Chapter 7. The Eigenvalue Problem

7. The Eigenvalue Problem, December 17, 2009 6
Normally in quantum mechanics we choose the basis set {|a 1 ,...,|a n }
and know the operator Â. This means that we can calculate A mn = a m | Â | a n.
We do not know |ψ or λ, and our mission is to find them from Â|ψ = λ|ψ.
We have converted this operator equation, by using the orthonormal basis
set {|a 1 ,...,|a N }, into the matrix equation Eq. 23 (or Eq. 25 or Eq. 26).
In Eq. 23, we know A mn but we do not know ψ = {ψ 1 , ψ 2 ,...,ψ N } or λ.
WemustcalculatethemfromEq.23.Onceweknowψ 1 , ψ 2 ,...,ψ N ,wecan
calculate |ψ from
|ψ =
N
|a n a n | ψ ≡
n=1
N
ψ n |a n (27)
§ 6 Use a computer. Calculating eigenvalues and eigenvectors of a matrix
“by hand” is extremely tedious, especially since the dimension N is often very
large. Most computer languages (including Fortran, C, C++, Mathematica,
Maple, Mathcad, Basic) have libraries of functions that given a matrix will
return its eigenvalues and eigenvectors.
I will not discuss the numerical algorithms used for finding eigenvalues.
Quantum mechanics could leave that to experts without suffering much harm.
The Mathematica file Linear algebra for quantum mechanics.nb shows
how to use Mathematica to perform calculations with vectors and matrices.
Nevertheless, it is important to understand some of the theory related
to the eigenvalue problem, since much of quantum mechanics relies on it. I
presentsomeofitinwhatfollows.
§ 7 The eigenvalue problem and systems of linear equations. The eigenvalue
problem Eq. 23 can be written as
n=1
or
⎛
⎜
⎝
N
(A mn − λδ mn )ψ n =0, m =1, 2,...,N (28)
n=1
A 11 − λ A 12 A 13 ··· A 1N
A 21 A 22 − λ A 23 ··· A 2N
A 31 A 32 A 33 − λ ··· A 3N
. . .
. . . .
A N1 A N2 A N3 ··· A NN − λ
⎞ ⎛
⎟ ⎜
⎠ ⎝
ψ 1
ψ 2
ψ 3
.
ψ N
⎞ ⎛
=
⎟ ⎜
⎠ ⎝
0
0
0.
0
⎞
⎟
⎠
d
= 0
(29)