Self-Consistent Field Theory and Its Applications by M. W. Matsen

1.5 Mathematical Techniques and Approximations 19
in quantum mechanics, the eigenfunctions are orthogonal and can be normalized to satisfy,
∫
1
dr f i (r)f j (r) =δ ij (1.60)
V
where V is the total volume of the system. Furthermore, the eigenfunctions form a complete
basis, which implies that any function of position, g(r), can be expanded as
g(r) =
∞∑
g i f i (r) (1.61)
i=0
The coefficients, g i , of the expansion are determined by multiplying Eq. (1.61) by f j (r),
integrating over r, and using the orthonormal condition from Eq. (1.60). This gives the
formula,
g i = 1 ∫
drf i (r)g(r) (1.62)
V
Once the orthonormal basis functions have been obtained, it is a trivial matter to evaluate
q(r,s). One starts by expanding it as
q(r,s)=
∞∑
q i (s)f i (r) (1.63)
i=0
and substituting the expansion into Eq. (1.24). Given the inherent property of the eigenfunctions,
Eq. (1.59), the diffusion equation reduces to
∞∑
∞∑
q i ′ (s)f i(r) =− λ i q i (s)f i (r) (1.64)
i=0
i=0
With the same steps used to derive Eq. (1.62), it follows that
q ′ i(s) =−λ i q i (s) (1.65)
for all i =0,1,2,.... Differential equations do not come any simpler. Inserting the solution into
Eq. (1.63) gives the final result,
q(r,s)=
∞∑
c i exp(−λ i s)f i (r) (1.66)
i=0
where the constant,
c i ≡ q i (0) = 1 ∫
V
dr f i (r)q(r, 0) (1.67)
is determined by the initial condition. Similarly, the expression for the complementary partition
function is
∞∑
q † (r,s)= c † i exp(−λ i(1 − s))f i (r) (1.68)
i=0