A simple proof of Born’s rule for statistical interpretation of quantum mechanics

Biswaranjan Dikshit

Abstract


The Born’s rule to interpret the square of wave function as the probability to get a specific value in measurement has been accepted as a postulate in foundations of quantum mechanics. Although there have been so many attempts at deriving this rule theoretically using different approaches such as frequency operator approach, many-world theory, Bayesian probability and envariance, literature shows that arguments in each of these methods are circular. In view of absence of a convincing theoretical proof, recently some researchers have carried out experiments to validate the rule up-to maximum possible accuracy using multi-order interference (Sinha et al, Science, 329, 418 [2010]). But, a convincing analytical proof of Born’s rule will make us understand the basic process responsible for exact square dependency of probability on wave function. In this paper, by generalizing the method of calculating probability in common experience into quantum mechanics, we prove the Born’s rule for statistical interpretation of wave function.


Keywords


Quantum mechanics; Born’s rule; Hilbert space; Eigen vectors

Full Text:

DOWNLOAD PDF

References


M. Born, “Quantenmechanik der Stoßvorgange”, Z. Phys., 38, 803-827 (1926)

Urbasi Sinha, Christophe Couteau, Thomas Jennewein, Raymond Laflamme and Gregor Weihs, “Ruling Out Multi-Order Interference in Quantum Mechanics”, Science, 329, 418-421 (2010)

Max Born, “The statistical interpretation of quantum Mechanics”, Nobel Lecture, (1954)

David Deutsch, “Quantum theory of probability and decisions”, Proc. R. Soc. Lond. A, 455, 3129-3137 (1999)

Edward Farhi, Jeffrey Goldstone and Sam Gutmann, “How Probability Arises in Quantum Mechanics”, Annals of physics, 192, 368-382 (1989)

J B Hartle, “Quantum mechanics of Individual systems”, American Journal of Physics, 36 (8), 704-712 (1968)

Carlton M. Caves, Christopher A. Fuchs and Rudiger Schack, “Quantum probabilities as Bayesian probabilities”, Physical Review A, 65, 022305 (2002)

Christopher A. Fuchs and Rudiger Schack, “Quantum-Bayesian coherence”, Reviews of modern physics, 85 (4), 1693-1715 (2013)

Christopher A. Fuchs and Asher Peres, “Quantum Theory Needs No ‘Interpretation’”, Physics Today, 70-71 (March 2000)

Meir Hemmo and Itamar Pitowsky, “Quantum probability and many worlds”, Studies in History and Philosophy of Modern Physics, 38, 333-350 (2007)

David J. Baker, “Measurement outcomes and probability in Everettian quantum mechanics”, Studies in History and Philosophy of Modern Physics, 38, 153-169 (2007)

Adrian Kent, “Against Many-Worlds Interpretations”, Int. J. Mod. Phys. A, 5, 1745-1762 (1990)

Andres Cassinello and Jose Luis Sanchez-Gomez, “On the probabilistic postulate of quantum mechanics”, Foundations of Physics, 26 (10), 1357-1374 (1996)

Carlton M. Caves and Rudiger Schack, “Properties of the frequency operator do not imply the quantum probability postulate”, Annals of Physics, 315, 123–146 (2005)

H. Barnum, C. M. Caves, J. Finkelstein, C. A. Fuchs and R. Schack, “Quantum probability from decision theory?”, Proc. R. Soc. Lond. A, 456, 1175-1182 (2000)

Euan J. Squires, “On an alleged “proof” of the quantum probability law”, Physics letters A, 145 (2-3), 67-68 (1990)

Veiko Palge and Thomas Konrad, “A remark on Fuchs’ Bayesian interpretation of quantum Mechanics”, Studies in History and Philosophy of Modern Physics, 39, 273–287 (2008)

Wojciech Hubert Zurek, “Probabilities from entanglement, Born’s rule from envariance”, Physical Review A, 71, 052105 (2005)

Wojciech Hubert Zurek, “Environment-Assisted Invariance, Entanglement, and Probabilities in Quantum Physics”, Physical review letters, 90 (12), 120404 (2003)

Wojciech Hubert Zurek, “Decoherence, einselection, and the quantum origins of the classical”, Reviews of modern physics, 75, 715-775 (2003)

Ruth E. Kastner, “ ‘Einselection’ of pointer observables: The new H-theorem?”, Studies in History and Philosophy of Modern Physics, 48, 56–58 (2014)

Maximilian Schlosshauer and Arthur Fine, “On Zurek’s Derivation of the Born Rule”, Foundations of Physics, 35 (2), 197-213 (2005)

Ulrich Mohrhoff, “Probabilities from envariance?”, Int. J. Quantum Inform., 02, 221 (2004)


Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

ISSN: 2394-3688

© Science Front Publishers