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

1.4 Analogy with Quantum/Classical Mechanics 17
1.4 Analogy with Quantum/Classical Mechanics
Every so often, mathematics and physics are bestowed with a powerful mapping that equates
two diverse topics. In one giant leap, years of accumulated knowledge on one topic can
be instantly transferred to another. Here, we encounter such an example by the fact that a
Gaussian chain in an external field maps onto the quantum mechanics of a single particle in a
potential. For those without a background in quantum mechanics, the section can be skimmed
over without seriously compromising one’s understanding of the remaining sections.
The mapping becomes apparent when examining the path-integral formalism of quantum
mechanics introduced by Feynman and Hibbs (1965). With incredible insight, they demonstrate
that the quantum mechanical wavefunction, Ψ(r,t), for a particle of mass, m, ina
potential, U(r), is given by the path integral,
∫
Ψ(r,t)= Dr p exp(iS/)δ(r p (t) − r) (1.47)
over all possible trajectories, r p (t), where the particle terminates at position, r, at time, t.
(Note that i ≡ √ −1.) In this definition, each trajectory contributes a phase factor given by the
classical action,
∫ [ m
S ≡ dt
2 |r′ p(t)| 2 − U(r p (t))]
(1.48)
which, in this case, is kinetic energy minus potential energy integrated along the particle
trajectory. Naturally, Feynman justified this by showing that Eq. (1.47) is equivalent to the
usual time-dependent Schrödinger equation,
i ∂ [
]
∂t Ψ(r,t)= − 2
2m ∇2 + U(r) Ψ(r,t) (1.49)
in much the same way Section 1.2 derived the diffusion equation for q(r,s).
Taking a closer look, the mathematics of this quantum mechanical problem is exactly that
of our polymer in an external field. A precise mapping merely requires the associations,
t ⇔ s (1.50)
m ⇔ 3
a 2 N
(1.51)
U(r) ⇔ −w(r) (1.52)
S ⇔ E[r α;0,s]
k B T
(1.53)
r p (t) ⇔ r α (s) (1.54)
⇔ −i (1.55)
Ψ(r,t) ⇔ q(r,s) (1.56)
Indeed, these substitutions transform Eq. (1.47) for the wavefunction into Eq. (1.25) for the
partition function of sN polymer segments, and likewise the Schrödinger Eq. (1.49) transforms
into the diffusion Eq. (1.24).