QM Lecture 6

LECTURE-6 The PRINCIPLE of COMPLEMENTARY
Quantum Behavior of Photons, Electrons and Atoms
The principle of complementarity:

Two observables are ‘complimentary’ if precise knowledge of one of them implies that all
possible outcomes of measuring the other one are equally probable.
Complementary distinguishes the world of quantum phenomena from the realm of classical physics.

Example 1. Let the observables be the position and momentum (along one direction) of a particle. If the position is predetermined
then the result of a momentum measurement cannot be predicted since all momentum values are equally probable (in a large
range.)

Example 2. Let the observables be two orthogonal spin components Sz and Sy of a spin-1/2 particle. If the vertical spin component Sz
has a definite value (‘up’ or ‘down’) then upon measuring its horizontal component (which can be equal to only ‘left’ or ‘right’)
each will be found with a probability of 50%.

Most examples have used the position (particle-like) and momentum (wave-like) attribute of a quantum mechanical object (be a photon
or a massive particle) to illustrate complementary, including Niels Bohr in his discussions with Einstein in 1927. This is the
historical reason why complementary is often superficially identified with the ‘wave-particle duality of matter’ (ability of
quantum-mechanical entities to behave as particles or waves under different experimental conditions.)
Complementarity is is more general and more fundamental to quantum mechanics than is the
uncertainty principle. Complementarity remains intact even when the uncertainty relation plays no role.

The Uncertainty Principle is not the only enforcer of complementarity.

References
- Richard Feynman, “The Feynman Lectures on Physics,” Volume III, Ch 1. Volume I, Ch 30 and Ch-38.
- Reuben S. Aspden and Miles J. Padgett. Video recording true single-photon double-slit interference. American Journal of Physics 84, 671 (2016).
https://doi.org/10.1119/1.4955173
- P. Grangier, G. Roger and A. Aspect, “Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences.”
Europhys. Lett., 1 (4), 173 (1986). It illustrates the wave-particle duality of light; addresses the concept of "single photon state"; uses a Mach-Zehnder interferometer.
- D. F. Walls. A simple field theoretic description of photon interference. American Journal of Physics 45, 952 (1977).
- M. O. Scully, B. G. Englert, and H. Walther. “Quantum Optical Test of Complementary.” Nature 351, 111 (1991).
- S. P. Walborn, M. O. Terra Cunha, S. Padua, and C. H. Monken. Double Slit Quantum Eraser Physical Review A 65, 033818 (2002)
This paper describes the optical implementation of Scully's 1991 Quantum Optical Test of Complementary.
- Berthold-Georg Englert, Marlan O. Scully and Herbert Walther. The Duality in Matter and Light. Scientific American 271, 86 (December 1994).
- S. Durr, T. Nonn & G. Rempe. Origin of quantum-mechanical complementarity probed by a `which-way' experiment in an atom interferometer. Nature 395, 33 (1998).

6.1 Shortcomings of classical concepts for describing the behavior of electrons passing through two slits
6.1.A The concept of Probability. An experiment with bullets
6.1.B An experiment with light
6.1.C An experiment with electrons
6.1.D On the restricted information about the motion of quantum particles
6.1.E Failed attempts to track the electrons’ trajectories. An understanding of the wave-particle duality
6.1.F Discarding the idea that electrons move along paths
6.1.G Double-slit experiments with more massive particles. The role of INFORMATION

6.2 The Heisenberg’s Uncertainty Principle
6.2.A Uncertainty in the position x and linear momentum p. IN 2021, WE JUMP FROM HERE TO LECTURE 7
6.2.B Relationship between the uncertainty of the energy content E of a pulse and the time t required
for the measurement
6.2.B.a Gratings, phasors (addition of multiple waves), conditions for constructive interference,
evaluation of the sharpness of the peak of order m.
6.2.B.b Gratings and spectral resolution.
6.2.B.c Condition for the minimum length time t required for the measurement of the energy E
with a resolution deltaE.
6.2.C. More examples.
6.2.C.a The resolving power of a microscope and the uncertainty principle
6.2.C.b The size of an atom

6.3 The Principle of Complementary
Investigation of other mechanisms, different than the uncertainty Heisenberg principle, that enforce Complementary
Objective of this section
6.3.A New lights on the two-slit experiment
What was believed before
New lights from recent experiments.
6.3.B A two-slit experiment with atoms and a which-way detection that does not fall prey to the position-momentum uncertainty relation
The “which-way” detection.
6.3.C Correlation between the atomic wavefunction and the states of the which-way detector
Case: Discarding the laser and the two maser cavities.
Case: Excitation laser but discarding the maser cavities.
Case: Laser is on, and cavities are present
6.3.D Quantum Eraser
Introduction
6.4 Contrasting the (deterministic) classical and (probabilistic) quantum mechanics descriptions
6.4.1 Deterministic character of classical mechanics
6.4.2 Absence of trajectories in the quantum behavior of particles.
6.4.3 Classical mechanics as a requirement and as a limiting case of quantum mechanics
6.4.4 The less detailed quantum mechanics description
6.5 The non-relativistic quantum mechanics description

Additional References