Dirac Notation in building Path Integrals

[Elwin.Martin]

Alright, so I was wondering if anyone could help me figure out from one step to the next...
So we have defined |qt>=exp(iHt/$\hbar$)|q>
and we divide some interval up into pieces of duration τ

Then we consider
<$q_{j+1}$$t_{j+1}$|$q_{j}$$t_{j}$>
=<$q_{j+1}$|e^{-iHτ/$\hbar$}|$q_{j}$>
=<$q_{j+1}$|1-(i/$\hbar$)Hτ+O(τ²)|$q_{j}$>
=$\delta$($q_{j+1}$-$q_{j}$)-(iτ/$\hbar$)<$q_{j+1}$|H|$q_{j}$>

Dumb question:
Where did the delta function come from? I know where the second term comes from and I'm assuming that the higher order terms are being tossed since the τ² and higher order terms have been deemed sufficiently small...but where does this delta function come from?

Is this something like <$q_{j+1}$|1|$q_{j}$> = 1 iff $q_{j+1}$=$q_{j}$, or something having to do with independence? But...then I don't see why after this we have:

(2$\pi$$\hbar$)^-1$\int$dp e^{(i/$\hbar$)p($q_{j+1}$-$q_{j}$})
coming from the delta function term .-. what?

 

[Bill_K]

Your last question... because that's what the Fourier transform of a delta function is:

δ(x) ≡ (1/2π) ∫e^ixy dy

and put x = q_j+1 - q_j, y = p/ħ

 

[Morgoth]

your 1st Q;
it is the normalization of q_i, q_j