  Following earlier work, we view two dimensional non-linear sigma model with
target space $\cM$ as a single particle relativistic quantum mechanics in the
corresponding free loop space $\cLM$. In a natural semi-classical limit
($\hbar=\alpha' \to 0$) of this model the wavefunction localizes on the
submanifold of vanishing loops which is isomorphic to $\cM$. One would expect
that the relevant semi-classical expansion should be related to the tubular
expansion of the theory around the submanifold and an effective dynamics on the
submanifold is obtainable using Born-Oppenheimer approximation. In this work we
develop a framework to carry out such an analysis at the leading order in
$\alpha'$-expansion. In particular, we show that the linearized tachyon
effective equation is correctly reproduced up to divergent terms all
proportional to the Ricci scalar of $\cM$. The steps leading to this result are
as follows: first we define a finite dimensional analogue of the loop space
quantum mechanics (LSQM) where we discuss its tubular expansion and how that is
related to a semi-classical expansion of the Hamiltonian. Then we study an
explicit construction of the relevant tubular neighborhood in $\cLM$ using
exponential maps. Such a tubular geometry is obtained from a Riemannian
structure on the tangent bundle of $\cM$ which views the zero-section as a
submanifold admitting a tubular neighborhood. Using this result and exploiting
an analogy with the toy model we arrive at the final result for LSQM.
