Non-Commuting Operators, Uncertainty Relations

Pritam Chattopadhyay, Ayan Mitra, Goutam Paul

Won Sang Chung

Kenjiro Yanagi, Kohei Sekikawa

In quantum mechanics it is well known that the Heisenberg-Schrödinger uncertainty relations hold for two non-commutative observables and density operator. Recently Dou and Du (J. Math. Phys. 54:103508, 2013 ; Int. J. Theor. Phys. 53:952-958, 2014 ) obtained several uncertainty relations for two non-commutative non-hermitian observables and density operators. In this paper, we show that their results can be given as corollaries of our non-hermitian extensions of Heisenberg type or Schrödinger type uncertainty relations for the generalized metric adjusted skew information or generalized metric adjusted correlation measures which were obtained in Furuichi and Yanagi (J. Math. Anal. Appl. 388:1147-1156, 2012 ).

Xiao Zheng, Shao‐Qiang Ma, Guo‐Feng Zhang +2 more

We unify the product and the sum form variance-based uncertainties to a Schr\{o}dinger-liked uncertainty relation. Some well-known uncertainties in either product or sum form can be recovered by the Schr\{o}dinger-liked uncertainty, i.e., the product form uncertainty is the Schr\{o}dinger-liked uncertainty relation for Hermitian operators and the sum form uncertainty is actually Schr\{o}dinger-liked uncertainty relation for non-Hermitian operators. Thus the Schr\{o}dinger-liked uncertainty relation provides a unified uncertainty relation. Also, the unified uncertainty reveals some quantum properties that the traditional uncertainty relation cannot do, and indicates that the uncertainty relation for the incompatible observables is bounded not only by the commutation relation between themselves, but also by the quantities related with the other operator, which is named as auxiliary operator. The importance of the unified uncertainty, as well as the concept of the auxiliary operator deduced from it, is demonstrated in explaining and fixing the trivial lower bound appearing in the traditional uncertainty relations.

Peter W. Milonni

Abstract The electromagnetic field is quantized for free space and for dispersive dielectric media. Different quantum states of the field (thermal, laser, squeezed) are discussed and their photon-statistical properties derived. The density matrix operator for the field is introduced along with a detailed discussion of the diagonal representation of the field density operator and its properties. Field commutation relations, correlation functions, and uncertainty relations are derived and discussed. Also discussed are the historically important Bohr-Rosenfeld analysis of field commutation and uncertainty relations; energy-time uncertainty relations; simultaneous (Arthurs-Kelly) measurement of non-commuting observables; and complementarity from the perspective of probability amplitudes and probabilities for distinguishable and indistinguishable processes.