Chapter 7. The Eigenvalue Problem

7. The Eigenvalue Problem, December 17, 2009 28
The left-hand side of Eq. 147 can be written as (use p |e(p)e(p)| = Î)
N
p=1 k=1
=
N
e(j) | Û −1 | e(p)e(p) | Ĥ | e(k)e(k) | Û | e(i)
N
p=1 k=1
N U
−1 H pk U ki = U −1 HU
jp ji
where we have denoted the matrix elements by
U
−1 jp ≡e(j) | Û −1 | e(p)
(149)
H pk ≡e(p) | Ĥ | e(k)
U ki ≡e(k) | Û | e(i)
and used the observation that the second double sum in Eq. 149 is the rule
for matrix multiplication. This allows us to write the last term (i.e. U −1 HU)
as the product of the matrices U −1 , H, andU. Combining Eq. 147 with
Eq. 149 proves the theorem.
We can go a step further and construct the matrix U from the eigenvectors
of Ĥ. Wehavedefined Û through Eq. 145,
Acting with e(j)| on this gives
|x(i) = Û|e(i)
e(j) | x(i) = e(j) | Û | e(i) = U ji (150)
Using the completeness relation for the basis set {|e(i)} N i=1, wehave
|x(i) =
N
|e(j)e(j) | x(i) (151)
j=1
This means that e(j) | x(i) are the components of the eigenket |x(i) in the
basis {|e(i)} N i=1. They are the numbers we obtain when we find the eigenvectors
of the matrix H; the eigenvectors are x(i) ={e(1) | x(i), e(2) | x(i),
..., e(N) | x(i) }. The matrix U is (see Eq. 150)
⎛
U =
⎜
⎝
e(1) | x(1) e(1) | x(2) ··· e(1) | x(N)
e(2) | x(1) e(2) | x(2) ··· e(2) | x(N)
e(3) | x(1) e(3) | x(2) ··· e(3) | x(N)
.
.
.. . .
e(N) | x(1) e(N) | x(2) ··· e(N) | x(N)
⎞
⎟
⎠
(152)