State Kets in QM and F. Path Integral

[Sina]

Greetings,

I know that position state ket is a continuous state ket satisfying X|x> = x|x>. There is however one notation I don't understand. What does it mean when we label the position ket with a discrete index and then use these to expand operators as <x_i|H|x_j>? What does it generally mean to label a continuous ket with a discrete index? I would also welcome any insightful comments about position kets. I have seen them before but I have realized I haven't completely understood it.

The reason is that I was studying f. path integrals and using some identity and inserting couple of identity operators integral (<x_i|x_i> dx_i) and using a operator identity we get the path integral representation of the matrix elements of the propagator. That integral contains terms dx_i*dx_(i-1)... and they explain it as integrating through all possible paths starting from x_o and ending at x_f (our initial and final state kets). I don't quite understand how to understand it that way.. The integral is over the space of functions (vectors) on Hilbert space (state kets in our case) but how do we see it as a sequence of state evaluation starting at x_o and ending at x_f?

Thanks..

 

[DrClaude]

In the case of the path integral, the index corresponds to the discretized time, not to a discretization of space: $x_i$ is $x(t_i)$. The time interval is then taken in the limit $\Delta t \rightarrow 0$.