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.

In units $\hslash =1$, a particle of mass $m$ with potential $V(q)=-Fq$ has classical path $q_c(t)=q_0+[(q_1-q_0)/T-FT/(2m)]t+F{t}^2/(2m)$. Its action and exact quantum-mechanical propagator areThe fluctuation operator is the free one because the potential is linear. This is a special case of classical-path factorization of a quadratic path integral.

For potential $V(x)=\alpha x$, momentum follows a characteristic with $\dot{p}=-\alpha$. The retarded quantum-mechanical propagator translates its momentum argument and accumulates the kinetic action along that characteristic. Its apparent division by $\alpha$ is removable on the delta-function support; at zero force it becomes the free diagonal momentum kernel. Fourier transform gives the usual constant-force position kernel with a cubic-time action term.

For unit mass and $\hslash =1$, this prepoint kernel tends to the identity and generates the Time-dependent Schrödinger equation. The square-root branch follows the damped Fresnel integral. Its normalized moments are $1$, $0$, and $i\Delta t$ at orders zero, one and two; these fix both the normalization and the kinetic sign.

Expand a smooth wavefunction and potential in the normalized short-time Schrödinger kernel. Its odd moment vanishes and its second moment is $i\Delta t$, so one step changes the wavefunction by $i\Delta t{\psi }^{\prime \prime }/2-i\Delta tV\psi +O(\Delta t^2)$. Division by the time step gives the Time-dependent Schrödinger equation.

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

Completing the square in the integral over an intermediate position shows that two free-particle propagators compose to the propagator for the sum of their positive time intervals. For equal slices, the quadratic identity isThe Fresnel integral contributes a factor proportional to $\sqrt{k/(k+1)}$; multiplying successive factors gives $1/\sqrt{N}$. A consistent convergence regulator fixes the phases of these square roots.

Wick rotation of the free-line propagator gives the displayed coincident-endpoint kernel for positive Euclidean duration $T$. Equivalently it is the path integral over Dirichlet fluctuations about the straight classical path, with its time-sliced Gaussian measure included. It is a propagator density, not a full thermal partition function: that trace additionally integrates position and, on a multiply connected configuration space, sums the winding sectors. For an angular coordinate replace $M$ by moment of inertia.

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.

Expanding a configuration-space path integral about a classical trajectory $q_c$ with fixed endpoints gives $S[q_c+f]=S[q_c]+\frac{1}{2}\int f{\mathcal{O}}_{c}fdt+O(f^3)$, where $f$ obeys homogeneous Dirichlet boundary conditions. The Gaussian functional integral contributes $(\det {\mathcal{O}}_c{)}^{-1/2}$ with the phase prescribed by continuation of the quantum-mechanical propagator. Relative to the free operator ${\mathcal{O}}_0=-m{d}^2/d{t}^2$, the prefactor is $D_0(T)[\det {\mathcal{O}}_c/\det {\mathcal{O}}_0{]}^{-1/2}$. In a general potential the operator, and thus the prefactor, depends on the endpoints through $q_c$. For a quadratic potential $V^{\prime \prime }$ is constant and all higher fluctuation terms vanish, so the approximation is exact away from singular focusing times. At a zero mode the simple determinant formula requires a limiting prescription or a different treatment of the zero mode.

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.