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Gedankenexperimente werden Wirklichkeit
The strange features of quantum mechanics
in the light of modern experiments
Peter Schmüser
Particle-wave complementarity
Double-slit experiments
(many examples)
Entanglement
Einstein-Podolsky-Rosen (EPR) paradox
Bell‘s inequality
(I omit the proof)
EPR experiments with entangled photons
Nonlocality in quantum mechanics
Schrödinger‘s cat
(very brief)
Decoherence and the quantum-classical
boundary (very brief)
Dienstag, 18. Januar 2011
Albert Einstein and Niels
Bohr
Solvay Congress 1927
Double slit experiment with particles and waves
For many decades this was a Gedankenexperiment in quantum theory
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Double-slit experiments with photons, electrons, C60-buckyballs
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Statistical interpretation of quantum mechanics
a) Compute the wave function by solving the Schrödinger equation
b) Square the wave function to get the probability for finding the particle
Erwin Schrödinger
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Max Born
Werner Heisenberg
Is the statistical interpretation correct?
Answer: yes
Double-slit experiment
few electrons and very many electrons
these experiments were impossible in 1925
Niels Bohr was the main proponent of the statistical interpretation
Copenhagen interpretation of quantum mechanics
Albert Einstein rejected it: „Gott würfelt nicht“
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What is light? Wave or stream of particles?
Answer: both
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Compton effect
Helmut Rauch
Physik in unserer Zeit,
Nov. 1998
Text
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Neutron interferometer: each neutron interferes with itself
perfect crystal interferometer
cut out of a Si single crystal
use Bragg reflection
put aluminum slab in one arm
acts like a glass plate in an
optical interferometer
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Influence of gravtiy on neutron interference
rotatable interferometer
fringes a a function
of rotation angle
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Dirac-spinors change sign upon a 2π rotation
Paul Dirac: Dirac equation and antiparticles
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Double-slit experiment with Na atoms and Na2 molecules
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Bose-Einstein condensation
Wolfgang Ketterle
Macroscopic interference of matter waves
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The quantum corral
(Don Eigler IBM)
Wave nature of electron is directly visible
Using scanning tunneling microscope
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Aharanov-Bohm effect: the wavelength of an electron is changed
by a magnetic vector potential
(q= charge)
essential for QED
Gottfried Möllenstedt
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Akira Tonomura
Hitachi Research Labs
Physics Today April 1990
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Evolution of the diffraction pattern
each electron makes just one spot
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Verification of Aharonov-Bohm effect
magnetic flux quantization
in Nb superconductor
magnetic flux lines emerging
from a superconductor
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Entanglement:
consequence of superposition principle, applied to 2 or more
particles
Wikipedia:
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Important application of superposition principle:
hybrid wave functions in molecular and solid state physics
ethylene
benzene
graphite
graphene
methane
diamond
silicon
germanium
GaAs
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Einstein-Podolsky-Rosen (EPR) paradox 1935
EPR answer: No
Conclusion by Einstein et al: quantum mechanics cannot be the
ultimate theory. There must exist an underlying deterministic theory.
This would be a local theory with hidden variables, and quantum mechanics might be
considered as an averaged version of the deeper theory.
Analogy: Statistical thermodynamics is the underlying theory of phenomenological
thermodynamics. The positions and momenta of a huge number of particles are the hidden
variables which are not measurable. Internal energy, pressure, entropy etc. are averaged
quantities that can be measured.
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Reponse by Niels Bohr, a few months later
Bohr claims that quantum mechanics is complete.
But it is a nonlocal theory, as we will see
Bohr‘s arguments are not very convincing (at least not to me)
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1957: David Bohm proposes an EPR experiment that seems feasible
Bohm and Aharonov consider a spin-0 molecule
that disintegrates into 2 spin-1/2 atoms
From then on the EPR paradox was usually discussed in the variant
proposed by Bohm
But for many years most physicists ignored the issue
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Example: ground state of hydrogen molecule
symmetric spatial wave function
antisymmetric spin function
(spins antiparallel)
energy as function of
distance between nuclei
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EPR experiment as proposed by Aharonov and Bohm
The 2 electrons in the H2 molecule have antiparallel spin vectors.
Molecule is split into 2 atoms by a method which does not affect the spin.
Suppose this is a classical system. Then the 2 hydrogen atoms must have antiparallel
spin vectors even at large separation
from angular momentum conservation
All 3 components of the spin vector of atom 2 are completeley determined
by measuring the spin vector of atom 1. No measurement on atom 2 is
needed, nor any long-range interaction between the atoms.
Quantum mechanics is much more complicated:
only one spin component can have a definite value, the other two are
completely unknown
spin singlet state
(reference is z axis)
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Important result: spin singlet along z axis is also spin singlet along x or y
(reference is z axis)
(reference is x axis)
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Consequences of a spatial limitation of entanglement
Einstein et al. assume a limited interaction range of the 2 systems.
Outside this range the 2 atoms have nothing to do with each other.
We apply the EPR view in a very naive way (Einstein was not that naive!).
Immediately after the disintegration of the H2 molecule the two H-atoms are still in the
entangled spin singlet state, but when they leave the interaction range, the atoms have to
„decide“ which spin orientation they want to choose.
within interaction range
outside interaction range
Now Experimenter does something terrible: he measures spin in x direction
in 50% of the cases he finds RR or LL:
angular momentum is not conserved!
Hence: entanglement must
be of unlimited range
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Two possibilities to understand the long-range spin correlation
(1) Niels Bohr: quantum mechanics is a „complete theory“.
An entangled wave function may be infinitely long
(2) Albert Einstein: the „spukhafte Fernwirkung“ is absurd.
Quantum mechanics is an incomplete theory.
There must exist a more fundamental local theory
Theoretical arguments alone cannot settle the dispute. The great scholars Einstein and
Bohr debated all their life about quantum theory but never came to an agreement.
1964: John Bell takes a fresh look at quantum mechanics and
derives the Bell inequality
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Nature March 1998
John Bell
theoretician
Alain Aspect
experimentalist
It is the great achievement of John Bell that
the discussion about the physical reality of
quantum systems has been transferred from
philosophy to experimental physics
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Experiment with entangled photons by Alain Aspect
two-photon cascade of Calcium atom
entangled 2-photon wave function
circular polarization
linear polarization
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Detection of photons at large distance from source S
polarizing beam splitters measure polarization parallel or perpendicular to unit vector a
resp. unit vector b
parallel +1 result, perpendicular -1 result
From entangled wave function
follows:
coincidence of the 2 photons
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Define expectation value
angle φ between the two polarizers
E(φ)
Aspect et al. PRL 49 (1982)
φ
The two photons are indeed correlated in polarization
Choose angle φ=0: the polarization detectors are then parallel,
and we get E(φ)=1. The photons are 100% correlated.
From the measurement on photon 1 we know everything about photon 2.
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Aim of Aspect‘s experiment: try to distinguish between the 2 possible
origins of long-range polarization correlation:
(1) entangled quantum state
(2) local theory with hidden variables
Measure with two orientations a, a‘ of detector A
and b, b‘ of detector B
Define linear combination of expectation values
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Experimental result:
quantum theory is correct, hidden variables do not exist
the quantum mechanical form of S is
Bell‘s inequality evaluated for S reads
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Bell-limit
2
0
30
60
90
angle φ
2
4
Bell‘s inequality is viotated by 40 standard deviations
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Intermezzo: Derivation of Bell‘s inequality (after K. Fredenhagen)
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statement: magnitude of S must be less than 2
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Aspect‘s experiment shows strong violation of Bell‘s inequality. There remains one final
loophole for protagonists of the idea of hidden variables: if the setting of the detectors A
and B is static they could communicate by some some mysterious mechanism
Can be excluded by changing the angles of detectors A and B during the flight of the
photons so rapidly that information transfer from A to B would require signal speed much
larger than speed of light
Experiments with entangled laser photons and time-varying detectors
violate Bell‘s inequality as well
Anton Zeilinger
1 UV photon is converted into 2 infrared photons in BBO crystal
The IR photons are in an entangled polarization state
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The nonlocal nature of quantum mechanics
The experiments (by Aspect, Zeilinger and many others) prove:
quantum theory is correct, hidden variables do not exist
So Bohr was right, Einstein was wrong
Unavoidable consequence: entangled wave functions may have very
large extension (more than 140 km measured)
Abstract of Nature article by Alain Aspect
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Schrödinger‘s cat
Inspired by the EPR paper, Erwin Schrödinger proposed in 1935 a Gedankenexperiment in
which a macrocopic object (the cat) is entangled with a microscopic system (radioactive
atomic nucleus). He wanted to demonstrate the incompleteness and weirdness of quantum
mechanics.
ψ=[ (excited nucleus/ cat alive)+(nucleus in ground state / cat dead) ]
The nucleus is in a superposition of excited and ground state, so the
cat should also be in a superposition of alive and dead, and a decision
would only be done in the moment when a measurement is performed
Result of many „Schrödinger cat“ experiments: a macroscopic system
has always a lot of interaction with the environment. This destroys the
quantum mechanical coherence very rapidly.
A cat is either alive or dead, but not both at the same time
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Serge Haroche et al.: experiments with „mini-cats“
mini-cat: microwave photons in a superconducting cavity
decoherence time = τ / n
τ=time constant of cavity=160 μsec
n number of photons ( n=3 - 10)
decoherence time of a cat is
unmeasurably short because n is huge
The cat is always a classical object,
it is never a quantum object
Physics Today July 1998
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