Page 153 - Numerical methods for chemical engineering

Eigenvalue problems in quantum mechanics 139

We next integrate each of these equations over [0, 2P] to obtain a set of algebraic
equations:
N N

b m h nm = E b m s nm (3.209)
m=−N m=−N
The coefficients s nm are
' 2P
∗
s nm = χ (x)χ m (x)dx (3.210)
n
0
and the coefficients h nm are
2P
'
ˆ
∗
h nm = χ (x)[Hχ m (x)]dx (3.211)
n
0
where the Hamiltonian operator is
h ¯ 2 d 2
ˆ
Hχ =− χ(x) + V (x)χ(x) (3.212)
2m e dx 2
We compute these integrals shortly, but first let us consider the structure of (3.209), which
we write as
N N

h nm b m = E s nm b m n = 0, ±1, ±2,..., ±N (3.213)
m=−N m=−N
Let us define the new indices
p = m + N + 1 m = 0, ±1, ±2,..., ±N p = 1, 2,..., 2N + 1
(3.214)
q = n + N + 1 n = 0, ±1, ±2,..., ±N q = 1, 2,..., 2N + 1
so that, with B = 2N + 1, we define a B × B overlap matrix S,a B × B Hamiltonian matrix
B
H, and a coefficient vector c ∈ C , such that
(3.215)
H pq = h mn S pq = s mn c p = b m
The system of equations (3.213) then becomes
B B

H qp c p = E S qp c p q = 1, 2,..., B (3.216)
p=1 p=1
This is the qth row of the generalized matrix eigenvalue problem

Hc = ESc (3.217)
H and S are both Hermitian and S is also positive-definite. Let the energy eigenvalues be
[k]
E k with Hc [k] = E k Sc . These E k are the allowable energy states of the system, whose
corresponding normalized wavefunctions are
B
[k]
ϕ (x)
[k] ϕ (x) = c χ m=p−N−1 (x) (3.218)
[k]
[k]
2P 2 p
ψ (x) = -
[k]
ϕ (x) dx
0 p=1
with the corresponding electron probability densities
[k]
[k]

ρ (x) = ψ (x) 2 (3.219)
e