1982/06/01 by Cleon E. Dean, S. A. Fulling · 12 citations
Physics and Astronomy · #Advanced Chemical Physics Studies #Amplitude #Bound state #Classical mechanics #Complex plane #Continuous spectrum #Eigenfunction #Eigenvalues and eigenvectors #Electric field #Hamiltonian (control theory) #Mathematical analysis #Mathematical physics #Physics #Plane wave #Quantum electrodynamics #Quantum mechanics #Quantum optics and atomic interactions #Sharpening #Spectroscopy and Quantum Chemical Studies #Wave function
paper · doi:10.1119/1.12818
published in American Journal of Physics 50(6), 540-544 (American Institute of Physics)
openalex publication_date 1982/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/26
A one-dimensional square-well model of an atom in an electric field is solved exactly. This problem is instructive for two reasons: (1) The Hamiltonian has a continuous spectrum, but the eigenfunctions do not reduce asymptotically to plane waves. Therefore the general Titchmarsh–Kodaira theory of eigenfunction expansions is needed to normalize the eigenfunctions properly. The normalization is expressed by a spectral density function, ρ(E). (2) ρ(E) exhibits ’’bumps’’ at values of the energy at which the electron wave function’s amplitude inside the well is particularly large. These are resonant states of the system. The gradual sharpening of the resonances into discrete bound states as the electric field is turned off is demonstrated.