2 research outputs found
Inner products of resonance solutions in 1-D quantum barriers
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.Comment: 13 pages, 2 figure
Probabilistic Interpretation of Resonant States
We provide probabilistic interpretation of resonant states. This we do by
showing that the integral of the modulus square of resonance wave functions
(i.e., the conventional norm) over a properly expanding spatial domain is
independent of time, and therefore leads to probability conservation. This is
in contrast with the conventional employment of a bi-orthogonal basis that
precludes probabilistic interpretation, since wave functions of resonant states
diverge exponentially in space. On the other hand, resonant states decay
exponentially in time, because momentum leaks out of the central scattering
area. This momentum leakage is also the reason for the spatial exponential
divergence of resonant state. It is by combining the opposite temporal and
spatial behaviors of resonant states that we arrive at our probabilistic
interpretation of these states. The physical need to normalize resonant wave
functions over an expanding spatial domain arises because particles leak out of
the region which contains the potential range and escape to infinity, and one
has to include them in the total count of particle number.Comment: 11 pages, 5 figures, to appear in Pramana Journal of Physics as an
article in the proceedings of Homi Bhabha Centenary Conference on
Non-Hermitian Hamiltonians in Quantum Physics PHHQP VIII; this version are
with added references as well as some rewording after reviewer's suggestion