On Hermitian and non-Hermitian flux conservation for quantum tunneling decay
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On Hermitian and non-Hermitian flux conservation for quantum tunneling decay Gastón García-Calderón
· Lorea Chaos-Cador
Received: 9 June 2020 / Accepted: 28 August 2020 © Chapman University 2020
Abstract We consider exact expansions of the decaying wave solution to the Schrödinger equation in terms of continuum energy solutions (Hermitian) and resonant states involving transient functions (non-Hermitian) discussed in Ref. (G. García-Calderón et al. Phys Scr T151(T151): 014076 https://doi.org/10.1088/0031-8949/2012/T151/ 014076, 2012), to analyze the conditions that guarantee flux conservation in quantum tunneling decay. For the Hermitian case, we find that flux conservation depends on the difference of two arbitrary energies of the continuum which establishes an intriguing correlation with the continuum wave solutions whereas for the non-Hermitian case, it provides a condition that relates the imaginary part of each complex pole of the outgoing Green’s function of the problem with the corresponding resonant state. Keywords Quantum tunneling decay · Hermitian flux conservation · Non-Hermitian flux conservation 1 Introduction The study of the energy continuum was originated at the beginning of quantum mechanics in connection with decay and scattering problems, and in fact, it is intimately related to fundamental aspects of the formalism as the probabilistic interpretation [2,3] and the phenomenon of tunneling decay [4]. For quantum tunneling decay, one may distinguish Hermitian and non-Hermitian formulations that go back to the early days of quantum mechanics. An exact treatment using the Hermitian formulation, where the decaying wave function is expanded in terms of the continuum wave solutions of the problem, as shown below by (16), provides no physical insight on the time evolution of decay unless a full numerical integration is performed. The non-Hermitian formulation originated in 1928 by Gamow, who imposed outgoing boundary conditions to the solutions of the time-dependent Schrödinger equation for the description of α-decay in nuclei. This approach led to complex energy eigenvalues and to the well-known analytical expression of the exponential decay law exp(−n t/h¯ ), where the decay rate n is given by the imaginary part of the complex energy eigenvalue [5–7]. The corresponding eigenstates, the so-called resonant Gastón García-Calderón (B) Instituto de Física, Universidad Nacional Autónoma de México, 04510 Mexico City, Mexico e-mail: [email protected] Lorea Chaos-Cador Universidad Autónoma de la Ciudad de México, 09709 Mexico City, Mexico e-mail: [email protected]
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states, increase exponentially with distance beyond the interaction potential, and as a consequence, the usual rules regarding normalization, orthogonality and completeness do not hold. Due to these features, the approach by Gamow has been considered by some authors as a phenomenological no fundamental approximation of
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