Time-dependence of Double-Slit experiment

I don't understand quantum mechanics.

Some wise person said that it you understand the double slit experiment
then you understand quantum mechanics. While trite, there is some truth
in that, I can follow the logic from the double slit experiment to the
path integral formulation heavily used by Feynmann.

I don't understand the double-slit experiment.

I understand parts of it, such as how a single photon can interfere
with itself, but I don't understand the time-dependent aspect.

Consider the following gedanken experiment. Have an asymmetric
arrangement of two slits, say 1 light second apart, one near and one
far. Arrange the experiment so that the light through the distant slit
has a significant relative amplitude.

Light travels at a finite speed, so the light passing through the
nearer slit will get to the screen before that passing through the
further slit. So my common sense says ... taking into account what
Einstein said about common sense being a prejudice I should rephrase
that ... my prejudice says that we should expect to see no interference
fringes for approximately the first second, until the light has had
time to take the longer path, and then after that interference fringes
should appear.

Has this ever been checked experimentally?

In quantum mechanics the amplitude to propogate from from point x_I to
point x_F in time T is governed by the operator e^(-iHT), where H is
the Hamiltonian. The amplitude in question is just <x_I | e^(-iHT) |
x_F>.

I don't yet understand how to get the switch from no fringes at short
times to fringes at longer times from <x_I | e^(-iHT) | x_F>. What have
I missed?

.

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