Uncertainty Principle Involving Mass and Momentum

C. G. DARWIN

B Bagchi, R Ghosh, P Goswami

Abstract We show in this paper that the basic representations of position and momentum in a quantum mechanical system, that are guided by a generalized uncertainty principle and lead to a corresponding one-parameter eigenvalue problem, can be interpreted in terms of an extended Schrödinger equation embodying momentum-dependent mass. Some simple consequences are pointed out.

S. Jalalzadeh, H. Moradpour

We show that implementing a generalized uncertainty principle (GUP) with both minimal length and maximal momentum directly on the reduced phase space of the Schwarzschild black hole (BH) leads to a finite and discrete mass spectrum, a strict upper bound on the BH mass, a bounded entropy, and a fully regulated Hawking temperature. We further construct a GUP-deformed lapse function that preserves the ADM mass and horizon radius while exactly reproducing the GUP temperature through the surface gravity. Using the most massive observed supermassive BHs, we derive the constraint on the GUP parameter, $β\lesssim 10^{-98}$, showing that present astrophysical data already impose robust bounds on minimal length quantum gravity.

B. Kivel

Jan Hilgevoord, Jos Uffink

The uncertainty principle of position and momentum is a mathematically more involved relation than is usually realised. The well-known Heisenberg uncertainty relation between standard deviations is much too weak to express the uncertainty principle. By employing other notions of width, relations are given which do express the uncertainty principle.

Jos Uffink, Jan Hilgevoord