Phase considerations in the Thomas-WKB theory: Film structures, quantum wells, and quantum wires

Some one-dimensional systems of current interest—quantum wires, quantum wells, and metal films—are discussed. The phase shift due to path shift is explicitly pointed out. The underlying physics is investigated by a combination of the Thomas-Fermi model and the conventional practical WKB method. We find an additional phase in our solutions and find that the potential is generally singular near junctions. Density-functional theory needs to be modified near these junctions. Physically these singularities are due to electron holes in these quantized circuits.