Given an initial and final quantum [state](State%20vector)s, $|a_F\rangle$, and $|a_I\rangle$ respectively and a time, $t$, we define the _transition amplitude_ as the [probability amplitude](probability%20amplitude) that characterizes the likelihood that the state $|a_I\rangle$ _transitions_ to state $|a_F\rangle.$ The [evolution](time%20evolution%20operators.md) of a state over time $t$ is modeled deterministically by an operator $\hat{U}=e^{i\hat{H}t}$ and thus the transition amplitude is in its general form as $\mathcal{A}(a_F,t_F;a_I,t_I)=\langle a_F|e^{i\hat{H}t}|a_I\rangle$ where $t=t_f-t_i.$ The associated probability of a transition from state $|a_I\rangle$ to $|a_F\rangle$ is $|\mathcal{A}(a_F,t_F;a_I,t_I)|^2$ following from [Born's rule](Born%20rule). # Dynamical Pictures [[Dynamical pictures]] # [[Path integral formulation]] This is a way of re-formulating quantum mechanic by generalizing the [[Least action principle]]. In [pre-quantum physics](Mechanics%20(index).md), one may construct the possible trajectories of a particle exactly and the path integral formulation generalizes this approach in quantum mechanics, where trajectories aren't deterministic, yet more fundamental principles apply. #QuantumMechanics/QuantumDynamics #QuantumMechanics/QuantumDynamics/PathIntegrals