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Inner products of resonance solutions in 1D quantum barriers

2010; Institute of Physics; Volume: 43; Issue: 17 Linguagem: Inglês

10.1088/1751-8113/43/17/175301

1751-8121

J. Julve, F. J. de Urríes,

Quantum Mechanics and Non-Hermitian Physics

The properties of a prescription for the inner products of the resonance (Gamow states), scattering (Dirac kets), and bound states for 1-dimensional quantum barriers are worked out. The divergent asypmtotic behaviour of the Gamow states is regularized using a Gaussian convergence factor first introduced by Zel'dovich. With this prescription, most of these states (with discrete complex energies) are found to be orthogonal to each other, to the bound states, and to the Dirac kets, except when they are neighbors, in which case the inner product is divergent. Therefore, as it happens for the continuum scattering states, the norm of the resonant ones remains non-calculable. Thus, they exhibit properties half way between the (continuum real) Dirac-delta orthogonality and the (discrete real) Kronecker-delta orthogonality of the bound states.