69 research outputs found
Teaching the hidden symmetry of the Kepler problem in relativistic quantum mechanics - from Pauli to Dirac electron
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
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 and exponentially decaying states of lifetime
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 , 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
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
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
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
(defined by a Breit-Wigner lineshape) and lifetime (defined by the
exponential law) always and exactly fulfill the relation .
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 of
the S-matrix pole position s_R. For the Z-boson this means that and where M_Z and
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
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 instead of being bounded from
below (``Fermi's approximation''). Then the Fourier transform
of the extended Lorentzian becomes the exponential, but only for times , 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 and have exponential
time evolution for only. This leads to probability predictions
that do not violate causality.Comment: 23 pages, no figures, accepted by Phys. Rev.
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