Local States in One-Dimensional Symmetrical Quantum Systems
Abstract Local quantum states, which play an important role in quantum dynamical treatments, are expanded analytically with respect to a basis of eigen functions of a symmetrical Hamiltonian ℋ̂(x) = ℋ̂(- x). Exact local states (ELS) in one-dimensional symmetrical quantum systems are therein defined as quantum states which are local eigenstates of the Hamiltonian ℋ̂(x) on one half space ℝ+ or ℝ- and are identically equal to zero on the other half space. Local properties like the projection operator on one half space can be given in terms of ELS-basis, but it is shown that the energy moments 〈(〈ℋ̂ 〉 - 〈ℋ̂)k〉 with respect to the ELS do not converge. Consequently, if one uses the ELS as quasistationary initial states, as has been done recently by some authors [5], the lifetimes of these states cannot be estimated from time energy uncertainty relation using the second energy moment as an energy uncertainty measure. A harmonic oscillator system and a symmetrical double oscillator are treated as examples.
