14,830 research outputs found
Born-Jordan Quantization and the Uncertainty Principle
The Weyl correspondence and the related Wigner formalism lie at the core of
traditional quantum mechanics. We discuss here an alternative quantization
scheme, whose idea goes back to Born and Jordan, and which has recently been
revived in another context, namely time-frequency analysis. We show that in
particular the uncertainty principle does not enjoy full symplectic covariance
properties in the Born and Jordan scheme, as opposed to what happens in the
Weyl quantization
Quantum Indeterminacy, Polar Duality, and Symplectic Capacities
The notion of polarity between sets, well-known from convex geometry, is a
geometric version of the Fourier transform. We exploit this analogy to propose
a new simple definition of quantum indeterminacy, using what we call
"hbar-polar quantum pairs", which can be viewed as pairs of position-momentum
indeterminacy with minimum spread. The existence of such pairs is guaranteed by
the usual uncertainty principle, but is at the same time more general. We use
recent advances in symplectic topology to show that this quantum indeterminacy
can be measured using a particular symplectic capacity related to action and
which reduces to area in the case of one degree of freedom. We show in addition
that polar quantum pairs are closely related to Hardy's uncertainty principle
about the localization of a function and its Fourier transform.Comment: Revised improved versio
Born--Jordan Quantization and the Equivalence of Matrix and Wave Mechanics
The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's
matrix mechanics on a firm mathematical basis. Born and Jordan showed that if
one wants to ensure energy conservation in Heisenberg's theory it is necessary
and sufficient to quantize observables following a certain ordering rule. One
apparently unnoticed consequence of this fact is that Schr\"odinger's wave
mechanics cannot be equivalent to Heisenberg's more physically motivated matrix
mechanics unless its observables are quantized using this rule, and not the
more symmetric prescription proposed by Weyl in 1926, which has become the
standard procedure in quantum mechanics. This observation confirms the
superiority of Born-Jordan quantization, as already suggested by Kauffmann. We
also show how to explicitly determine the Born--Jordan quantization of
arbitrary classical variables, and discuss the conceptual advantages in using
this quantization scheme. We finally suggest that it might be possible to
determine the correct quantization scheme by using the results of weak
measurement experiments.Comment: Errors corrected; slightly expanded. arXiv admin note: substantial
text overlap with arXiv:1404.677
On the (Non)Equivalence of the Schr\"odinger and Heisenberg Pictures of Quantum Mechanics
The aim of this short Note is to show that the Schr\"odinger and Heisenberg
pictures of quantum mechanics are not equivalent unless one uses a quantization
rule clearly stated by Born and Jordan in their famous 1925 paper. This rule is
sufficient and necessary to ensure energy conservation in Heisenberg's matrix
mechanics. It follows, in particular, that Schr\"odinger and Heisenberg
mechanics are not equivalent if one quantizes observables using the Weyl
prescription.Comment: Funded by Austrian Research grant FWF P20442-N1
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