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

1.5 Mathematical Techniques and Approximations 21
for the entropic free energy. The logarithmic term can be dropped in the large-N limit. An
integration by parts is then performed to reexpress the equation as
∫
f e
k B T = a2 N
dr |∇f 0 (r)| 2 (1.77)
6V
where the boundary term vanishes as a result of the assumption, ∇φ α (r) =0, along the edge
of V. For the final step of the derivation, the eigenfunction, f 0 (r), is replaced by the segment
concentration, φ α (r), using Eqs. (1.73) and (1.74). This produces the Landau-Ginzburg free
energy expression,
∫
f e
k B T = a2 ρ 0
24
dr |∇φ α(r)| 2
φ α (r)
(1.78)
which gives the entropic energy without the need to evaluate the underlying field, w(r). The
derivation assumes only that the molecular weight (i.e., N) of the polymer is large.
1.5.3 Fourier Representation
Quantum mechanical problems are often made more tractable by reexpressing them in socalled
momentum space. The principle is very similar to that of the spectral method, where
quantities are expanded in terms of energy eigenstates. The main difference is that the expansions
are now in terms of eigenfunctions, exp(ik · r), of the momentum operator, ˆp ≡−i∇.
(Here, i ≡ √ −1.) The other difference is that the momentum eigenvalue, k, is continuous,
which implies that sums are now replaced by integrals and Kronecker delta functions are
substituted by Dirac delta functions. For example, the orthogonality condition in Eq. (1.60)
becomes
∫
1
(2π) 3 dr exp(i(k 1 − k 2 ) · r) =δ(k 1 − k 2 ) (1.79)
Furthermore, the expansion of an arbitrary function, g(r), becomes the integral,
g(r) = 1 ∫
(2π) 3 dk g(k)exp(ik · r) (1.80)
and the coefficient of the expansion,
∫
g(k) ≡ dr g(r)exp(−ik · r) (1.81)
is now a continuous function, referred to as the Fourier transform of g(r). Although g(k)
is complex, its complex conjugate satisfies g ∗ (k) =g(−k), which implies that |g(k)| 2 =
g(−k)g(k). Before moving on, we quote a useful identity,
∫
dr g 1 (r)g 2 (r) = 1 ∫
(2π) 3 dk g 1 (−k)g 2 (k) (1.82)