Here is an experiment that can't be described by our natural (classical)
senses!
Suppose we have an experiment setup consisting of a photon source, a beam
splitter (which was once implemented using a half-silvered mirror), and a
pair of photon detectors. Have a look at the figure!
Suppose we send a series of individual photons along a path from the
photon source towards the beam splitter. We observe the photon arriving at
the detector on the right on the beam splitter half of the time, and arriving
at the detector above the beam splitter half of the time, as illustrated! The
simplest way to explain this behaviour in a theory of physics is to model the
beam splitter as effectivel flipping a fair coin, and choosing whether to
transmit or reflect the photon based on the result of the coin-flip, whose
outcome determines whether the photon is transmitted or reflected.
Now let's modify the setup a little bit ...
... we've used beam splitters and fully reflecting mirrors to direct the
photons along either of the paths.
Now, what the output of the experiment should look like? Classical thinking
suggests that they should arrive at one detector half of the time and on the
other for the rest. But this is a wrong reality! The photons are found arriving
at only one of the detectors, 100% of the time!
The result of the experiment may astonish you! But, we've quantum model to the
rescue. The non-intuitive behaviour results from features of quantum mechanics
called superposition and interference.
Clearly, photon can take 1 of 2 possible paths, reflected and transmitted. Let's
represent the transmitted path by 0 and the other by 1.
Let's denote the state of a photon in path 0 by the vector: 
, and path of
photon in path 1 by the vector: 
The photon leaving the source starts out in path 0. Crossing the beam splitter,
has a probability of being either on path 0 or path 1. Quantum mechanics
says beam splitter causes the photon to go into a superposition of taking both
the 0 and 1 paths. Mathematically, we can describe such superposition by a
linear combination of the state vectors for the paths. So, the general path
state will be described by a vector:
where, 
and 
are the probabilities of photons being on path 0 and 1
respectively.
So, state of our photons become:
When the photon passes through the beam splitter, we multiply its state vector
by the matrix:
... since in quantum mechanics, any linear operator can be represented as a matrix.
So, for the photon starting out in path 0, after passing through the first beam
splitter it comes out in state:
This result corresponds with the observed behaviour that, after going through the first
beam splitter, we would measure the photon in path 0 with probability 
and in path 1 with probability 
.
If we do not measure which path the photon is in, immediately after it passes through
the first, the its state remains:
Now if the photon is allowed to pass through the second beam splitter (before
making any measurements of the photon's path), its new state vector is:
If we measure the path after the second splitter. We find it coming out in the
path 1 with probability 
. Thus after the second beam splitter the
photon is entirely in the path 1, which is what is observed in experiments!
This new mathematical framework is called quantum mechanics!
