Download “Can Quantum-Mechanical Description of Physical Reality Be

“Can Quantum-Mechanical
Description of Physical Reality Be
Considered Complete?”
-EPR paper
-Bohr’s reply
Outline
• Overview of QM
• Positivism vs Realism
• EPR paper
• Bell’s Inequality
Einstein-Podolsky-Rosen---1935
• Complete Theory: “Every element of the
physical reality must have a counter-part in
the physical theory”
• Reality: “If, without in any way disturbing a
system, we can predict with certainty the
value of a physical quantity, then there exists
an element of physical reality corresponding
to this physical quantity”
Postulates of Quantum Mechanics
1920’s
• A quantum mechanical system is completely
described by the wavefunction ψn
• Observable quantities are represented by
mathematical operators that are used for
computational purposes
• The expectation value of an observable
represents the mean value of an observable
for a given ψ
Expectation Value
 X   x    * xdx
 Y   y    * y dy
 X  Y   ( x  y )    * ( x  y )d   * xdx   * ydy
 X    Y  X  Y 
Heisenberg Uncertainty Principle
 2  ( Aop  A ) ( Aop  A )  f f
A
f  ( Aop  A )
g  ( Bop  B )
   f f g g  f g
2
A
2
B
2
1

z  [Re( z )]2  [Im( z )]2  [Im( z )]2   ( z  z*)
 2i

2
2
Heisenberg Uncertainty Principle
1
 f g  g f 
 2i

2
 A2 B2  
f g  AopBop  A B
g f  BopAop  A B
Aop, Bop  AopBop  BopAop
1

2 2
 A B   Aop, Bop 
 2i

2
Where do you stand
Realism
Positivism
• Reality is independent of the
observer and the instruments
used to make observations
• Theories attempt to describe
an observer-independent
reality
• Science reveals facts about
nature revealed through sensory
perceptions
– goes beyond merely registering
the fact that instrument A will
give effect B under conditions C
• What we observe is in total
and direct correspondence
with what actually exists
– attempts to understand their
relationship through observation
and experimentation.
• Any statement about the world
that is not empirically verifiable is
meaningless
• There is no way to observe an
observer-independent reality so
we cannot verify its’ existence
• Theories should be economical:
Ptolemy vs Copernicus
Copenhagen Interpretation
• QM does not describe an objective reality “out
there.” It offers probabilities of observing
various values for observables when
measured
• The act of measurement “collapses the
wavefunction” so that the set of probabilities
immediately assumes only one value with
probability equal to unity
The Battle Begins----1930’s
Einstein
• There exists a reality “out
there” independent of our
observation (particles have
properties whether we
observe them or not)
• Determinism
• Locality, Spooky action at a
distance
• QM is incomplete
Bohr
• The only things that are real
are those which we observe
(it is meaningless to assign
properties to a unobserved
system)
• Probabilistic
• Complementarity
• Collapse of wavefunction
• QM is complete
What’s the difference?
Einstein
• Quantum mechanics is very
impressive. But an inner
voice tells me that it is not
yet the real thing. The
theory produces a good
deal but hardly brings us
closer to the secret of the
Old One. I am at all events
convinced that He does not
play dice.
• -Letter to Bohr
Bohr
• There is no quantum world.
There is only an abstract
physical description. It is
wrong to think that the task
of physics is to find out how
nature is. Physics concerns
what we can say about
nature.
• -Bulletin of the Atomic
Scientists
“Can Quantum-Mechanical
Description of Physical Reality Be
Considered Complete?”
Einstein-Podolsky-Rosen
Complete Theory
 e
i
( ) pox

qop  x  (constant)
Element of Reality
 
pop 
 po
i x
b
b
a
a
P(a, b)    *dx   dx  b  a
EPR

 ( x1 , x2 )   e

i
( x1  x2  x0 ) p

dp
EPR

 ( x1 , x2 )   n ( x2 )un ( x1 )
n 1
• Observable A is measured has the value ak
 k ( x2 )uk ( x1 )
• Consider another observable B

 ( x1 , x2 )    s ( x2 )vs ( x1 )
s 1
• Measure B
obtain the value br
r ( x2 )vr ( x1 )
EPR

 ( x1 , x2 )   e

i
( x1  x2  x0 ) p

dp

( x1 , x2 )    p ( x2 )u p ( x1 )dp

 p ( x2 )  e
i
( )( x2  x0 ) p

u p ( x1 )  e
  p
Pop p 
  p p
i x2
i
( ) px1

EPR

 ( x1 , x2 )   e


i
( x1  x2  x0 ) p

dp
( x1 , x2 )    x ( x2 )vx ( x1 )dx

vx ( x1 )   ( x1  x)

 x ( x2 )   e
i
( )( x  x2  x0 ) p


PopQop  Qop Pop 

i
dp  h ( x  x2  x0 )
2
2
1

2 2
 A B   Aop, Bop  
4
 2i

John Von Neumann----1932
• “Mathematical Foundations of Quantum
Mechanics”
• Impossibility proof: There must be some
observable with a non-zero variance
• QM predictions cannot be reproduced by a
deterministic theory.
Von Neumann’s “Proof”
• For an ensemble of many identical systems,
it is found that
 X    Y  X  Y 
• Let us introduce a state with a “Hidden
Vector”, labeled as ϕh. This hidden vector, if
known, will make the theory deterministic
Von Neumann’s “Proof”
• The expectation value of X on a system in
the state ϕh is denoted by < Xh>
• For an ensemble of many identical systems,
Von Neumann assumed:
 Xh    Yh  Xh  Yh 
“Lack of Imagination”
• “What is proved by impossibility proofs is lack
of imagination” – J.S. Bell (Theory of Local
Beebles)
State
Xmeasured
Ymeasured
(X+Y)measured
System #1
ϕφh1
2
5
5
System #2
ϕφh2
4
3
9
System #3
ϕφh1
2
5
5
System #4
ϕφh2
4
3
9
Dr. Bertlmann
•
“ ...The philosopher in the street, who has
not suffered a course in quantum
mechanics, is quite unimpressed by
Einstein–Podolsky–Rosen correlations. He
can point to many examples of similar
correlations in everyday life. The case of
Bertlmann’s socks is often cited. Dr.
Bertlmann likes to wear two socks of
different colours. Which colour he will
have on a given foot on a given day is
quite unpredictable. But when you see
that the first sock is pink you can be
already sure that the second sock will not
be pink. Observation of the first, and
experience of Bertlmann, gives immediate
information about the second. There is no
accounting for tastes, but apart from that
there is no mystery here. And is not the
EPR business just the same?...”
Bertlmann’s Socks
• Test a  washing for 1hr at 0oC
• Test b  washing for 1hr at 22.5oC
• Test c  washing for 1hr at 45oC
Bertlmann’s Socks
n[ab ]  n[abc ]  n[abc ]
n[b c ]  n[ab c ]  n[ab c ]
n[a c ]  n[abc ]  n[ab c ]
n[ab ]  n[abc ]
n[b c ]  n[ab c ]
n[ab ]  n[b c ]  n[abc ]  n[ab c ]
n[ab ]  n[b c ]  n[a c ]
Bertlmann’s Socks
N   (a, b)  n[ab ]
N   (b, c)  n[b c ]
N   (a, c)  n[a c ]
Experiment
Test-Sock A
Test-Sock B
1
a
b
2
b
c
3
a
c
N  (a, b)  N  (b, c)  N  (a, c)
P (a, b)  P (b, c)  P (a, c)
Quantum Socks?
Socks
Washing machines
Temperatures
Photons
Polarization analyzers
Polarizer orientations
P (a, b)  P (b, c)  P (a, c)
Aspect-Grangier-Roger
Choose a basis
L , R
v , h
v L
2
 h L
2
 v R
2
 v' ,  h'
 h R
 v '  v   h'  h  cos( )
 h'  v    v '  h  sin( )
2
1

2
Finding Projection Amplitudes
 v'   L  L  v'   R  R  v'
 v  v'   v  L  L  v'   v  R  R  v'
cos   
v L
1

2
 L  v'
1 i

e
2
1 i
1 i
e 
e
2
2
v R
 R  v'
1

2
1 i

e
2
Projection Amplitudes
v
v
1
h
0
h
0
 v'
cos 
 sin  
 h'
sin  
cos 
L
1
2
i
2
R
1
2
1
i
2
 v'
 h'
L
R
sin  
1
2
1
2
cos 
i
2
i
2
0
e  i
2
e i
2
0
1
ie  i
2
 ie i
2
e i
2
 ie i
2
1
0
0
1
cos 
 sin  
1
e
 i
2
ie  i
2
Entangled States
Alice
Eigenstates
 vA
 hA
Eigenvalue
RvA
RhA
Bob
Eigenstates
 v'B
 h'B
 

1
 LA  LB   RA  RB
2
Joint Measurement
Eigenstates
 '    vA  vB'
Eigenvalue
RvB'
RhB'
 '   hA  vB'
 '    vA  hB'
 '   hA  hB'

Entangled States
   '    '     '   '      '    '     '   '  
 '   
 '   
 ' 

1
 vA  vB'  LA  LB   RA  RB
2

1
 vA  LA  vB'  LB   vA  RA  vB'  LB
2
1  1 ei (b a )
1 e i (ba ) 


 

2 2
2
2
2 
 ' 

1
 
cos(b  a)
2

1
 '   
sin(b  a)
2
1
 '   
sin(b  a)
2
1
 '    
cos(b  a)
2
Entangled States
1
 ' cos(b  a)   ' sin(b  a)   ' sin(b  a)   ' cos(b  a)
 
2
P  (a, b)   '  
2
1
 cos 2 (b  a)
2
P (a, b)   '  
2
1
 sin 2 (b  a)
2
2
P  (a, b)   '  
P (a, b)   '  
2
1
 sin 2 (b  a)
2
1
 cos 2 (b  a)
2
Bells Inequality
Experiment
Photon A
PA1 orientation
Photon B
PA2 orientation
Difference
1
a(0o)
b(22.5o)
b-a=22.5o
2
b(22.5o)
c(45o)
c-b=22.5o
3
a(0o)
c(45o)
c-a=45o
P (a, b)  P (b, c)  P (a, c)
1 2
1
1
sin (22.5o )  sin 2 (22.5o )  sin 2 (45o )
2
2
2
0.1464  0.2500
Only Assume Locality
A(a,  )  1
B(b,  )  1
JM (a, b,  )  A(a,  ) B(b,  )
JM (a, b)   A(a,  B(b,  )  ( )d
JM (a, b)  JM (a, d )    A(a,  B(b,  )  A(a,  ) B(d ,  )  ( )d
JM (a, b)  JM (a, d )   B(b,  )  B(d ,  )  ( )d
JM (a, b)  JM (a, d )  JM (c, b)  JM (c, d )  2
Aspect-Dalibard-Roger
Only Assume Locality
Experiment
Photon A
PA1 orientation
Photon B
PA2 orientation
Difference
1
a(0o)
b(22.5o)
b-a=22.5o
2
a(0o)
d(67.5o)
d-a=67.5o
3
c(45o)
b(22.5o)
b-c=-22.5o
4
c(45o)
d(67.5o)
d-c=22.5o
JM (a, b)  JM (a, d )  JM (c, b)  JM (c, d )  2
2.828  2
Who was right?
Realist
• Einstein was right in
doubting quantum theory
but wrong about using local
hidden variables
• Copenhagen can’t be the
end
Positivist
• Einstein was wrong
• Bohr was right
Noteworthy
•
•
•
•
•
Schrödinger's cat
David Bohm
Delayed choice experiments
Decoherence
Griffiths
!!!THANK YOU ALL!!!
•SPECIAL THANKS TO
EVERYONE IN THE
PHYSICS
DEPARTMENT!
Citations
• “The Meaning of Quantum Theory”
--Jim Baggot
• “Can Quantum Mechanical Description of
Physical-Reality Be Considered Complete?”
--EPR
• “Can Quantum Mechanical Description of
Physical-Reality Be Considered Complete?”
--Niels Bohr
Citations
• “Speakable and Unspeakable in Quantum
Mechanics”
--John Bell
• “Einstein, Bohr and the Quantum Dilemma”
--Andrew Whitaker
• “Where does the weirdness go?”
--David Lindley