“The famous thought experiment in which Heisenberg imagined measuring the position of an electron using a gamma-ray microscope, finally leads to the concept of a minimum uncertainty product (of the position and momentum uncertainties of the electron under observation) of the order of the Planck’s constant.”
“Though Heisenberg asserted that, ‘this relation is a straight forward mathematical consequence of the quantum mechanical commutation rule for the position and the corresponding momentum operators pq – qp = –ih’, he actually derived the relation using a semi-quantitative definition of imprecision/indeterminacy in the position coordinate q and the corresponding momentum p in terms of ‘spreading’ of the Gaussian ‘probability–amplitude packet’ of a microparticle (like electron) and dubbed it ‘a slight generalization’ of the Dirac–Jordan formulation.” “it should be noted that Heisenberg’s view on time was both ambiguous and contradictory.”
“According to Heisenberg1, the uncertainty relations created ‘room’ (1927, p. 180) or ‘freedom’ (1931, p. 43) for the introduction of some non-classical mode of description of experimental data. Although Bohr accepted the conclusions of the paper, he disagreed with Heisenberg’s conception of indeterminacy as a limitation of the applicability of classical notions.”
“For Bohr, the central idea was wave–particle duality and in spite of the fact that the views on QM of the two founding fathers, Heisenberg and Bohr, are often clubbed as the ‘Copenhagen interpretation’, there is considerable difference between their views on uncertainty relation, wave–particle duality and Bohr’s complementarity principle (BCP). Bohr pointed out that the uncertainties in the experiment did not exclusively arise from the discontinuities (existence of quantum of action), but also from the fact that the position and the momentum of the electron cannot be simultaneously defined in the microscope experiment” “On the other hand, Bohr has always defended the uncertainty relations against the objections raised by Einstein in his famous thought experiments christened as Einstein’s slit [fenda] and Einstein’s box.”
“There is no doubt that Heisenberg’s notion of uncertainty played an important role in the initial stages of the development of QM and even now it offers in many cases a quick glimpse of some peculiar quantum results. The use of a mixture of classical and quantum concepts is its real strength, because one can get some picture of what may be happening to the individual ‘particles’ and in most cases it offers a semi-classical explanation for some quantum phenomena.”
“In the general formulation of QM, any pair of non-commuting operators is subject to similar lower bounds for the uncertainty products”
“For a classical particle it is always possible to know both position and momentum with finite errors, i.e. both position and momentum density functions must be compact. This is impossible in the statefunction description because band limited functions cannot be spatially compact.”
“For Bohr, the indeterminacy relations are essentially an expression of wave–particle duality in QM, which he further expounded as the ‘complementarity Principle’ in his Como lecture. However, the ‘wave–particle trade-off’ is now expressed in terms of an inequality, known as Englert–Greenberger duality or simply wave–particle duality relation.”
“This was ridiculed in Lev Landau’s joke: <To violate the time–energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!>.”
“this misconception [quebra da lei da termodinâmica clássica] is based on the false axiom that the energy of the universe is an exactly known parameter at all times.” Therefore it is not that the conservation of energy is violated when quantum field theory uses temporary electron–positron pairs in its calculations, but that the energy of quantum systems is not known with enough precision to limit various possibilities.”
“we note that the quantum measurements are generically invasive and measuremental disturbance cannot be calculated accurately. But non-invasive measurements are possible in classical mechanics (CM) and even for invasive measurements, the measuremental disturbance may be calculated and accounted for accurately.”
“Eventually, the recent experimental claims regarding the general failure of the naive error-disturbance and error–error relations, have sparked a stimulating debate concerning the proper mathematical definition of ‘error’ and ‘disturbance’.”
“In view of the absence of a full-fledged theory of time measurements, it may be relevant to point out here that an appropriate time–energy UVUR is yet to be developed.”
FOOL FOOL FOOL: “Any tentative explanation using uncertainty relations provides only naive semi-classical arguments. Nevertheless, it appears that though ‘uncertainty’ and ‘complementarity’ are two independent notions, in some cases they are inextricably related.”
“The first high-precision experimental test for the uncertainty relations came about only in 1969 from Shull’s single-slit neutron diffraction experiment. Later in the 1980s followed the neutron interferometric experiments by Kaiser et al. and Klein et al.”
“Here, the momentum uncertainty is inferred semiclassically from the measured position distribution of the particles at the detection screen using far-field approximation. With this observation, Busch et al. have pointed out that these analyses of the experimental data do not provide model-independent, direct confirmation of the uncertainty relation and discussed its possible model-independent validation.”
“It is finally concluded that, ‘although correct for uncertainties in states, the form of Heisenberg’s precision limit is incorrect if naively applied to measurement’ and that the experiment ‘highlights an important fundamental difference between uncertainties in states and the limitations of measurement in quantum mechanics’.”
“Time indeterminacy was also demonstrated in quantum beats experiment in neutron interferometry by Badurek et al.”
“We conclude this critique with a brief discussion of Popper’s experimental proposal aimed at falsifying uncertainty relations which generated a great deal of interest and controversy. (…) However, following some crucial objections from von Weizsäcker and Einstein, Popper accepted their criticisms and withdrew the proposition.” “Popper’s experiment was realized in 1999 by Kim and Shih using a spontaneous parametric down-conversion (SPDC) photon source. They did not observe an extra spread in the momentum of particle 2 due to particle 1 passing through a narrow slit. In fact, the observed momentum spread was narrower than that contained in the original beam. This observation seemed to imply that Popper was right. However, Kim and Shih asserted that this result does not constitute a violation of the uncertainty principle and observed: ‘Popper and EPR were correct in the prediction of the physical outcomes of their experiments. However, Popper and EPR made the same error by applying the results of two-particle physics to the explanation of the behavior of an individual particle. The two-particle entangled state is not the state of two individual particles. Our experimental result is emphatically NOT a violation of the uncertainty principle which governs the behaviour of an individual quantum’.” “In spite of some basic flaws in the original analysis, Popper’s intuitive recognition of the problem shows great insight. However, it should be pointed out in the same breath that his challenge to the foundation of QM has turned out to be misplaced.”
NOTAS
“Most present-day textbooks emphasize that space and time play fundamentally different roles in quantum mechanics. … time poses no fundamental problem for quantum mechanics. If by space and time one understands the coordinates of a given space and time background, none of these coordinates are operators in quantum mechanics. If, on the other hand, one thinks of position and time as dynamical variables (obeying equations of motion) of a specific physical system situated in space–time, the representation of such variables by quantum mechanical operators is possible.”
Seclusão Anagógica 