Quantum-mechanical propagator ($K(q_f,t_f;q_i,t_i)$)

A quantum-mechanical propagator is the position-space kernel $K(q_f,t_f;q_i,t_i)=⟨q_f|U(t_f,t_i)|q_i⟩$ of the time-evolution operator. It is the amplitude to evolve between the specified endpoint configurations.

For a free particle of mass $m$ on the line,

A semiclassical propagator sums $e^{i{S}_{\mathrm{cl}}/\hslash }$ over classical paths joining its endpoints, weighted by their quadratic fluctuation determinants and phase indices. It is exact for quadratic actions.

The Van Vleck determinant is the determinant of mixed endpoint derivatives of a classical action. It measures how nearby classical trajectories focus and supplies the fluctuation prefactor of the semiclassical propagator.