On the definition of the time evolution operator for time-independent Hamiltonians in non-relativistic quantum mechanics

Abstract

The usual definition of the time evolution operator e(-iHt/(h) over bar) = Sigma(infinity)(n-0) 1/h! (-i/(h) over bar Ht)(n), Where H is tha Hamiltonian of the system, as given in almost every book on quantum mechanics, causes problems in some situations. The operators that appear in quantum mechanics are either bounded or unbounded. Unbounded operators are not defined for all the vectors (wave functions) of the Hilbert space of the system; when applied to some states, they give a non-normalizable state. Therefore, if H is an unbounded operator, the definition in terms of the power series expansion does not make sense because it may diverge or result in a non- normalizable wave function. In this article, we explain why this is so and suggest, as an alternative, another definition used by mathematicians. (C) 2017 American Association of Physics Teachers.