- #1
PORFIRIO I
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- TL;DR Summary
- I'm trying to derive the path integral, but got stuck on this part.
I'm trying to derive the path integrals, but this step got me confused:
Consider the propagator
$$K_{q_{j+1},q_j}=\langle q_{j+1}|e^{-iH\delta t}|q_j\rangle $$
Knowing that ##\delta t## is small, we can expand it as
$$K_{q_{j+1},q_j}=\langle q_{j+1}|(1-iH\delta t-\frac 1 2 H² \delta t²+...)|q_j\rangle = \langle q_{j+1}|q_j\rangle -i\delta t\langle q_{j+1}|H|q_j\rangle +o(\delta t²)$$
I don't understand why the third term in the expansion ##-\frac 1 2 H²\delta t²## became this ##o(\delta t²)## and why it isn't inside the brackets like the other terms. what am I missing? Thanks in advance.
Consider the propagator
$$K_{q_{j+1},q_j}=\langle q_{j+1}|e^{-iH\delta t}|q_j\rangle $$
Knowing that ##\delta t## is small, we can expand it as
$$K_{q_{j+1},q_j}=\langle q_{j+1}|(1-iH\delta t-\frac 1 2 H² \delta t²+...)|q_j\rangle = \langle q_{j+1}|q_j\rangle -i\delta t\langle q_{j+1}|H|q_j\rangle +o(\delta t²)$$
I don't understand why the third term in the expansion ##-\frac 1 2 H²\delta t²## became this ##o(\delta t²)## and why it isn't inside the brackets like the other terms. what am I missing? Thanks in advance.