Quadratic polynomially deformed $su(1,1)$ and $su(2)$ algebras are utilised in model Hamiltonians to show how the gravitational system consisting of a black hole, infalling radiation and outgoing (Hawking) radiation can be solved exactly. The models
allow us to study the long-time behaviour of the black hole and its outgoing modes. In particular, we calculate the bipartite entanglement entropies of subsystems consisting of a) infalling plus outgoing modes and b) black hole modes plus the infalling modes,using the Janus-faced nature of the model.The long-time behaviour also gives us glimpses of modifications in the character of Hawking radiation. Lastly, we study the phenomenon of superradiance in our model in analogy with atomic Dicke superradiance.
We analyze $e^{+}e^{-}rightarrow gammagamma$, $e^{-}gamma rightarrow e^{-}gamma$ and $gammagamma rightarrow e^{+}e^{-} $ processes within the Seiberg-Witten expanded noncommutative scenario using polarized beams. With unpolarized beams the leading or
der effects of non commutativity starts from second order in non commutative(NC) parameter i.e. $O(Theta^2)$, while with polarized beams these corrections appear at first order ($O(Theta)$) in cross section. The corrections in Compton case can probe the magnetic component($vec{Theta}_B$) while in Pair production and Pair annihilation probe the electric component($vec{Theta}_E$) of NC parameter. We include the effects of earth rotation in our analysis. This study is done by investigating the effects of non commutativity on different time averaged cross section observables. The results which also depends on the position of the collider, can provide clear and distinct signatures of the model testable at the International Linear Collider(ILC).
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to o
ther coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenbergs equation of motion for the radial component of the momentum for the Hydrogen atom.
