Quantum Time of Arrival

Undergraduate thesis and notebook calculations in C++11 & Mathematica. Plots are written as CSV plus simple SVG, so no Mathematica, gnuplot, or any C++ plotting library is required.

Thesis Summary

This project studies the quantum time-of-arrival problem for a free particle wavefunction. Because a quantum particle has spatial spread and measurements change the wavefunction, its arrival at a detector is computed probabilistically rather than as a single classical trajectory. The thesis first treats time as a forward-directed external parameter and compares small-detector and random-position detector approximations against the classical arrival time. It then explores a dynamic, non-directional treatment of time that places time on more equal footing with position; under the default parameters, this approach produces arrival-time estimates close to the classical value. You can find teh full thesis inside thesis folder.

Cpp code

physics.cpp: psi0(...) from thesis Eq. 2.18 and propagator(...) from thesis Eq. 2.24.
arrival.cpp: small_detector(...) from thesis Eqs. 3.32-3.37, random_detector(...) from thesis Eqs. 3.38-3.39, and dynamic_arrival_time(...) from thesis Eqs. 4.3-4.4.
plotting.cpp: CSV and self-contained SVG output helpers.
generation.cpp: figure recipes, verification summaries, and default thesis parameter runs.
qtoa.cpp: command-line entry point.
evolved_state(...): notebook post-measurement wavefunction recurrence for plotting |psi_n(x,t)|^2.

Build

make

Run

make run

Outputs are written under ./output/.

Results

The default run recreates the arrival-time checks for a free particle moving from x0=-5 to a detector at xd=-3 with classical arrival time t=0.1. The small-detector recurrence gives tbar=0.100793868305 for n=100 and delta1=0.005. Wider detector windows increase approximation error, while the random-position detector remains close to the small-detector and exact-integral behavior for narrow windows. The dynamic, non-directional time calculation also stays close to the classical result, with representative checks around tbar=0.10051.

The generated SVG plots below are stored in output/; matching CSV data files are generated beside them and ignored by git.