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Eigenvalue problems in quantum mechanics 137

factorization of B,
B = LL T (3.195)

and write
T
Aw = λLL w (3.196)
We now define the transformation
T
z = L w w = L T(−1) z (3.197)
to obtain the corresponding eigenvalue problem
−1 T(−1)
L AL z = λz (3.198)
−1
If A is symmetric, so is L AL T (−1) .As L is lower triangular, we can compute L −1 very
quickly column-by-column by forward substitution
   
| | | | ˜ [ j] [ j]
LL −1 = L ˜ [1] ... l ˜ [N]  = I =  [1] ... e [N]  Ll = e
l
e
j = 1, 2,..., N
| | | |
(3.199)
−1
Once we obtain the eigenvalues λ j and eigenvectors z [j] of L AL T (−1) , we compute the
corresponding generalized eigenvectors
[ j] T(−1) [ j]
w = L z (3.200)
eig and eigs allow the use of an optional nonsingular matrix B in the problem Aw = λBw
(type help eig or help eigs for further details). For
2 1 −1 2 −1 0
   
A =  1 4 −2  B =  −1 2 −1  (3.201)
−1 −2 6 0 −1 2
the generalized eigenvalues satisfying Aw = λBw are computed by

A = [2 1 -1; 1 4 -2; -1 -2 6];
B=[2-10;-12-1;0-12];
e = eig(A,B),
e=
0.5664
3.1128
4.8207

Eigenvalue problems in quantum mechanics

Eigenvalue analysis lies at the heart of quantum mechanics. Here we consider only a simple
example involving a single electron in one dimension, but the numerical approach is the
same as that used in more realistic 3-D calculations of atoms and molecules. We wish to