2009/05/28 by Tanwa Arpornthip, Carl M. Bender
Mathematics · Physics and Astronomy · #Basis (linear algebra) #Conduction band #Particle (ecology) #Particle in a box #Potential energy #Quantum #Quantum and electron transport phenomena #Quantum chaos and dynamical systems #Quantum tunnelling #Spectral Theory in Mathematical Physics #Thermal conduction #math-ph #math.MP
paper · pdf · doi:10.1007/s12043-009-0117-5
11 pages, 10 figures
arxiv created 2009/05/28 · openalex publication_date 2009/08/01 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time in which to perform the measurement. This paper considers the possibility that ΔE, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behavior of a classical particle in a one-dimensional periodic potential V(x)=-cos(x) is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitatively different behaviors: (i) The particle may hop from classically-allowed site to nearest-neighbor classically-allowed site in the potential, behaving as if it were a quantum particle in an energy gap and undergoing repeated tunneling processes, or (ii) the particle may behave as a quantum particle in a conduction band and drift at a constant average velocity through the potential as if it were undergoing resonant tunneling. The classical conduction bands for this potential are determined numerically with high precision.