Mathematical Models for Unstable Quantum Systems and Gamow States

Abstract

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.

Non-Kolmogorovian Probabilities and Quantum Technologies

“…The rejection of collapse does not necessarily lead to unitary dynamics. For example, the idea of describing non-unitary dynamics without appealing to collapse was studied extensively by I. Prigogine and the Brussels–Austin school [ 52 , 53 ] (related to this, see also [ 54 ], for a more recent review on quantum unstable systems). However, in the philosophy of physics community, the irreversible approaches are usually not taken into account, favoring alternatives that appeal to unitary evolutions only.…”

Averages of observables on Gamow states

“…The construction relies on some results of Mathematical Analysis that we do not want to mention here, since we do not want to involve the reader with mathematical details that, although important, distract from the objective of this presentation. They will be published in a forthcoming paper [43].…”

Accelerating Quantum Decay by Multiple Tunneling Barriers

“…Perhaps the simplest one-dimensional quantum mechanical model possessing quasi-stationary (resonance) states, decaying via tunneling leakage, is the double rectangular potential barrier model [ Figure 1 a], which was introduced in a famous paper by Gamov to model decay [ 7 ]. When the barrier height is infinite, the system sustains a set of stationary (non-decaying) bound states at some quantized energies; however, when the barrier height is not infinite, some of these states, those with energies close to the bottom of the barriers, become metastable, i.e., they become resonance states (also known as Gamow or Siegert states, or quasi-bound states; see, e.g., [ 9 , 10 , 11 , 12 , 13 , 14 ] and references therein). This means that an initial wave function prepared in a bound state of the infinite barrier approximately maintains its shape but decays in time in a nearly exponential manner through tunneling leakage across the barriers, generating small-amplitude outgoing waves that spread outward the barrier region [ 9 , 10 , 11 ].…”