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\intdp 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