Now, according to the above discussion, the interference pattern is built
up one photon at a time: *i.e.*, the pattern is not due to the interaction
of different photons. Moreover, the point at which a given photon
strikes the film is not influenced by the points at which previous photons
struck the film, given that there is only one photon in the
apparatus at any given time. Hence, the only way in which the
classical interference pattern can be reconstructed, after a great many photons have passed through the apparatus, is if each photon has a
greater *probability* of striking the film at points where the classical
interference pattern is bright, and a lesser probability of striking the film at points where the
interference pattern is dark.

Suppose, then, that we allow photons to
pass through our apparatus, and then count the number of photons which
strike the recording film between and
, where
is a relatively small division. Let us call this number . Now, the number of
photons which strike a region of the film in a given time interval is equivalent to the intensity of the light illuminating that region of the film multiplied by the area of the region, since
each photon carries a fixed amount of energy. Hence, in order to
reconcile the classical and quantum viewpoints, we need

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Note that, in the quantum mechanical picture, we can only predict
the *probability* that a given photon strikes a given point on the
film. If photons behaved classically then we could, in principle, solve their
equations of motion and predict *exactly* where each photon was going to strike
the film, given its initial position and velocity. This loss of determinancy
in quantum mechanics is a direct consequence of *wave-particle duality*.
In other words, we can only reconcile the wave-like and particle-like
properties of light in a *statistical* sense. It is impossible to reconcile
them on the individual particle level.

In principle, each photon which passes through our apparatus is equally
likely to pass through one of the two slits. So, can we determine
which slit a given photon passed through? Well, suppose that our
original interference experiment involves sending photons
through our apparatus. We know that we get an
interference pattern in this experiment. Suppose that we perform a modified interference
experiment in which we close off one slit, send photons
through the apparatus, and then open the slit and close off
the other slit, and send photons through the apparatus. In this
second
experiment, which is virtually identical to the first on the individual photon
level,
we know exactly which slit each photon passed through.
However, the wave theory of light (which we expect to agree
with the quantum theory in the limit ) tells us that our
modified interference experiment will *not* result in the formation of an interference pattern. After all, according to wave theory, it is impossible to obtain a two-slit interference
pattern from a single slit. Hence, we conclude that any attempt to measure
which slit each photon in our two-slit interference experiment
passes through results in the destruction of the interference pattern. It follows
that, in the quantum mechanical version of the two-slit interference experiment, we must think of each photon
as essentially passing through *both* slits simultaneously.