“Can Quantum-Mechanical Description of Physical Reality Be … · “Can Quantum-Mechanical...
Transcript of “Can Quantum-Mechanical Description of Physical Reality Be … · “Can Quantum-Mechanical...
“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
dxxxX *
dyydxxdyxyxYX **)(*)(
dyyyY *
YXYX
Heisenberg Uncertainty Principle
ffAAAA opopA )()(2
)( AAf op )( BBg op
222 gfggffBA
2
2222*)(
2
1)][Im()][Im()][Re(
zz
izzzz
Heisenberg Uncertainty Principle
2
22
2
1
fggf
iBA
BABAgf opop BAABfg opop
opopopopopop ABBABA ,
2
22 ,2
1
opopBA BA
i
Where do you stand
Realism
• Reality is independent of the observer and the instruments used to make observations
• Theories attempt to describe an observer-independent reality – 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
Positivism • Science reveals facts about
nature revealed through sensory perceptions – 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?”
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Einstein-Podolsky-Rosen
Complete Theory Element of Reality
EPR
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EPR
• Observable A is measured has the value ak
• Consider another observable B
• Measure B obtain the value br
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EPR
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EPR
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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
• Let us introduce a state with a “Hidden Vector”, labeled as ϕh. This hidden vector, if known, will make the theory deterministic
YXYX
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:
hhhh YXYX
“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 φ 2 5 5
System #2 φ 4 3 9
System #3 φ 2 5 5
System #4 φ 4 3 9
State Xmeasured Ymeasured (X+Y)measured
System #1 ϕh1 2 5 5
System #2 ϕh2 4 3 9
System #3 ϕh1 2 5 5
System #4 ϕh2 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
][][][ cbancbanban
][][][ cbancbancbn
][][][ cbancbancan
][][ cbanban ][][ cbancbn
][][][][ cbancbancbnban
][][][ cancbnban
Bertlmann’s Socks
][),( banbaN ][),( cbncbN ][),( cancaN
Experiment Test-Sock A Test-Sock B
1 a b
2 b c
3 a c
),(),(),( caNcbNbaN
),(),(),( caPcbPbaP
Quantum Socks?
Socks
Washing machines
Temperatures
Photons
Polarization analyzers
Polarizer orientations
),(),(),( caPcbPbaP
Aspect-Grangier-Roger
Choose a basis
hv , RL , '' , hv
2
12222
RhRvLhLv
)sin('' hvvh
)cos('' hhvv
Finding Projection Amplitudes
''' vRRvLLv
''' vRRvvLLvvv
ii ee 2
1
2
1cos
2
1Lv
2
1Rv
i
vL e2
1'
i
vR e2
1'
Projection Amplitudes
R
L
'h
'v
h
v
v h 'v 'h L R
1
1
1
1
1
1
0
0
0
0
0
0
cos
cos
cos
cos sin
sin
sin
sin
2
1
2
1
2
1
2
1
2
i
2
i
2
i
2
i
2
ie
2
ie
2
ie
2
ie
2
iie
2
iie
2
iie
2
iie
Entangled States
Alice
A
v
Eigenstates Eigenvalue
A
vR
A
h A
hR
Bob
Eigenstates Eigenvalue
B
v '
B
h 'B
hR '
B
vR '
Joint Measurement
Eigenstates B
v
A
v ''
B
v
A
h ''
B
h
A
v ''
B
h
A
h ''
B
R
A
R
B
L
A
L 2
1
Entangled States
''''''''
B
R
A
R
B
L
A
L
B
v
A
v '2
1'
B
L
B
v
A
R
A
v
B
L
B
v
A
L
A
v ''2
1'
22
1
22
1
2
1'
)()( abiabi ee
)cos(2
1' ab
)sin(2
1' ab
)sin(2
1' ab
)cos(2
1' ab
Entangled States
)cos(')sin(')sin(')cos('2
1abababab
)(cos2
1'),( 22
abbaP )(sin2
1'),( 22
abbaP
)(sin2
1'),( 22
abbaP )(cos2
1'),( 22
abbaP
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
),(),(),( caPcbPbaP
)45(sin2
1)5.22(sin
2
1)5.22(sin
2
1 222 ooo
2500.01464.0
Only Assume Locality
1),( aA 1),( bB ),(),(),,( bBaAbaJM
dbBaAbaJM )(),(,(),(
ddBaAbBaAdaJMbaJM )(),(),(),(,(),(),(
ddBbBdaJMbaJM )(),(),(),(),(
2),(),(),(),( dcJMbcJMdaJMbaJM
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
2),(),(),(),( dcJMbcJMdaJMbaJM
2828.2
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