3 A probability found through quantum mechanics. In the test of Bell's theorem, experiment 6.2, what is the probability given by quantum mechanics that, if the orientation settings are different, the two detectors will flash different colors?
3 A probability found through quantum mechanics. In the test of Bell's theorem, experiment 6.2, what is the probability given by quantum mechanics that, if the orientation settings are different, the two detectors will flash different colors?
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Transcribed Image Text:42
6 The Einstein-Podolsky-Rosen Paradox
the University of Geneva and his collaborators performed the
experiment with detectors in the Swiss villages of Bellevue and
Bernex, separated by nearly seven miles.
6.2 Experiment 6.2: Random distant measurements
This experiment is called the "test of Bell's theorem". The reasoning is
intricate, so I give an outline here before plunging into the details. We
will build an apparatus much like the previous one with a central source
that produces a pair of atoms, and with two detector boxes. Mounted
atop each detector box are a red lamp and a green lamp. Every time
the experiment is run, a single lamp on each detector box lights up. On
some runs the detector on the left flashes red and the detector on the
right flashes green, on other runs both detectors flash red, etc. When the
of each detector flashing a different color is. But we can also analyze
apparatus is analyzed by quantum mechanics, we find that the probability
the apparatus under the assumption of local determinism. This analysis
shows that the probability of each detector flashing a different color is
or more. (Exactly how much more depends on exactly which local
deterministic scheme is employed, see problem 6.4.) Experiment agrees
with quantum mechanics, so the assumption of local determinism, natural
though it may be, is false. Any local deterministic scheme, including the
second alternative interpretation mentioned on page 38, must be wrong.
The apparatus
This experiment uses the same source as the previous experiment, but
now the detectors are not regular Stern-Gerlach analyzers, but the tilting
Stern-Gerlach analyzers described in section 5.3 (page 33). Each of the
two analyzers has probability of being oriented as A, B, or C. If you
wish, you may set the detector orientations and then have the source
generate its pair of atoms, but you will get the same results if you first
launch the two atoms and then set the detector orientations while the
atoms are in flight. Mounted on each detector are two colored lamps. If
an atom comes out of the + exit, the red lamp flashes; if an atom comes
out the exit, the green lamp flashes.
<<-0
source
0
6.2 Experiment 6.2: Random distant measurements
The prediction of quantum mechanics
If the two detectors happen to have the same orientation, then this
experiment is exactly the same as the previous one, so exactly the same
results are obtained: the two detectors always flash different colors. On
the other hand, if the two detectors have different orientations, then they
might or might not flash different colors.
What is the probability that the two detectors flash different colors in
general, that is, when the two detectors might or might not have the
same orientation? Suppose the detector on the left is closer to the source
than the detector on the right. If the left detector were set to A and
flashed green (that is, -), then the atom on the right has m = +mB.
In the previous chapter we saw that when such an atom enters the right
detector, it has probability of causing a red flash and probability of
causing a green flash. You can readily generalize this reasoning to show
that regardless of orientation, the two detectors flash different colors with
probability.
We conclude that:
43
(1) If the orientation settings are the same, then the two detectors
flash different colors always.
(2) If the orientation settings are ignored, then the two detectors
flash different colors with probability.
And these results are indeed observed!
The prediction of local determinism
In any local deterministic scheme, each atom must leave the source already
supplied with an instruction set that determines which lamp flashes for
each of the three orientation settings. For example, an instruction set
might read (if set to A then flash red, if set to B then flash red, if set
to C then flash green), which we abbreviate as (RRG). One natural way
to implement an instruction set scheme would be through the atom's
associated magnetic arrow: if the detector is vertical (orientation A) and
the atom's arrow points anywhere north of the equator, then the atom
leaves through the + exit, while if the atom's arrow points anywhere south
of the equator, then the atom leaves through the exit. Similar rules hold
for orientations B and C: the atom always leaves through the exit towards
which its arrow most closely points. The argument that follows holds for
This postulated scheme is inconsistent with quantum mechanics because it assumes that an atom's
magnetic arrow points in the same manner that a classical stick does, with definite values for all
three projections my, my, and m, simultaneously.
Transcribed Image Text:6.3 A probability found through quantum mechanics. In the test of Bell's
theorem, experiment 6.2, what is the probability given by quantum
mechanics that, if the orientation settings are different, the two
detectors will flash different colors?
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