Classical limit of the propagator

[crises]

TL;DR

When taking $$ \hbar \rightarrow 0 $$ shouldnt the propagator result in a delta distribution?

I am currently starting with my first qft lectures and i am trying to see for the free particle that the propagator $$ <x_i | e^{-i\frac{p}{2m} T|x_f}>$$ will equal to one if x_f = 1, x_i=0 m=1 u=1 p=1, T=1 and $$\hbar \rightarrow 0$$ or 0 otherwise. I understand that this limit will result in an exponential term, but it is not one.

Any help, explanation why

 

[Demystifier]

Summary: When taking $$ \hbar \rightarrow 0 $$ shouldn't the propagator result in a delta distribution?

Why do you think so? You can think of propagator as a correlation of the positions of a single particle at two different times. In classical physics the position at one time depends on the position of an earlier time, so they are correlated. So the classical correlation at different times shound not vanish.

 

[crises]

What if the Hamiltonian is such that for a given time difference the particle can not be at position $x_2$ starting from position $x_1$ ?

For example, a particle with $p=1$ and $m=1$ has a Hamiltonian $H=\frac{1}{2}$. Suppose 1-D. Then if $x_1=0$ and $x_2=1$ $ \Delta T=1$. So the integral is a constant up to normalizaton factor. However, if T is not as before then in the classical limit there is no contribution, because there is no path that can correspond to those B.C

 

[Demystifier]

What if the Hamiltonian is such that for a given time difference the particle can not be at position $x_2$ starting from position $x_1$ ?

Then for those particular values the classical correlator vanishes. In fact, the classical correlator vanishes at all points except those at the classical trajectory, so it is a kind of a delta function. And I guess you want to understand why. Here is a hint.

What is $f(\varphi)=\lim_{\hbar\rightarrow 0}e^{i\varphi/\hbar}$?
Can you draw the real part of $f(\varphi)$?
From this drawing can you conclude what is $\int_{\varphi_1}^{\varphi_2}d\varphi\, f(\varphi)$?