exact same inner product integrals; the difference is just which electron you number 1 and which one you number 2 that determines whether the wave functions are listed as or .
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
and then multiplied out into inner products of matching spin components to give
The other 15 matrix coefficients can be found similarly, and most will be zero.
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.