Gedankenexperimente werden Wirklichkeit The strange features of quantum mechanics in the light of modern experiments

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Gedankenexperimente werden Wirklichkeit The strange features of quantum mechanics in the light of modern experiments Particle-wave complementarity Double-slit experiments (many examples) Entanglement

1 Gedankenexperimente werden Wirklichkeit The strange features of quantum mechanics in the light of modern experiments 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) Peter Schmüser Decoherence and the quantum-classical boundary (very brief) Albert Einstein and Niels Bohr Solvay Congress 1927

2 Double slit experiment with particles and waves For many decades this was a Gedankenexperiment in quantum theory

3 Double-slit experiments with photons, electrons, C60-buckyballs

4 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 Werner Heisenberg Max Born

5 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 5

6 What is light? Wave or stream of particles? Answer: both Compton effect

7 Helmut Rauch Physik in unserer Zeit, Nov Text 7

single crystal use Bragg reflection put aluminum slab

8 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

9 Influence of gravtiy on neutron interference rotatable interferometer fringes a a function of rotation angle 9

10 Dirac-spinors change sign upon a 2π rotation Paul Dirac: Dirac equation and antiparticles 10

11 Double-slit experiment with Na atoms and Na2 molecules 11

12 Bose-Einstein condensation Wolfgang Ketterle Macroscopic interference of matter waves

13 The quantum corral (Don Eigler IBM) Wave nature of electron is directly visible Using scanning tunneling microscope

14 Aharanov-Bohm effect: the wavelength of an electron is changed by a magnetic vector potential (q= charge) essential for QED Gottfried Möllenstedt 14

15 Akira Tonomura Hitachi Research Labs Physics Today April

$diffraction$ pattern each

16 Evolution of the diffraction pattern each electron makes just one spot 16

17 Verification of Aharonov-Bohm effect magnetic flux quantization in Nb superconductor magnetic flux lines emerging from a superconductor 17

18 Entanglement: consequence of superposition principle, applied to 2 or more particles Wikipedia:

19 Important application of superposition principle: hybrid wave functions in molecular and solid state physics ethylene benzene graphite graphene methane diamond silicon germanium GaAs 19

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.

20 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.

21 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) 21

spin-1/2 atoms From then on the EPR paradox was usually discussed in

22 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 22

23 Example: ground state of hydrogen molecule symmetric spatial wave function antisymmetric spin function (spins antiparallel) energy as function of distance between nuclei 23

24 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) 24

25 Important result: spin singlet along z axis is also spin singlet along x or y (reference is z axis) (reference is x axis) 25

26 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 26

27 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 27

28 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 28

29 Experiment with entangled photons by Alain Aspect two-photon cascade of Calcium atom entangled 2-photon wave function circular polarization linear polarization

30 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

31 Define expectation value E(φ) angle φ between the two polarizers 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. 31

32 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 32

33 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 4 2 Bell-limit angle φ 2 4 Bell s inequality is viotated by 40 standard deviations

34 Intermezzo: Derivation of Bell s inequality (after K. Fredenhagen) 34

35 statement: magnitude of S must be less than 2 35

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

36 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

37 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 37

38 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 38

decoherence time = τ / n τ=time constant of cavity=160 μsec n number of photons (

39 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

Wave properties of matter & Quantum mechanics I. Chapter 5

Wave properties of matter & Quantum mechanics I Chapter 5 X-ray diffraction Max von Laue suggested that if x-rays were a form of electromagnetic radiation, interference effects should be observed. Crystals