Understanding the Path Integral Formulation

Question

Currently, I am reading chapter 3 of Condensed Matter Field Theory, which is on the Path Integral formulation of quantum mechanics. The book denotes $\Delta t = t/N$, where $t$ is the total time considered, and $N$ is the total number of time segments. Equation 3.3 gives a Path integral approximation:

$$\langle q_f |(e^{-i\hat{H}\Delta t/\hbar})^N|q_i\rangle = \langle q_f|I_1 e^{-i\hat{T}\Delta t/\hbar}e^{-i\hat{V}\Delta t/\hbar}I_2....I_Ne^{-i\hat{T}\Delta t/\hbar}e^{-i\hat{V}\Delta t/\hbar} |q_i \rangle$$

where $I_n$ is the identity operator $$I_n=\int dp_n\int dq_n |q_n\rangle\langle q_n|p_n\rangle\langle p_n| .$$

I understand this part; I also understand that $$\langle q_n|p_n\rangle = \frac{1}{\sqrt{2\pi \hbar}}e^{ip_n q_n/\hbar}.$$

However, I do not understand how the following formula is obtained. I understand that I should plug in the definitions for $I_n$ and $\langle q_n|p_n\rangle$, but when I do so I do not get the following. For instance I do not see why there should be a $(2\pi \hbar)^N$; shouldn't there be a $(2\pi \hbar)^\frac{N}{2}$instead? Also, where does $V(q_n)$ and $T(p_{n+1})$ come from? Also, where does the last term in the exponent come from?

$$\langle q_f |(e^{-i\hat{H}\Delta t/\hbar})^N|q_i\rangle = \int \Pi_{n=1}^{N-1} d{q_n} \Pi_{n=1}^N \frac{dp_n}{2\pi \hbar} e^{-\frac{it}{\hbar} \sum_{n=0}^{N-1}\big(V(q_n) + T(p_{n+1}) - p_{n+1}\frac{q_{n+1} - q_n}{t})\big)}.$$

Answer 1

Let's consider $k$-th term \begin{equation} U(q_{k+1,q_k,\epsilon}) = \langle q_{k+1}|e^{-i H \epsilon} | q_k\rangle \end{equation} We assume that \begin{equation} H = \frac{p^2}{2}+V(q) \end{equation} Then, \begin{equation} U(q_{k+1,q_k,\epsilon})=\int dp_k\langle q_{k+1}|p_k\rangle \langle p_k |e^ {-i\epsilon \left (\frac{p^2}{2}+V(q) \right )}| q_k\rangle =\int dp_k\langle q_{k+1}|p_k\rangle \langle p_k | q_k\rangle e^{-i\epsilon \left (\frac{p_k^2}{2}+V(q_k) \right )} = \int dp_k\frac{e^{i p_kq_{k+1}}}{\sqrt{2\pi}} \frac{e^{-i p_kq_{k}}}{\sqrt{2\pi}} e^{-i\epsilon \left (\frac{p_k^2}{2}+V(q_k) \right )}+O(\epsilon^2) =\int \frac{dp_k}{2\pi} e^{i\epsilon \left (p_k\frac{q_{k+1}-q_k}{\epsilon}-\frac{p_k^2}{2}-V(q_k) \right )}+O(\epsilon^2) \end{equation}