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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 i 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. ii 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. iii 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. iv