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Journal of Mathematical Physics

Decaying states in the rigged Hilbert space formulation of quantum mechanics

Arno Böhm

Published May 1, 1980• 42 citations
Publisher

Abstract

Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit–Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.

DOI
10.1063/1.524553

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arXiv:hep-th/0301158•Jan 21, 2003•3 citations
Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit–Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.
After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.
Resonances and decaying states have been associated with first-order poles of the S-matrix and represented by exponentially decaying Gamow kets using the Rigged Hilbert Space formulation of quantum mechanics. In analogy, higher-order S- matrix poles, represented by a mixture of Gamow-Jordan vectors, are associated with higher-order decaying states. The exact Golden Rule for decaying states associated to a higher-order pole of the S-matrix is calculated. These higher-order decaying states operators also obey the exponential decay law, but they have a more complicated linewidth than the Breit-Wigner energy distribution. This linewidth depends on the interaction V. To test whether these higher-order decaying state operators represent possible physical states, we discuss their linewidth for various assumptions about the matrix elements of V.
The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,’’in,’’ and ’’out’’ eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ’’out’’ eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ’’complete’’ sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ’’out’’ eigenvectors. The free, ’’in’’ and ’’out’’ eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential.
In the first part, Gamow vectors from S-matrix poles of higher multiplicity are derived in analogy to the Gamow vectors describing resonances from first-order poles. With these vectors a density operator is constructed that describes resonances associated with higher-order poles that obey an exponential decay law. It turns out that this operator formed by higher-order Gamow vectors has a unique structure. In the second part, the Gamow vectors are used in order to describe decaying and resonance states at a potential barrier of finite width. The exact numerical solutions of the S-matrix poles are presented and the out-going boundary condition for the Gamow vector description and consequences from that are discussed.
It is well known that a state with complex energy cannot be the eigenstate of a self-adjoint operator, such as the Hamiltonian. Resonances, i.e., states with exponentially decaying observables, are not vectors belonging to the conventional Hilbert space. One can describe these resonances in an unusual mathematical formalism based on the so-called rigged Hilbert space (RHS). In the RHS, the states with complex energy are denoted as Gamow vectors (GVs), and they model decay processes. We study the GVs of the reversed harmonic oscillator, and we analytically and numerically investigate the unstable evolution of wave packets. We introduce the background function to study initial data that are not composed only by a summation of GVs, and we analyze different wave packets belonging to specific function spaces. Our work furnishes support for the idea that irreversible wave propagation can be investigated using rigged-Hilbert-space quantum mechanics and provides insight for the experimental investigation of irreversible dynamics.
In this second in a series of four articles we create a mathematical formalism sufficient to represent nontrivial hamiltonian quantum dynamics, including resonances. Some parts of this construction are also mathematically necessary. The specific construction is the transforming of a pair of quantized free oscillators into a resonant system of coupled oscillators by analytic continuation which is performed algebraically by the group of complex symplectic transformations, thereby creating dynamical representations of numerous semi-groups. The quantum free oscillators are the quantum analogue of classical action angle variable solutions for the coupled oscillators and quantum resonances, including Breit-Wigner resonances. Among the exponentially decaying Breit-Wigner resonances represented by Gamow vectors are hamiltonian systems in which energy transfers from one oscillator to the other. There are significant mathematical constraints in order that complex spectra be accommodated in a well defined formalism which represents dynamics, and these may be met by using the commutative real algebra $\C (1,i)$ as the ring of scalars in place of the field of complex numbers. These mathematical constraints compel use of fundamental (spinor) representations rather than UIR's. By including distributional solutions to the Schr{ö}dinger equation, placing us in a rigged Hilbert space, and by using the Hamiltonian as a generator of canonical transformation of the space of states, the Schr{ö}dinger equation is the equation for parallel transport of generalized energy eigenvectors, explicitly establishing the Hamiltonian as the generator of dynamical time translations in this formalism.