The Time-Dependent Schr¨ odinger Equation - The Need for the Hamiltonian to be Self-Adjoint.pdf

178 Brazilian Journal of Physics, vol. 38, no. 1, March, 2008
The Time-Dependent Schrodinger¨ Equation: The Need for the Hamiltonian to be Self-Adjoint
Vanilse S. Araujo,
Escola de Engenharia Maua, Sao˜ Paulo, Brazil and Faculdade de Engenharia da Fundac¸ao˜ Santo Andre, Sao˜ Paulo, Brazil
F. A. B. Coutinho,
Faculdade de Medicina da Universidade de Sao˜ Paulo, Sao˜ Paulo, 01246-903, Brazil
F. M. Toyama
Department of Information munication Sciences, Kyoto Sangyo University, Kyoto 603-85555 Japan
Received on 20 December, 2007
We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint.
We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of
the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of
some well-known simple problems – the infinite square well, the finite square well, and the harmonic oscillator.
We show that these elementary illustrations can be enriched by using more general boundary conditions, which
are patible with self-adjointness. In particular, we show that a puzzling problem associated with the
Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness.
We e to Stone´s theorem, which is the main topic of this paper, and which is s