69 research outputs found

    Teaching the hidden symmetry of the Kepler problem in relativistic quantum mechanics - from Pauli to Dirac electron

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    Hidden symmetry in Coulomb interaction is one of the mysterious problems of modern physics. Additional conserved quantities associated with extra symmetry govern wide variety of physics problems, from planetary motion till fine and hyperfine structures of atomic spectra. In this paper we present a simple derivation of hidden symmetry operator in relativistic quantum mechanics for the Dirac equation in the Coulomb field. We established that this operator may be reduced to the one introduced by Johnson and Lippmann. It is worthwhile to notice that this operator was discussed in literature very rarely and so is not known well among physicists and was omitted even in the recent textbooks on relativistic quantum mechanics and/or quantum electrodynamics.Comment: 5 page

    Time Asymmetric Quantum Mechanics

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    The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone--von Neumann theorem, the solutions of the dynamical equations, the Schr\"odinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)). If one requires that scattering resonances of width Γ\Gamma and exponentially decaying states of lifetime τ=Γ\tau=\frac{\hbar}{\Gamma} should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t0t<t_{0}\leq t<\infty, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics

    Mapping of strongly correlated steady-state nonequilibrium to an effective equilibrium

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    By mapping steady-state nonequilibrium to an effective equilibrium, we formulate nonequilibrium problems within an equilibrium picture where we can apply existing equilibrium many-body techniques to steady-state electron transport problems. We study the analytic properties of many-body scattering states, reduce the boundary condition operator in a simple form and prove that this mapping is equivalent to the correct linear-response theory. In an example of infinite-U Anderson impurity model, we approximately solve for the scattering state creation operators, based on which we derive the bias operator Y to construct the nonequilibrium ensemble in the form of the Boltzmann factor exp(-beta(H-Y)). The resulting Hamiltonian is solved by the non-crossing approximation. We obtain the Kondo anomaly conductance at zero bias, inelastic transport via the charge excitation on the quantum dot and significant inelastic current background over a wide range of bias. Finally, we propose a self-consistent algorithm of mapping general steady-state nonequilibrium.Comment: 15 pages, 9 figure

    Irreversible Quantum Mechanics in the Neutral K-System

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    The neutral Kaon system is used to test the quantum theory of resonance scattering and decay phenomena. The two dimensional Lee-Oehme-Yang theory with complex Hamiltonian is obtained by truncating the complex basis vector expansion of the exact theory in Rigged Hilbert space. This can be done for K_1 and K_2 as well as for K_S and K_L, depending upon whether one chooses the (self-adjoint, semi-bounded) Hamiltonian as commuting or non-commuting with CP. As an unexpected curiosity one can show that the exact theory (without truncation) predicts long-time 2 pion decays of the neutral Kaon system even if the Hamiltonian conserves CP.Comment: 36 pages, 1 PostScript figure include

    On the Mass and Width of the Z-boson and Other Relativistic Quasistable Particles

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    The ambiguity in the definition for the mass and width of relativistic resonances is discussed, in particular for the case of the Z-boson. This ambiguity can be removed by requiring that a resonance's width Γ\Gamma (defined by a Breit-Wigner lineshape) and lifetime τ\tau (defined by the exponential law) always and exactly fulfill the relation Γ=/τ\Gamma = \hbar/\tau. To justify this one needs relativistic Gamow vectors which in turn define the resonance's mass M_R as the real part of the square root ResR\rm{Re}\sqrt{s_R} of the S-matrix pole position s_R. For the Z-boson this means that MRMZ26MeVM_R \approx M_Z - 26{MeV} and ΓRΓZ1.2MeV\Gamma_R \approx \Gamma_Z-1.2{MeV} where M_Z and ΓZ\Gamma_Z are the values reported in the particle data tables.Comment: 23 page

    Relating the Lorentzian and exponential: Fermi's approximation,the Fourier transform and causality

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    The Fourier transform is often used to connect the Lorentzian energy distribution for resonance scattering to the exponential time dependence for decaying states. However, to apply the Fourier transform, one has to bend the rules of standard quantum mechanics; the Lorentzian energy distribution must be extended to the full real axis <E<-\infty<E<\infty instead of being bounded from below 0E<0\leq E <\infty (``Fermi's approximation''). Then the Fourier transform of the extended Lorentzian becomes the exponential, but only for times t0t\geq 0, a time asymmetry which is in conflict with the unitary group time evolution of standard quantum mechanics. Extending the Fourier transform from distributions to generalized vectors, we are led to Gamow kets, which possess a Lorentzian energy distribution with <E<-\infty<E<\infty and have exponential time evolution for tt0=0t\geq t_0 =0 only. This leads to probability predictions that do not violate causality.Comment: 23 pages, no figures, accepted by Phys. Rev.
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