The path integral – The Cornu-spiral

If the photon is sent from A through an empty room and arrives at B, we do not know how it reached B.

We can, however, calculate the detection probability. For that purpose, we consider a selection of 11 potential paths. The paths become longer compared to the direct, straight path in the centre when moving outwards, leading to an increasing phase shift.

Summing over the amplitudes of all paths, we obtain the so-called “Cornu spiral”. The resulting red vector corresponds to the amplitude for the transition probability from point A to point B. We see that the direct, straight path and the directly adjacent paths are dominant. The contributions of the other paths mutually cancel one another out.

Using the Cornu spiral, we can also depict any further interference patterns that photons can generate. Should, for example, a single or double slit exist in space as a scattering centre, some of the potential paths between Point A and Point B are cut off. Thus, other contributions from the Cornu spiral, which then no longer cancel one another out, suddenly become relevant.