Recently, I have been learning about path integrals and I was wondering, can the probability of a certain path be weighted more in a path integral? Said in another way, can certain paths have more probability in a path integral?
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In general the "weighting" of each path $q$ in a path integral is given by $e^{\frac{i}{\hbar}S[q]}$. Then paths for which the action $S$ is stationary with respect to small deviations from the path are the only ones which really contribute because the contributions from those with non stationary $S$ get averaged out as the phase changes very rapidly (because $\hbar$ is very small).
The number $S[q]$ is defined as:
$$S[q] = \int_{t_0}^{t_1}L(q(t),\dot{q}(t), t)dt$$
Where $t$ is some parameter that varies along the path and $L$ is the lagrangian. The Langrangian will depend on the details of your system but for a free particle it looks like the classical kinetic energy $L = \frac{1}{2}\dot{q}^2$.
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