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AIP conference proceedings

On a Class of Hairy Square Barriers and Gamow Vectors

Nicolás Fernández-García

Published Jan 1, 2007• 1 citation
Publisher

Abstract

The second order Darboux‐Gamow transformation is applied to deform square one dimensional barriers in non‐relativistic quantum mechanics. The initial and the new “hairy” potentials have the same transmission probabilities (for the appropriate parameters). In general, new Gamow vectors are constructed as Darboux deformations of the initial ones.

DOI
10.1063/1.2563172

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The second order Darboux‐Gamow transformation is applied to deform square one dimensional barriers in non‐relativistic quantum mechanics. The initial and the new “hairy” potentials have the same transmission probabilities (for the appropriate parameters). In general, new Gamow vectors are constructed as Darboux deformations of the initial ones.
Abstract We develop the Darboux procedure for the case of the two‐level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two‐level system, transforming only one real potential into another real potential. We apply the obtained Darboux transformation to known exact solutions of the two‐level system. Thus, we find three classes of new solutions for the two‐level system and the corresponding new potentials that allow such solutions.
Gamow vectors in non-relativistic quantum mechanics are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues. Like the Dirac kets, they are mathematically well defined in the Rigged Hilbert Space. Gamow kets are derived from the resonance poles of the S-matrix. They have a Breit-Wigner energy distribution, an exponential decay law, and are members of a basis vector expansion whose truncation gives the finite dimensional effective theories with a complex Hamiltonian matrix. They also have an asymmetric time evolution described by a semigroup generated by the Hamiltonian, which expresses a fundamental quantum mechanical arrow of time. These Gamow kets are generalized to relativistic Gamow vectors by extrapolating from the Galilei group to the Poincare group. This leads to semigroup representations of the Poincare group which are characterized by spin j and complex invariant mass square. In these non-unitary representations the Lorentz subgroup is unitarily represented and the four-momenta are "minimally complex" in the sense that the four-velocity is real. The relativistic Gamow vectors have all the properties listed above for the non-relativistic Gamow vectors and are therefore ideally suited to describe relativistic resonances and quasistable particles.