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QUANTUM MECHANICAL LAWS
Bogdan Mielnik and Oscar Rosas-Ortiz
Departamento de Física, Centro de Investigación y de Estudios Avanzados,
A.P. 14740, México D.F. 07000, Mexico
Summary
The present day quantum laws unify simple empirical facts and fundamental principles describing
the behaviour of micro-systems. A parallel but different component is the symbolic language
adequate to express the ‘logic’ of quantum phenomena. This report is the sum of both elements. The
Sections 1-7 present the decline of the classical theory and the evidence of the particle-wave duality
on an elementary level. In the Sections 8-9 the origin of the Schrödinger’s wave mechanics and
Born’s statistical interpretation are presented. The following Sections 10-23 are basically addressed
to the adepts of exact sciences. They outline the formal and fundamental problems of Quantum
Mechanics, interpretational paradoxes, concepts of complementarity, teleportation and the idea of
quantum computing. Though using the symbolic language, Quantum Mechanics is not a hermetic
science. It is an open domain, crucial for the present day particle physics, quantum field theory and
contemporary informatics, though carrying a luggage of unsolved interpretational problems.
Keywords: Indeterminism, Quantum Observables, Probabilistic Interpretation, Schrödinger’s cat,
Quantum Control, Entangled States, Einstein-Podolski-Rosen Paradox, Cannonical Quantization,
Bell Inequalities, Path integral, Quantum Criptography, Quantum Teleportation, Quantum
Computing.
CONTENTS:
1. Introduction
2. Black body radiation: the lateral problem becomes fundamental.
3. The discovery of photons
4. Compton’s effect: collisions confirm the existence of photons
5. Atoms: the contradictions of the planetary model
6. The mystery of the allowed energy levels
7. Luis de Broglie: particles or waves?
8. Schrödinger’s wave mechanics: wave vibrations explain the energy levels
9. The statistical interpretation
9.1 Polarized photons: absence of deterministic rules?
9.2 Born’s statistical interpretation.
10. The Schrödinger’s picture of quantum theory
10.1 States, superpositions, probabilities
10.2 Quantum geometry
10.3 Linear operators: the spirit of interference
10.4 Quantum observables: old and new ideas
10.5 Observables as operators: the anatomy of the measurement
10.6 General formalism; the extended Born’s hypothesis
10.7 Functions of an observable
10.8 Transition probabilities: the most elementary statistical law
10.9 Construction of traditional observables
11. The uncertainty principle: instrumental and mathematical aspects.
12. Typical states and spectra
12.1 Harmonic oscillator: the spectral ladder, eigenstates and coherent states
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12.2 The angular momentum
12.3 Hydrogen and hydrogen-like atoms
12.4 Pauli’s exclusion principle
12.5 Particle spin: a controversial, non-classical property
12.6 The spectral bands
13. Unitary evolution
14. Canonical quantization: scientific or magic algorithm?
15. The mixed states
16. Quantum control: how to manipulate the particle?
17. Measurement theory and its conceptual consequences
17.1 State reduction (collapse of the wave packet)
17.2 The strong uncertainty principle
17.3 The idea of complementarity
17.4 “Quantum logic”
17.5 Quantum bit (qubit)
18. Interpretational polemics and paradoxes
18.1 Quest for alternative theories
18.2 The measurement problem
18.3 Borderline doctrine
18.4 The paradox of the Schrödinger’s cat
18.5 The doctrine of decoherence
18.6 Ensemble interpretation
18.7 The realistic view
18.8 Many worlds interpretation of Everett
18.9 Quantum jumps and quantum Zeno effect
18.10 The macroscopic superposition
19. Entangled states
19.1 General concepts
19.2 Spin-configuration entanglement
19.3 Many particle entangled states
20. Dirac’s theory of the electron as the square root of the Klein-Gordon law
20.1 The Klein-Gordon equation
20.2 Dirac’s equation
20.3 The hypotesis of positron
21. Feynman: the interference of virtual histories
22. Locality problems
22.1 The EPR paradox
22.2 Hidden variables and Bell’s inequalities
22.3 “Seeing in the dark”
22.4 Teleportation
23. The idea of quantum computing and future perspectives
23.1 Quantum cryptography
23.2 Quantum computing
24. Open questions
Glossary
Bohr magneton: The quantity g = eh
2mc
= 0.927×10-20 erg/oersted, where e and m are the
electron’s charge and rest mass, c is the light velocity, h = h 2π and h = 6.626×10-34
joules·sec, represents the (anomalous) magnetic moment of the electron observed in
the Zeeman effect, explained by the Dirac’s theory.
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Boson: Named after Satyendra Nath Bose, bosons are elementary particles with integer spin
and symmetric wave functions obeying the Bose-Einstein statistics (any number of
bosons can share the same quantum state). The quanta of light (photons) are the most
common and abundant bosons.
Complementarity: The fact that certain properties (aspects) of quantum systems cannot be
observed at the same time and make no sense simultaneously.
Copenhagen School: A pragmatic interpretational current following the ideas of N. Bohr, W.
Heisenberg, A. Rosenfeld, and others, centered around the concepts of uncertainty and
complementarity in quantum theories.
Determinism: An idea or demand that the initial conditions should determine uniquely the
final result of a certain process or experiment.
Electron: The elementary particle of negative electric charge e = 1.0602176462(63)×10-19
Coul, mass me = 9.10938188(72)×10-31 Kg and spin ½ crucial for the development
of quantum mechanics.
Entangled state: The state describing correlations between several independent properties
or components (e.g., different particles) of a quantum system. Expressed in terms
of formal (tensor) products of its component states, plays a central role in quantum
information, teleportation and computing.
Fermion: Named after Enrico Fermi, fermions are micro particles with half-integer spin and
antisymmetric wavefunctions obeying the Fermi-Dirac statistics (no two fermions
can occupy the same quantum mechanical state at the same time). The most
elementary examples are the electron, proton and neutron.
Hidden parameters: The hypothetical classical variables permitting to recover the
deterministic picture of quantum experiments.
Indeterministic effect (phenomenon) : An effect which cannot be precisely predicted on the
basis of fundamentally available initial data.
Magnetic anomaly: The fact that the magnetic moment of the electron (Bohr magneton) is
twice as great as could be explained by the electron spin. It indicates that neither the
spin nor the electron’s magnetic moment admit the simple mechanical model of an
internal electron rotation.
Measurement paradox: The state of a quantum system evolves according to a linear,
deterministic law; yet, the results of the measurements, in general, are unpredictable
(indeterministic).
Photon: The light corpuscle of vanishing rest mass and spin 1 discovered in photoelectric
effects. One of principal actors in the development of life and science.
Quantum logic: An abstract approach to quantum theory looking for an analogy between
the simplest quantum mechanical experiments (filters) and propositions of a certain
logical system.
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Quantum observable: A non-classical concept of an observable property represented by an
operator, i.e., an operation preformed on quantum states.
Quantum state: A collection of non-contradictory data (information) permitting to predict
in probabilistic terms the behaviour of a quantum system. If the information is
maximal, the state is pure, represented by a wavefunction (wave packet), more
generally, by a vector in a linear space (state vector).
Quantum superposition: An “undecided”, pure state of a micro-system involving several
other states as tentative options (components) and represented by their linear
combination.
Reduction (collapse) of the wave packet: A hypothetical, unpredictable jump of the microsystem state to one of the eigenstates of the measured quantity.
Schrödinger’s cat: An emblematic creature in a superposed state of being simultaneously
alive and dead.
Spin: An intrinsic quantized property of micro particles generalizing (but not identical to)
the classical rotational momentum.
Uncertainty principle: The law limiting the simultaneous
measurability
of
noncommutingobservables. The phenomenon characteristic for the complementary
quantities.
Wave-particle duality: The micro-particles propagate as waves but are detected as
corpuscles.
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