Conditional probabilities in quantum systems which have both initial and final boundary conditions are commonly evaluated using the Aharonov–Bergmann–Lebowitz rule. In this short note, we present a seemingly disturbing paradox that appears when applying the rule to systems with slightly broken degeneracies. In these cases, we encounter a singular limit—the probability “jumps” when going from perfect degeneracy to negligibly broken one. We trace the origin of the paradox and solve it from both traditional and modern perspectives in order to highlight the physics behind it: the necessity to take into account the finite resolution of the measuring device. As a practical example, we study the application of the rule to the Zeeman effect. The analysis presented here may stress the general need to first consider the governing physical principles before heading to the mathematical formalism, in particular, when exploring puzzling quantum phenomena.
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August 2019
PAPERS|
August 01 2019
Why physical understanding should precede the mathematical formalism—Conditional quantum probabilities as a case-study
Yakir Aharonov;
Yakir Aharonov
a)
Institute for Quantum Studies, Chapman University
, Orange, California 92866; Schmid College of Science and Technology, Chapman University
, Orange, California 92866; and School of Physics and Astronomy, Tel Aviv University
, Tel Aviv 6997801, Israel
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Eliahu Cohen;
Eliahu Cohen
b)
Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar Ilan University
, Ramat Gan 5290002, Israel
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David H. Oaknin
David H. Oaknin
c)
Rafael Advanced Defense Systems Ltd.
, Haifa 61532, Israel
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a)
Electronic mail: yakir@post.tau.ac.il
b)
Electronic mail: eliahu.cohen@biu.ac.il
c)
Electronic mail: d1306av@gmail.com
Am. J. Phys. 87, 668–673 (2019)
Article history
Received:
February 23 2019
Accepted:
June 04 2019
Citation
Yakir Aharonov, Eliahu Cohen, David H. Oaknin; Why physical understanding should precede the mathematical formalism—Conditional quantum probabilities as a case-study. Am. J. Phys. 1 August 2019; 87 (8): 668–673. https://doi.org/10.1119/1.5115980
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