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

18 1 Self-consistent field theory and its applications
Extending this analogy further, the strong-stretching theory (SST) developed in Section
1.3 is found to be equivalent to classical mechanics. In Feynman’s formulation, the connection
between quantum mechanics and classical mechanics becomes exceptionally transparent.
When the action, S, of a particle is large relative to , the path integral becomes dominated by
the extremum of S, because all paths close to the extremum contribute the same phase factor,
and thus add constructively. (This is the basis of Fermat’s principle in optics by which the
full wave treatment of light can be reduced to classical ray optics (Hecht and Zajac 1974).)
However, the procedure of finding the extremum of S is precisely the Lagrangian formalism
of classical mechanics (Goldstein 1980). Thus, in the limit of large S, the particle follows the
classical mechanical trajectory determined by Newton’s equation of motion (i.e, ma = F),
mr ′′
p(t) =−∇U(r p (t)) (1.57)
Equally familiar in classical mechanics is the equation for the conservation of energy,
m
2 |r′ p (t)|2 + U(r p (t)) = constant (1.58)
In terms of the analogy, classical mechanics corresponds to the case where E[r α ; s 1 ,s 2 ]/k B T
is large relative to one, which is the condition for the strong-stretching theory (SST) discussed
in the preceding section. Indeed, Eq. (1.57) matches precisely with Eq. (1.41) for the groundstate
polymer trajectory, as does Eq. (1.58) with Eq. (1.44). It is this analogy from which the
ground-state trajectory acquires its name, the classical path.
1.5 Mathematical Techniques and Approximations
Quantum mechanics has experienced a massive activity of research since its introduction midway
through the last century. The accumulated knowledge is staggering, and the fact that our
problem maps onto it means that we can make significant progress without having to reinvent
the wheel. This section demonstrates several useful calculations that draw upon what are now
standard techniques in quantum mechanics.
1.5.1 Spectral Method
One of the most widely used and fruitful techniques in quantum mechanics is the spectral
method for solving the time-dependent wavefunction, Ψ(r,t). Here, the method is adapted to
the evaluation of the partial partition functions, q(r,s) and q † (r,s), for a homopolymer in an
external field, w(r). The first step is to solve the analog of the time-independent Schrödinger
equation,
[ a 2 ]
N
6 ∇2 − w(r) f i (r) =−λ i f i (r) (1.59)
for all the eigenvalues, λ i , and eigenfunctions, f i (r), where i =0,1,2,... For reasons that will
be discussed at the end of this section, the eigenvalues are ordered from smallest to largest. As