Exchange effects in a quasi-one-dimensional electron gas

We calculate the electron exchange of a quasi-one-dimensional electron gas in a quantum-well wire of radius ${\mathit{R}}_0$. A two-subband model is considered and the exchange self-energy for the first and second subband is calculated under the assumption that only the lowest subband is partially filled with electrons. Band-bending effects are also discussed. Results for the total energy per electron including kinetic and exchange energy are presented.