Integral transforms of the quantum mechanical path integral: Hit function and path-averaged potential

We introduce two integral transforms of the quantum mechanical transition kernel that represent physical information about the path integral. These transforms can be interpreted as probability distributions on particle trajectories measuring respectively the relative contribution to the path integral from paths crossing a given spatial point (the hit function) and the likelihood of values of the line integral of the potential along a path in the ensemble (the path-averaged potential).

Figure 1

$\mathcal{H}\left(z\right|y,x;T)$, for the free particle in one dimension. The solid line plots (27) and the data points represent a numerical sampling. Translating the initial and final point shifts the hit function due to translational symmetry.

Figure 2

$\mathcal{H}\left(z\right|y,x;T)$ for the harmonic oscillator in one dimension, $\omega =1$. The solid line is a numerical evaluation of the hit function distribution and the data points represent a numerical sampling; increasing $T$, the distribution resembles ${\left|{\psi }_0\left(z\right)\right|}^2$ as in (14).

Figure 3

$\mathcal{P}\left(v\right)$ for the linear potential $V\left(x\right)=kx$ for various parameters $x$, $y$, $T$, and $k$ in one dimension. The points represent a computational sampling of the distribution using worldline numerics; lines are theoretical fits based upon (31).

Figure 4

$\widehat{\mathcal{P}}\left(v\right)$ for a constant magnetic field ($e=1$ and in Fock-Schwinger gauge) and $x=y$ for illustrative values of $T$ and $B$. Lines are analytic curves using (36) overlayed on a sampling of the distribution from a worldline numerics simulation.