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Physics 6210/Spring 2007/Lecture 30
Lecture 30
Relevant sections in text:
§
3.9, 5.1
Bell’s theorem (cont.)
Assuming suitable hidden variables coupled with an assumption of locality to determine
the spin observables with certainty we found that correlation functions must satisfy
h
A
(ˆ
n
1
)
B
(ˆ
n
2
)
i  h
A
(ˆ
n
1
)
B
(ˆ
n
3
)
i ≤
1 +
h
A
(ˆ
n
2
)
B
(ˆ
n
3
)
i
.
We now show that quantum mechanics is not compatible with this inequality. We compute
the expectation value of the product of the two observers’ measurements (in units of ¯
h/
2)
using quantum mechanics:
h
A
(ˆ
n
1
)
B
(ˆ
n
2
)
i
=
±
1
¯
h/
2
²
2
h
S
1
S
2
i
=
4
¯
h
2
h
ψ

(ˆ
n
1
·
~
S
1
)(ˆ
n
2
·
~
S
2
)

ψ
i
.
(Again, this quantity is the
correlation function
of the two spins.) With
z
chosen along
ˆ
n
1
, this quantity is easily computed (exercise):
h
ψ

(ˆ
n
1
·
~
S
1
)(ˆ
n
2
·
~
S
2
)

ψ
i
=
¯
h
4
(
h
+
   h
+

)ˆ
n
2
·
~
S
2
(

+
i
+
 
+
i
)
=

¯
h
2
4
cos
θ,
where
θ
is the angle between ˆ
n
1
and ˆ
n
2
. To get the last equality we assume that ˆ
n
1
and
ˆ
n
2
are in the
x

z
plane with
z
along and ˆ
n
1
. Using
θ
to denote the angle between ˆ
n
1
and
ˆ
n
2
we have
(
h
+
h
+

)ˆ
n
2
·
~
S
2
(

+
i
+

+
i
) = (
h
+
h
+

)[cos
θS
2
z
+sin
θS
2
x
](

+
i
+

+
i
) =

cos
θ
Of course the result is geometric and does not depend upon the choice of coordinates.
Thus, deﬁning
θ
ij
= ˆ
n
i
·
ˆ
n
j
, Bell’s inequality –
if it applied in quantum mechanics
–
would imply

cos
θ
13

cos
θ
12
 ≤
1

cos
θ
23
,
which is not true.
†
Thus quantum mechanics is not consistent with all observables
having local deﬁnite values based upon some (unknown) “hidden variables”. On the other
hand, if reality is such that all observables for the individual particles are compatible
and locally deﬁned (with QM just giving an incomplete statistical description), then this
†
To see this, just let ˆ
n
1
point along
y
, let ˆ
n
3
point along
x
, and let ˆ
n
2
lie at 45
◦
from
x
(or
y
) in the
x

y
plane, so that
θ
12
=
π/
4 =
θ
23
,
θ
13
=
π/
2.
1
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inequality should be valid, experimentally speaking. (Assuming of course that the correct
description can be obtained using some hidden variables
λ
as described above.)
Experiments to check the Bell inequality have been performed since the 1960’s. Many
regard the “Aspect experiment” of the early 1980’s as deﬁnitive. It clearly showed that the
Bell inequality was violated, while being consistent with quantum mechanical predictions.
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.
 Spring '07
 M
 Physics, mechanics

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