Foundations of Quantum Mechanics

Dr Willem M. de Muynck associate professor, retired from and presently guest at Theoretical Physics Department of Applied Physics Eindhoven University of Technology P.O.B. 513, 5600 MB Eindhoven the Netherlands Phone: +31 40 2474351 Fax: +31 40 2445253 E-mail: W.M.d.Muynck@tue.nl

Contents of this website

Quantum mechanics the way I see it

Main publications

Page significantly updated January 10, 2010 , small updates February 9, 2010 

• Preamble
  • Axiomatic approach of quantization
  • Measurement in the microscopic domain
  • Physics and philosophy
    • Ontology versus epistemology
    • `Microscopic reality' versus the `phenomena'
    • Empiricist versus rationalist influences
    • Theory-ladenness of observation
  • Inadequacy of the standard formalism

• Standard formalism of quantum mechanics
  • Quantum mechanical standard observable
  • Quantum mechanical state, the superposition principle
  • Probability distribution; Born rule
  • Expectation value
  • Time evolution
  • Liouville-von Neumann equation
  • Remarks on the standard formalism
    • Schrödinger and Heisenberg picture
    • Von Neumann's projection postulate
      • Strong von Neumann projection
      • Weak von Neumann projection
      • Objections to the projection postulate
      • Remarks on the projection postulate
        • Its experimental origin
        • Nondisturbing character of the Compton-Simon experiment
        • Von Neumann projection is a `preparation principle' rather than a `measurement principle'
    • `Von Neumann projection' versus `faithful measurement'
      • Remarks on `Von Neumann projection versus faithful measurement'
    • Relative frequencies and probability distribution
      • Remark on `relative frequencies and probability distribution'
    • (In)compatibility of observables
      • The Heisenberg-Kennard-Robertson inequality
      • Entropic uncertainty relation
    • Constants of the motion
      • Remarks on `constants of the motion'
  • Comparison with classical (statistical) mechanics

• Generalized formalism of quantum mechanics
  • Generalized quantum mechanical observable (POVM)
  • Probability distribution of POVM
  • Remarks on the `generalized formalism of quantum mechanics'

• Quantum mechanics and `reality'
  • Ontological realism
  • Epistemological realism
  • Remarks on `Quantum mechanics and reality'

• Quantum mechanics and `observation'

• Quantum mechanics and `measurement'
  • Quantum mechanical description of pre-measurement
  • Measurements of the first and second kind
  • The Heisenberg cut
  • Conditional preparation
    • The quantum Zeno effect

• Interpretations of quantum mechanics
  • Interpretation as a mapping
  • Prejudice

• Realist versus empiricist interpretation of quantum mechanics
  • Realist interpretations of quantum mechanics
    • Objectivistic-realist versus contextualistic-realist interpretations of quantum mechanics
  • Empiricist interpretation of quantum mechanics
  • Reasons to prefer a realist interpretation of quantum mechanics
    • Ontological reason:
      • Quantum reality
    • Epistemological reasons:
      • The classical paradigm
      • Platonism
    • Methodological reason:
      • Inference to the best explanation
  • Reasons to prefer an empiricist interpretation of quantum mechanics
    • Ontological reason:
      • Experimental data are exclusively `pointer positions of measuring instruments'
    • Epistemological reason:
      • Analogous cases
    • Methodological reasons:
      • Evading the paradoxes of a realist interpretation
      • Escaping from suggestions implied by the standard formalism
      • Acknowledging the essential role of measurement

• Realist versus instrumentalist interpretation of quantum mechanics
  • Remarks on the `instrumentalist interpretation'
    • Empiricist versus instrumentalist interpretation

• `Objectivity versus subjectivity' in quantum mechanics
  • `Objectivity versus subjectivity' in `realist interpretations of quantum mechanics'
  • `Objectivity versus subjectivity' in the `empiricist interpretation of quantum mechanics'
  • `Irreducibility versus reducibility' of quantum probability
    • Remarks on `irreducibility versus reducibility'
    • `Probabilistic versus statistical' interpretation of the Born rule
    • Remarks on the `probabilistic versus statistical' dichotomy
    • `Objective versus subjective probability' in quantum mechanics
    • Remarks on `objective versus subjective probability'

• `Determinism versus indeterminism' in quantum mechanics
  • Ambiguous use of the notion of `(in)determinism'
    • i) (In)determinism versus (a)causality
    • Remarks on `(in)determinism versus (a)causality'
    • ii) (In)determinism during free evolution and/or measurement
    • Remarks on `(in)determinism during free evolution and/or measurement'
    • iii) Ontological and epistemological senses of `(in)determinism during measurement'
    • Remarks on `ontological and epistemological senses of (in)determinism during measurement'

• `Individual-particle interpretation' versus `ensemble interpretation' of quantum mechanics
  • Interference with the `realist versus empiricist' dichotomy
  • Individual-particle interpretation of the quantum mechanical state vector
  • Critique of the `individual-particle interpretation'
    • The Schrödinger cat paradox
    • Remarks on the `Schrödinger cat paradox'
    • Additional arguments against the `individual-particle interpretation'
  • Ensemble interpretations of the quantum mechanical state vector
    • Observational evidence supporting the `ensemble interpretation'
    • Remarks on the `ensemble interpretation'
    • Possessed values principle
    • Counterfactual definiteness
    • Explanation by means of subensembles
    • `Ensemble interpretation' is more general than `statistical interpretation'
    • `Ensemble interpretation' and `incompleteness of quantum mechanics'
  • Homogeneous and inhomogeneous ensembles
    • The `minimal interpretation'
    • Critique of the `minimal interpretation'
    • Von Neumann's interpretation
    • Criticisms of von Neumann's interpretation
    • The `statistical interpretation'
    • Critique of the `statistical interpretation'
    • Quantum ensembles
    • Ontological aspects of quantum ensembles

• Ontic versus epistemic interpretation of quantum mechanics

• Quantum mechanics and logical positivism/empiricism
  • `Logical positivism/empiricism' and `completeness of quantum mechanics'
  • Theory-(in)dependence of measurement
  • `Empiricist versus operationalist' interpretation of quantum mechanics
  • Quantum mechanics after `logical positivism/empiricism'

• The "measurement problem"
  • Some proposals to solve the "measurement problem"
    • Solution by means of `von Neumann ensembles'
      • Criticism of `solution by means of von Neumann ensembles'
    • The "unobservability" solution
      • Criticism of the "unobservability" solution
    • The `decoherence solution'
      • Criticism of the `decoherence solution'
    • Solution by adopting an ensemble interpretation
      • Empiricist perspective on the `ensemble interpretation'

• Necessity of the `generalized formalism of quantum mechanics'
  • Some illustrations of the necessity of the generalized formalism
    • Inefficient photon detection
    • The double-slit experiment
    • Polarization measurements
    • Joint nonideal measurement of incompatible polarization observables
  • Nonideal measurements
  • Joint measurement of generalized observables
    • Commeasurability of (generalized) observables
    • Joint nonideal measurement of (generalized) observables
    • Complementarity in a joint nonideal measurement of incompatible standard observables
      • The Martens inequality
        • Proof of the `Martens inequality'
        • `Martens inequality' versus `entropic uncertainty relation'
        • `Martens inequality' versus `Heisenberg-Kennard-Robertson inequality'
        • The `Martens inequality' and `complementarity'
        • Remarks on `Martens inequality and complementarity'

• The Copenhagen (orthodox) interpretation
  • Preamble to the `Copenhagen interpretation'
    • Impossibility of an unambiguous definition
    • Between `instrumentalism/empiricism' and `scientific realism'
    • Heisenberg versus Bohr: physics versus philosophy
    • Bohr, Heisenberg, and `interpretations'
    • Essential role of measurement
    • `Copenhagen interpretation' versus `empiricist interpretation'
    • The `Copenhagen interpretation' and `experiment'
    • Negative and positive features of the `Copenhagen interpretation'
  • The quantum postulate
    • Remarks on the quantum postulate
    • Criticisms and appraisal of the `quantum postulate'
      • Ambiguity of the notion of `quantum phenomenon'
      • Criticisms of `unanalyzability of the quantum phenomenon'
        • Influence of the `classical paradigm'
        • Einstein's challenge of the `unanalyzability of the quantum phenomenon'
      • Reliance of the `Copenhagen interpretation' on the notion of `measurement'
      • The `quantum phenomenon' in the `EPR experiment'
      • Bohr versus Einstein on the `quantum postulate'
  • The Copenhagen completeness thesis
    • Completeness in the restricted sense
    • Completeness in the wider sense
    • Criticisms of the `Copenhagen completeness thesis'
      • Failure to distinguish `completeness in the wider sense' from `completeness in the restricted sense'
      • Metaphysical character of `Copenhagen completeness'
  • The correspondence principles
    • Different forms of the `correspondence principle'
      • i) Correspondence in the Old quantum theory
      • ii) Weak form of the correspondence principle
      • iii) Strong form of the correspondence principle
        • Remarks on `strong correspondence'
        • Latitude of definition of an observable (physical quantity)
        • Criticisms and appraisal of `strong correspondence'
          • Reliance of `strong correspondence' on `classical mechanics'
          • `Strong correspondence' without `reliance on classical mechanics'
      • iv) Empiricist form of the correspondence principle
  • Complementarity
    • `Complementarity of preparation' versus 'complementarity of measurement'
    • Bohr's notion of `complementarity'
      • The `uncertainty principle'
      • Incompatible pictures supplementing each other
    • Remarks on Bohr's notion of `complementarity'
      • `Complementarity' and `completeness'
      • `Particle-wave duality' and `particle-wave complementarity'
      • Critique of `particle-wave complementarity'
    • Heisenberg's notion of `complementarity'
      • Heisenberg's disturbance theory of measurement
    • Differences between Bohr and Heisenberg regarding the `uncertainty principle'
    • Critique of `Heisenberg's disturbance theory of measurement'
  • Critique of `complementarity'
  • `Heisenberg-Kennard-Robertson inequality' and `complementarity'
    • i) `Quantum mechanical acausality' versus `classical determinism'
    • ii) `Free evolution' versus `measurement'
    • iii) Ontological versus epistemological meaning
    • iv) `Completeness in the wider sense' versus `completeness in the restricted sense'
    • v) `Realist' versus `empiricist' interpretation
  • Copenhagen confusion of `preparation' and `measurement'
    • Criticisms of `measurement as preparation'

• Sketch of a `neo-Copenhagen interpretation'
  • Lists of positive and negative features of the `Copenhagen interpretation'
  • Remarks on `positive and negative features of the Copenhagen interpretation'

• Comparing Copenhagen and neo-Copenhagen interpretations

• The Einstein-Podolsky-Rosen (EPR) problem
  • The EPR reasoning
  • Remarks on the EPR reasoning
  • Bohr's answer to EPR
    • Critique of Bohr's answer to EPR
    • The nonlocality conundrum
    • `Bohr's answer to EPR' is not consistent with his `correspondence principle (strong form)'
    • Fundamental difference between EPR and EPR-Bell experiments
  • Einstein's reaction to Bohr's answer
    • Critique of Einstein's reaction
    • Possibility of `additional assumptions (loopholes)'
  • `EPR nonlocality' and experiment
  • Remarks on `EPR nonlocality'
    • Origin of `EPR nonlocality'
    • Dubious role of the `possessed values principle'
  • Role of `strict correlations' in the `EPR experiment'
    • Two possible explanations of `strict correlations'
  • Criticism of the assumption of `nonlocality'
    • `Nonlocality' versus `incompatibility'
    • How `nonlocality' replaced `incompatibility'
    • Return to metaphysics?
    • Generalized observables and EPR-Bell experiments

• The Bell inequality in quantum mechanics
  • The (standard) Bell inequality
  • The BCHS inequality
  • `Bell inequality' as an expression of `classicality'
  • Assumptions allowing derivation of the Bell inequality within quantum mechanics
    • i) Existence of a `quadrivariate probability distribution'
      • Criticism and appraisal of `existence of quadrivariate probability distributions'
    • ii) Compatibility of all (standard) observables that are involved
    • iii) Existence of `quadruples of measurement results'
      • iiia) Quadruples and the `possessed values principle'
      • iiib) Quadruples and `counterfactual definiteness'
      • iiic) Quadruples and `generalized measurements'
    • iv) The assumption of `locality'
      • Early developments
      • Later developments
        • Within the standard formalism
        • Within the generalized formalism
      • Methodological remark on `nonlocality versus (local) Heisenberg disturbance'
    • v) Non-entanglement (separability)
      • `Nonlocality' versus `entanglement'
  • The Bell inequality in the generalized formalism
    • A generalized Aspect measurement
    • Relation to `the standard Aspect measurements'
    • `Bell inequality' and `Heisenberg disturbance'
  • Bell inequality and interpretations of quantum mechanics
    • `Bell inequality' within a `contextualistic-realist interpretation'
    • `Bell inequality' within the `empiricist interpretation'

• Hidden variables or subquantum theories
  • Preamble
  • Bohm's hidden variables theory
    • Critique of Bohm's hidden variables theory
  • Stochastic hidden variables theories
    • Critique of stochastic hidden variables theories
    • The Bell inequality in stochastic hidden variables theories
  • Quantum mechanics as a theory of equilibrium processes

• Some relevant books