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3

The TD-Schrodinger equation is $$H(\lambda)|\psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle$$ So, we just need to plug in \eqref{4} and remove all terms that aren't first-order in $\dot\lambda$. Let's start with the left-hand side, since its a little simpler. $$\begin{align}H(\lambda)|\psi(t)\rangle&=e^{i\phi(\lambda(t))}e^{-i\gamma(t)}\...


8

It looks Doi makes some extra simplifications (beyond expanding the exponential) that are valid when the external field is weak. Let's start with what you wrote and make one simplification, $$ \begin{eqnarray} \overline{\delta \phi _a} &= \frac{\langle\delta \phi_a \rangle -\langle \delta \phi _a \beta U_{ext}\rangle}{\langle 1-\beta U_{ext}\rangle} \tag{...


5

For context to future readers, Doi starts with a model of an $N$ unit polymer formed by a random walk along a uniform grid of lattice length $b$. It seems to be implicitly assumed in the described derivation that the lattice used is $b\mathbb{Z}^3$, that is the uniform 3D grid. Note there are other types of lattices for which Eq.$\,(\ref{2})$ is not true. ...


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