Page 152 - Numerical methods for chemical engineering

138 3 Matrix eigenvalue analysis

compute the energy states of a single electron in a 1-D external potential field that has the
spatial periodicity
V (x + n2P) = V (x) n = 0, ±1, ±2,... (3.202)

Such a periodic system may be interpreted to be a 1-D “crystal.”
2
The probability of finding an electron in [x, x + dx]is |ψ(x)| dx, where ψ(x)isthe wave-
function of the electron, satisfying the Schr¨odinger equation,
2
h ¯ 2 d ψ
− 2 + V (x)ψ(x) = Eψ(x) (3.203)
2m e dx
¯
m e is the mass of an electron and h = h/2π where h is Planck’s constant. E is the energy of
the electron. For a more detailed discussion, consult Leach (2001) and Atkins & Friedman
(1989).

Numerical solution of a differential equation eigenvalue problem
In (3.203), we have a differential equation eigenvalue problem, but we have been study-
ing techniques to solve matrix eigenvalue problems. To convert this problem into a matrix
one, we expand ψ(x) as a linear combination of basis functions that satisfy the appro-
priate boundary conditions; here the periodicity condition ψ(x + n2P) = ψ(x), since
if V(x) is periodic, we expect ψ(x) to be also. We choose a plane wave basis set with
members

χ m (x) = e iq m x = cos(q m x) + i sin(q m x) (3.204)

The reasoning behind this choice of basis set is discussed in Chapter 9. Periodicity is satisfied
if the allowable wavenumbers q m satisfy
mπ
q m = m = 0, ±1, ±2,... (3.205)
P
Using this basis, we write a trial form of the wavefunction as
N N
iq m x
ψ(x) = b m χ m (x) = b m e (3.206)
m=−N m=−N
We wish to compute the coefficients {b m } that satisfy (3.203) best and thus truncate the
expansion to order N to make the problem dimension finite. Substituting this expansion into
(3.203), we have
N 2 N
h ¯ 2 d
b m − χ m (x) + V (x)χ m (x) = E b m χ m (x) (3.207)
2m e dx 2
m=−N m=−N
We now multiply this differential equation by the complex conjugates of each basis function,
∗
χ (x), to obtain a set of equations:
n
N ¯ 2 2 N
h d
∗ ∗
b m χ (x) − 2 χ m (x) + V (x)χ m (x) = E b m χ (x)χ m (x) (3.208)
n
n
m=−N 2m c dx m=−N