# Various expressions for total amplitude in Frederic Schuller's German QM lectures

In this lecture at around 53:31, the equation for total amplitude in terms of elementary amplitude along a path is written down.

$$\phi_{\text{total} } ( \overline{x_0} , \overline{x_n} , t_N- t_0) = \lim_{N \to \infty} \prod_{i=1}^n \int_{\overline{x_0}}^{\overline{x_n } } dx_i \phi_{path(x_1,x_2...,x_N)}^N \tag{1}$$

The set up is that a particle is shot by a source to a detector screen and the total probably of the particle being at position $$\overline{x_n}$$ starting at position $$x_0$$ starting at time $$t_0$$ is found by consider the integral of the elementary probability amplitude over all possible paths. Later the elementary probability amplitude is shown to be given as

$$\phi_{path(x_1,x_2...,x_N)}^N = e^{ i \frac{S \left[\text{path} \right]}{\overline{h}}}.\tag{2}$$

Now my confusion happens due to the successive lecture-3 at 15:41. There he introduces two factors $$A$$ and $$\epsilon$$ into the equation. $$A$$ is a normalization factor and $$\epsilon$$ is the total time and calls the equation the path integral. Here is a picture:

$$\phi( x_A,x_B; t_A,t_B) = \lim_{N \to \infty} \left( \prod_{i=1}^{N-1} \int dx_i \right) \frac{1}{A(\epsilon)^N} \exp \{ \frac{i}{ \hbar } S\left[ \text{path} (x_0,...,x_N ) \right] \} \tag{3}$$

What is the difference between (1) and (3)? Has prof maybe made typo in (1)?

Yes, $$A(\epsilon)~=~\sqrt{\frac{2\pi i\hbar \epsilon}{m}}, \qquad \epsilon~=~\frac{t_N-t_0}{N},$$ in eq. (3) is the famous Feynman fudge factor, which needs to be included in the path integral measure. For details, see e.g. this, this & this Phys.SE posts.