Sources of quantum waves

Due to the space and time dependence of the wavefunction in the time-dependent Schrodinger equation, different boundary conditions are possible. The equation is usually solved as an 'initial-value-problem', by fixing the value of the wavefunction in all space at a given instant. We compare this standard approach to 'source boundary conditions' that fix the wave at all times in a given region, in particular at a point in one dimension. In contrast to the well known physical interpretation of the initial-value-problem approach, the interpretation of the source approach has remained unclear, since it introduces negative energy components, even for 'free motion', and a time-dependent norm. This work provides physical meaning to the source method by finding the link with equivalent initial-value problems.