The space of analytical test functions ξ, rapidly decreasing on the real axis (i.e., Schwartz test functions of the type 𝒮 on the real axis), is used to construct the rigged Hilbert space (RHS) (ξ,ℋ,ξ′). Gamow states (GS) can be defined in RHS starting from Dirac’s formula. It is shown that the expectation value of a self-adjoint operator acting on a GS is real. We have computed exactly the probability of finding a system in a GS and found that it is finite. The validity of recently proposed approximations to calculate the expectation value of self-adjoint operators in a GS is discussed.
The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states
ϕ+ and the out-states (observables)
ψ− of the S-matrix elements
(ψ−,ϕ+)=(ψout,Sϕin) any elements of the HS $\H$,
{ψ−}={ϕ+}(=)˝, the RHS theory chooses the boundary conditions~:
ϕ+∈Φ−⊂⊂˝Φ−×,
ψ−∈Φ+⊂⊂˝Φ+×, where
Φ− (
Φ+) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states
ϕ+ and observables
ψ− provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets
∣E±∈Φ∓× (solutions of the Lippmann-Schwinger equation with
±iε respectively) and the Gamow vectors
∣ER−iΓ/2±∈Φ∓×. The Gamow vectors
∣ER−iΓ/2− have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.
Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.
There is compelling evidence that, when a continuous spectrum is present, the natural mathematical setting for quantum mechanics is the rigged Hilbert space rather than just the Hilbert space. In particular, Dirac's bra-ket formalism is fully implemented by the rigged Hilbert space rather than just by the Hilbert space. In this paper, we provide a pedestrian introduction to the role the rigged Hilbert space plays in quantum mechanics, by way of a simple, exactly solvable example. The procedure will be constructive and based on a recent publication. We also provide a thorough discussion on the physical significance of the rigged Hilbert space.