“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?”

xpi

o

e)(

oop pxi

p

)constant( xqop b

a

b

aabdxdxbaP *),(

Einstein-Podolsky-Rosen

Complete Theory Element of Reality

EPR

dpexxpxxx

i

)(

21

021

),(

EPR

• Observable A is measured has the value ak

• Consider another observable B

• Measure B obtain the value br

)()( 12 xux kk

)()(),( 1

1

221 xvxxxs

ss

)()( 12 xvx rr

1

1221 )()(),(n

nn xuxxx

EPR

dpexxpxxx

i

)(

21

021

),(

1)(

1)(px

i

p exu

dpxuxxx pp )()(),( 1221

pxxi

p ex))((

2

02

)(

p

p

pop pxi

P

2

EPR

dpexxpxxx

i

)(

21

021

),(

dxxvxxx xx )()(),( 1221

)()( 11 xxxvx

)()( 02

))((

2

02

xxxhdpexpxxx

i

x

4

,2

1 22

22

opopBA BA

ii

PQQP opopopop

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