Chapter 7: Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State

Detlef Dürr, Robert Grummt, Martin Kolb

Gamow's explanation of the exponential decay law uses complex "eigenvalues" and exponentially growing "eigenfunctions". This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wave functions. Observing that the time evolution of any wave function is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover the presentation can well be used in physics lectures to students.

Dennis Morris

Arno Böhm, Manuel Gadella, G. B. Mainland

Gamow vectors are generalized eigenvectors of the Hamiltonian with complex eigenvalues that describe exponentially decaying (or growing) states. The energy wavefunctions corresponding to Gamow vectors have a pole immediately below (or above) the real axis in the complex energy plane. Although complex energy values were introduced more than half a century ago for the theory of alpha decay, they have become disreputable and have been banished from quantum mechanics textbooks because of mathematical problems and incorrect physical interpretation. Developments in modern mathematics have now provided a mathematical foundation that has led to the correct physical interpretation. In this article, energy eigenvectors with complex eigenvalues are first introduced by explicitly considering a specific simple decaying system. Then, an elementary account of the properties of Gamow vectors is given. The results from the decaying system provide a motivation for the introduction of Gamow vectors and some of the background information needed to understand their properties.

Tim Maudlin

Jonathan Schonfeld

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).