1. Electron waveguideWe consider a

1. Electron waveguideWe consider a conducting channel in a 2DEG (an “electron waveguide”), defined by a lateral confining potential V (x), in the presence of a perpendicular magnetic field B (in the z-direction). In the Landau gauge A = (0,Bx, 0) the hamiltonian has the form 公式3.1for a single spin component (cf. Section II.F.1). Because the canonical momentum py along the channel commutes with H, one can diagonalize py and H simultaneously. For each eigenvalue ¯hk of py, the hamiltonian (3.1) has a discrete spectrum of energy eigenvalues En(k), n = 1, 2, . . ., corresponding to eigenfunctions of the form公式3.2In waveguide terminology, the index n labels the modes, and the dependence of the energy (or “frequency”) En(k) on the wave number k is the dispersion relation of the nth mode. A propagating mode at the Fermi level has cutoff frequency En(0) below EF. The wave function (3.2) is the product of a transverse amplitude profile n,k(x) and a longitudinal plane wave eiky. The average velocity vn(k)along the channel in state |n, ki is the expectation value of the y-component of the velocity operator p + eA:公式3.3For a zero magnetic field, the dispersion relation En(k) has the simple form (1.4). The group velocity vn(k) is then simply equal to the velocity ¯hk/m obtained from the canonical momentum. This equality no longer holds in the presence of a magnetic field, because the canonical momentum contains an extra contribution from the vector potential. The dispersion relation in a nonzero magneticfield was derived in Section II.F.1 for a parabolic confining potential V (x) = 12mω20x2. From Eq. (2.59) one calculates a group velocity ¯hk/M that is smaller than ¯hk/m by a factor of 1 + (ωc/ω0)2.