Probabilistic interpretation of resonant states

Abstract

We provide probabilistic interpretation of resonant states. We do this 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 behaviours 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 particles.

Effective Non-Hermitian Hamiltonians for Studying Resonance Statistics in Open Disordered Systems

“…Thus, sin(Re ka) > 0, which holds automatically due to (5). As was originally discussed in [3] (and later extended in [4]), we see that as time goes by, we can maintain the numerical value of the spatial integral of |Ψ n (t)| 2 (the probability) , provided we allow the integration domain to expand at a constant speed to the left, which is nothing but the ballistic velocity of the ejected particle.…”

“…The Schrödinger equation with this boundary condition leads to complex eigenvalues E n which correspond to resonances [1,2]. For a recent lucid discussion of resonances in quantum systems, with particular emphasis on the latter approach, see [3,4].…”

Resonant-state Expansion of the Green’s Function of Open Quantum Systems

“…The spatially diverging wave function is obviously outside the Hilbert space and hence can accommodate a complex eigenvalue. We can also show that the spatial divergence is physically necessary for particle-number conservation in an extended sense [2,3]. When we count the number of particles appropriately, the spatial divergence is cancelled by the temporal decay and thereby the number of particles is conserved.…”

“…Many measurements of the conductance of quantum dots connected to quantum wires have motivated us to carry out a series of recent work [1][2][3][4] on the transmission coefficient of open quantum systems in one dimension. The main purpose of the present article is to review the work.…”

Equivalence of the effective Hamiltonian approach and the Siegert boundary condition for resonant states

“…which is spatially divergent [22,24,[55][56][57][58][59][60][61][62][63][64][65]. The point here is that the wave functions of the resonant, antiresonant and anti-bound states are not normalizable and exist outside the Hilbert space.…”