LOUIS DE BROGLIE AND THE SINGLE QUANTUM PARTICLE By A. O. Barut We have abundant evidence and testimony that Louis de Broglie deeply cared about the foundations, the meaning, and our understanding of quantum theory in general and of wave mechanics in particular. So, too, did Erwin Schrodinger, along with Einstein, Bohr, Dirac, and Heisenberg. For de Broglie and Schrodinger this preoccupation meant not simply the acceptance of a novel set of rules, but a constant struggle and a search for complete clarity about the way in which the new theory fits into the great classical traditions of an objective physical world view. We may call this a striving for "physical rigor," rigor in reasoning, or intellectual rigor. There is not only mathematical rigor inside an axiomatic system with which everybody agrees, but there is, and there should be, rigor also in our concepts and methods. To this kind of rigor belongs the unity, the economy and simplicity, and the consistency of physical theories; naturally along with as complete and as clear an understanding of phenomena as possible. No loose ends, no proliferation of poorly tested and phenomenological entities, no bending of logic and compromise, and no handwaiving arguments can be tolerated. Unfortunately this kind of rigor seems to be missing in today's forefront of fundamental physical theories, viz. , particle or high-energy physics.

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One (19501951) On Heisenberg's Uncertainties and the Probabilistic Interpretation of Wave Mechanics.- 1. Principles of Wave Mechanics.- 1. Classical Mechanics of the Point Mass. Theory of Jacobi.- 2. Wave Propagation in an Isotropic Medium.- 3. Transition from Classical Mechanics to Wave Mechanics.- 4. General Equation for the Wave Mechanics of a Point Mass.- 5. Automatic Procedure for Obtaining the Wave Equation.- 2. Probabilistic Interpretation of Wave Mechanics.- 1. Interpretation of the ? Wave.- 2. Principle of Interference.- 3. Precise Statement of the Principle of Interference. Probability Fluid.- 4. The Uncertainty Relations of Heisenberg.- 5. The Principle of Spectral Decomposition (Born).- 6. New Ideas Resulting from the Preceding Conceptions.- 7. Return of Wave Mechanics to Classical Mechanics..- The Theorem of Ehrenfest. Group Velocity.- 3. Wave Mechanics of Systems of Particles.- 1. Old Dynamics of Systems of Point Masses.- 2. Wave Mechanics of Systems of Particles.- 3. Interpretation of Wave Mechanics for a System of Particles.- 4. General Formalism of Wave Mechanics.- 1. New Conception of the Quantities Attached to a Particle (or to a System).- 2. Eigenvalues and Eigenfunctions of a Linear Hermitian Operator.- 3. Continuous Spectrum of the Free-Particle Hamiltonian. Dirac's Delta Function.- 5. General Principles of the Probabilistic Interpretation of Wave Mechanics.- 1. General Ideas.- 2. The Algebraic Matrices and Their Properties.- 3. Operators and Matrices in Wave Mechanics.- 4. Mean Values and Dispersions in Wave Mechanics.- 5. First Integrals in Wave Mechanics.- 6. Angular Momentum in Wave Mechanics.- 6. Theory of the Commutation of Operators in Wave Mechanics.- 1. General Theorems.- 2. Corollaries of the Foregoing Theorems.- 3. Simultaneous Measurement of Two Quantities in Wave Mechanics.- 4. Examples of Quantities That Are Not Simultaneously Measurable. Distinction Between Two Kinds of Non-Commutation.- 7. Physical Impossibility of Simultaneously Measuring Canonically Conjugate Quantities.- 1. Necessity to Examine the Impossibility of Simultaneously and Precisely Measuring Two Canonically Conjugate Quantities.- 2. The Heisenberg Microscope.- 3. Measurement of the Speed of an Electron by Means of the Doppler Effect.- 4. Passage of a Particle through a Rectangular Aperture.- 5. Important Remark on the Measurement of Speed.- 6. The Case of Two Operators with a Nonzero Commutator.- 7. Bohr's Complementarity.- 8. Bohr's Calculation for Young's Slits.- 8. Precise Form of the Uncertainty Relations.- 1. Theorem About the Dispersion of Non-Commuting Quantities.- 2. Optimal Nature of the Gaussian Wave Packet.- 3. Comparison of the Theorem on Dispersions with the Qualitative Uncertainty Relations of Heisenberg (Pauli and Robertson).- 4. Various Considerations about Uncertainties. Sharp-edged Uncertainties.- 9. Heisenberg's Fourth Uncertainty Relation.- 1. The Absence of Symmetry between Space and Time in Wave Mechanics.- 2. Correct Formulation of the Fourth Uncertainty Relation.- 3. Illustration of the Preceding Definition.- 4. Various Remarks about the Fourth Uncertainty Relation.- 5. Method of Variation of Constants. Transition Probability.- 6. Transition Probabilities.- 7. Uncertainty Relations and Relativity Theory.- 8. Formulas of Mandelstam and Tamm.- 10. Examination of Some Difficult Points in Wave Mechanics.- 1. Reduction of the Probability Packet by Measurement.- 2. Impossibility of Discovering the Anterior State of a Measurement from its Posterior State. Effacement of Phases by Measurement.- 3. Possibility of Discovering the Past, Starting from a Measurement Made at a Given Instant (Postdiction).- 4. Interference of Probabilities.- 5. Some Consequences of the Disappearance of the Trajectory Concept.- 6. Discussions Concerning Correlated Systems.- 7. Complementary Remarks on the Einstein-Bohr Controversy.- Two (19511952) On the Probabilistic Interpretation of Wave Mechanics and Various Related Questions.- 11. Summary of Some General Concepts of Probability Calculus.- 1. Probability Laws for One Variable. Distribution Function.- 2. Probability Laws for Two Variables.- 3. Very Important Remark Concerning the Foregoing Results.- 12. Recalling the General Concepts of Wave Mechanics.- 1. The Interference Principle. Theory of the Pilot Wave.- 13. Introduction of the Characteristic Function into the Probability Formalism of Wave Mechanics.- 1. Characteristic Function for a Single Quantity.- 2. Characteristic Function for Two Commuting Quantities.- 3. Correlation Coefficient. Marginal Laws.- 4. General Theorems of Wave Mechanics Considered from the Characteristic-Function Point of View.- 5. Case of Two Noncommuting Quantities.- (a) Reminder About the Reduction of the Probability Packet by Measurement.- (b) Interference of Probabilities.- (c) The Distribution Function ?(x,?x) of Wigner and Bass.- (d) The Wigner-Bass Density ?(x,?x) and the Hydrodynamic Interpretation of Wave Mechanics.- (e) Yvon's Theory.- 14. Theory of Mixtures and von Neumann's Theory of Measurement.- 1. Mixtures and Pure Cases.- 2. Von Neumann's Statistical Matrix for a Pure Case.- 3. The Statistical Matrix for a Mixture of Pure Cases.- 4. Irreducibility of Pure Cases.- 5. Impossibility of Reducing the Laws of Wave Mechanics to a Hidden Determinism (von Neumann).- 15. Measurement Theory in Wave Mechanics.- 1. Generalities.- 2. Statistics of Two Interacting Systems.- 3. Correlation Coefficients in the Interaction between Two Quantum Systems.- 4. Measurement of a Quantity.- 5. Example of a Measuring Experiment.- 6. Diverse Remarks on Measurement.- 7. Thermodynamical Considerations (von Neumann).- 8. Reversible and Irreversible Evolutions.- 9. The Statistical Matrix P0.- 16. The Role of Time in Wave Mechanics.- 1. Retrodiction According to Costa de Beauregard.- 2. Special Role of Time in Quantum Mechanics. The Fourth Uncertainty Relation.- 3. Correct Statement of the Fourth Uncertainty Relation.- 4. The Fourth Uncertainty Relation and Perturbation Theory.- 5. The Operators H and (h/2?i)?/?t.- 6. Application of the Formalism of Arnous to the Operators Acting on Time.- 7. Multitemporal Formalism. Multiplicity Curves in Spacetime.- Works of Louis de Broglie.