Abstract
After some years of observed radioactivity, Gamow (Gamow (1928)) wrote down a formula in quantum mechanics that was supposed to govern the decay of the nucleus of an atom into a lighter nucleus with the emission of an alpha particle (helium atoms without their electron cloud, i.e. ionized helium). This formula, a Schrödinger equation for a stable particle of energy E, was written with a complex energy (with negative imaginary part), implying that the quantum wave function would go to zero exponentially fast in time, as was oberved for the survival probability of a radioactive nucleus.
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Notes
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Based on the argument that one can stop the evolution at any moment and then proceed as if starting from the new initial conditions, with a result equivalent to letting the system develop undisturbed for the entire time, an essentially Markovian hypothesis.
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We use angular brackets for generalized vectors and round brackets for proper normalized vectors in the Hilbert space. The set \(\{<E|, \phi \}\) is complete in \(\mathcal H\).
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Horwitz, L., Strauss, Y. (2020). Spectral Analysis in Hilbert Space. In: Unstable Systems. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-030-31570-2_1
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DOI: https://doi.org/10.1007/978-3-030-31570-2_1
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