An efficient method is developed for an iterative solution of the Poisson and Schrödinger equations, which allows systematic studies of the properties of the electron gas in linear deep-etched quantum wires. A much simpler two-dimensional ͑2D͒ approximation is developed that accurately reproduces the results of the 3D calculations. A 2D Thomas-Fermi approximation is then derived, and shown to give a good account of average properties. Further, we prove that an analytic form due to Shikin et al.

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... is a good approximation to the electron density given by the self-consistent methods.