Retrodiction beyond the Heisenberg uncertainty relation

In quantum mechanics, the Heisenberg uncertainty relation presents an ultimate limit to the precision by which one can predict the outcome of position and momentum measurements on a particle. Heisenberg explicitly stated this relation for the prediction of “hypothetical future measurements”, and it does not describe the situation where knowledge is available about the system both earlier and later than the time of the measurement. Here, we study what happens under such circumstances with an atomic ensemble containing 1011 rubidium atoms, initiated nearly in the ground state in the presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, $${\hat{x}}_{\text{A}}$$ x ̂ A and $${\hat{p}}_{\text{A}}$$ p ̂ A that satisfy $$[{\hat{x}}_{\text{A}},{\hat{p}}_{\text{A}}]=i\hslash$$ [ x ̂ A , p ̂ A ] = i ℏ . Quantum non-demolition measurements of $${\hat{p}}_{\text{A}}$$ p ̂ A before and of $${\hat{x}}_{\text{A}}$$ x ̂ A after time t allow precise estimates of both observables at time t. By means of the past quantum state formalism, we demonstrate that outcomes of measurements of both the $${\hat{x}}_{\text{A}}$$ x ̂ A and $${\hat{p}}_{A}$$ p ̂ A observables can be inferred with errors below the standard quantum limit. The capability of assigning precise values to multiple observables and to observe their variation during physical processes may have implications in quantum state estimation and sensing.

De Broglie Waves: Are Electrons Waves or Particles?

In 1924, de Broglie postulated that particles can behave like waves, thus complementing the observation by Einstein in 1905 that light can behave like particles. This wave–particle duality aspect for both particles and waves had a deep impact on the subsequent development of quantum mechanics. Some highly counterintuitive results, like the Heisenberg uncertainty relation and the Bose–Einstein condensation, that were motivated by wave–particle duality are discussed in this chapter. Following de Broglie’s hypothesis, a wave packet description for a particle is described. An analysis of the Heisenberg microscope is presented, thus motivating the Heisenberg uncertainty relation. The Davisson–Germer experiment that showed that electrons can behave like waves and the Compton effect that provided early conclusive evidence that light can behave like particles are also discussed.

On the Semiclassical Approach of the Heisenberg Uncertainty Relation in the Strong Gravitational Field of Static Blackhole

Heisenberg Uncertainty and Equivalence Principle are the fundamental aspect respectively in Quantum Mechanic and General Relativity. Combination of these principles can be stated in the expression of Heisenberg uncertainty relation near the strong gravitational field i.e. pr and Et . While for the weak gravitational field, both relations revert to pr and Et. It means that globally, uncertanty principle does not invariant. This work also shows local stationary observation between two nearby points along the radial direction of blackhole. The result shows that the lower point has larger uncertainty limit than that of the upper point, i.e. . Hence locally, uncertainty principle does not invariant also. Through Equivalence Principle, we can see that gravitation can affect Heisenberg Uncertainty relation. This gives the impact to our’s viewpoint about quantum phenomena in the presence of gravitation. Key words: Heisenberg Uncertainty Principle , Equivalence Principle, and gravitational field

Mathematical Uncertainty Relations and their Generalization for Multiple Incompatible Observables

We show that the famous Heisenberg uncertainty relation for two incompatible observables can be generalized elegantly to the determinant form for N arbitrary observables. To achieve this purpose, we propose a generalization of the Cauchy-Schwarz inequality for two sets of vectors. Simple consequences of the N-ary uncertainty relation are also discussed. Keywords: Generalized uncertainty relation, Generalized uncertainty principle, Generalized Cauchy-Schwarz inequality.