Question about notation in the Feynman Lectures on Physics III 3-1

[anuttarasammyak]

TL;DR Summary

Feynman Lectures on Physics III 3-1 notation of probability amplitude <x|s> meets Dirac's notation of bra ket inner product ?

I have a question on formula (3.1) and (3.2) in Feynman Lectures on Physics III 3-1, available online,
https://www.feynmanlectures.caltech.edu/III_03.html

<x|s> here can be interpreted also as inner product of bra <x| and ket |s>, following usual Dirac notation ?

For example, $<r_1|r_2>$ in formula (3.7), if we take it as inner product, it should be zero because bra and ket are position eigenvecors of different eigenvalues.　Feynman treats it as a kind of Green function. Is Green function noted in the form of < | > as a usual way?

I do not find this notation of probability amplitude in other textbooks. Your teaching will be highly appreciated.

 

[atyy]

Summary:: Feynman Lectures on Physics III 3-1 notation of probability amplitude <x|s> meets Dirac's notation of bra ket inner product ?

I have a question on formula (3.1) and (3.2) in Feynman Lectures on Physics III 3-1, available online,
https://www.feynmanlectures.caltech.edu/III_03.html

<x|s> here can be interpreted also as inner product of bra <x| and ket |s>, following usual Dirac notation ?

Yes. Feynman's notation is the same the the usual Dirac braket inner product.

For example, $<r_1|r_2>$ in formula (3.7), if we take it as inner product, it should be zero because bra and ket are position eigenvecors of different eigenvalues.　Feynman treats it as a kind of Green function. Is Green function noted in the form of < | > as a usual way?

The inner product $<r_2|r_1>$ is not zero, because if you read Feynman's text he means $<r_2|\text{the state at the time of measurement that evolved from a state localized at r_1 at an earlier time}>$

 

[anuttarasammyak]

I have got it. For clarification of time difference or evolution I add suffix of time explicitly to <r2|r1>, i.e.
$$< \mathbf{r_2}_{\ t}|\mathbf{r_1}_{\ t0}>$$
where $t>t_0$.

In later lines I found he mentions clearly
$$<r,t=t_1|{P,t=0}>$$

Thank you so much.