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Gamow-Jordan vectors and non-reducible density operators from higher-order S-matrix poles (1997)

Bohm, A. ; Loewe, Mark ; Maxson, S. ; [et al.]

College Park, Md. : American Institute of Physics (AIP)

Journal of Mathematical Physics 38 (1997), S. 6072-6100

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ISSN: 1089-7658

Source: AIP Digital Archive

Topics: Mathematics , Physics

Notes: In analogy to Gamow vectors that are obtained from first-order resonance poles of the S-matrix, one can also define higher-order Gamow vectors which are derived from higher-order poles of the S-matrix. An S-matrix pole of r-th order at zR=ER−iΓ/2 leads to r generalized eigenvectors of order k=0,1,...,r−1, which are also Jordan vectors of degree (k+1) with generalized eigenvalue (ER−iΓ/2). The Gamow-Jordan vectors are elements of a generalized complex eigenvector expansion, whose form suggests the definition of a state operator (density matrix) for the microphysical decaying state of this higher-order pole. This microphysical state is a mixture of non-reducible components. In spite of the fact that the k-th order Gamow-Jordan vectors has the polynomial time-dependence which one always associates with higher-order poles, the microphysical state obeys a purely exponential decay law. © 1997 American Institute of Physics.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1063/1.532203

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Hilbert Space or Gel'fand Triplet. Time-Symmetric or Time-Asymmetric Quantum Mechanics (1999)

Bohm, A. ; Kaldass, H. ; Patuleanu, P.

Springer

International journal of theoretical physics 38 (1999), S. 115-130

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ISSN: 1572-9575

Source: Springer Online Journal Archives 1860-2000

Topics: Physics

Notes: Abstract Intrinsic microphysical irreversibility is thetime asymmetry observed in exponentially decayingstates. It is described by the semigroup generated bythe Hamiltonian H of the quantum physical system, not by the semigroup generated by a Liouvillian Lwhich describes the irreversibility due to the influenceof an external reservoir or measurement apparatus. Thesemigroup time evolution generated by H is impossible in the Hilbert space (HS) theory, which allowsonly time-symmetric boundary conditions and a unitarygroup time evolution. This leads to problems with decayprobabilities in the HS theory. To overcome these and other problems (nonexistence of Dirac kets)caused by the Lebesgue integrals of the HS, one extendsthe HS to a Gel'fand triplet, which contains not onlyDirac kets, but also generalized eigenvectors of the self-adjoint H with complex eigenvalues(ER – iΓ/2) and a Breit-Wignerenergy distribution. These Gamow statesψG have a time-asymmetric exponentialevolution. One can derive the decay probability of the Gamow state into the decay productsdescribed by Λ from the basic formula of quantummechanics ℘(t) = Tr(|ψG ›‹ψG|Λ), which in HS quantum mechanicsis identically zero. From this result one derives the decay rate ℘(t) and all the standard relations between℘(0), Γ, and the lifetimeτR used in the phenomenology of resonancescattering and decay. In the Born approximation oneobtains Dirac's Golden Rule.