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arXiv:quant-ph/0209147 · Physics Letters A

The Gamow functional

Mario Castagnino, Manuel Gadella, R. Id Betán, Roberto Laura

Published Apr 1, 2001• 16 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(01)00194-3

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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.
In this review, we present a rigorous construction of an algebraic method for quantum unstable states, also called Gamow states. A traditional picture associates these states to vectors states called Gamow vectors. However, this has some difficulties. In particular, there is no consistent definition of mean values of observables on Gamow vectors. In this work, we present Gamow states as functionals on algebras in a consistent way. We show that Gamow states are not pure states, in spite of their representation as Gamow vectors. We propose a possible way out to the construction of averages of observables on Gamow states. The formalism is intended to be presented with sufficient mathematical rigor.
For single-particle nonrelativistic quantum mechanics, a Gamow state is an eigenfunction of the Hamiltonian with complex eigenvalue. Gamow states are not normalizable; they depend on time via the usual multiplier exp(-iEt) supplemented by a cutoff at an expanding wavefront. Gamow states have been used to extend nuclear shell models; they are to metastable states as normalizable eigenfunctions are to bound states. In this paper we generalize Gamow states to decays with multiple outgoing particles. We derive the exact form of the expanding wavefront, even for relativistic outgoing particles. In the non-relativistic limit we derive the exact form of the multi-particle Gamow state contribution to the system propagator (Green's function).