The Importance of Boundary Conditions in Quantum Mechanics

We discuss the role of boundary conditions in determining the physical content of the solutions of the Schrödinger equation. We study the standing-wave, the "in," the "out," and the purely outgoing boundary conditions. As well, we rephrase Feynman's +I varepsilon prescription as a time-asymmetric, causal boundary condition, and discuss the connection of Feynman's +I varepsilon prescription with the arrow of time of Quantum Electrodynamics. A parallel of this arrow of time with that of Classical Electrodynamics is made. We conclude that, in general, the time evolution of a closed quantum system has indeed an arrow of time built into the propagators.