Insight into the nature of the wave

Insight into the nature of the wave functions in a magnetic field can be obtained from the correspondence with classical trajectories. These are most easily visualized in a square-well confining potential, as we now discuss (following Ref.266). The position (x, y) of an electron on the circle with center coordinates (X, Y ) can be expressed in terms of its velocity v by公式3.4with ωc ≡ eB/m the cyclotron frequency. The cyclotron radius is (2mE)1/2/eB, with E ≡ 12mv2 the energy of the electron. Both the energy E and the separation X of the orbit center from the center of the channel are constants of the motion. The coordinate Y of the orbit center parallel to the channel walls changes on each specular reflection. One can classify a trajectory as a cyclotron orbit, skipping orbit, or traversing trajectory, depending on whether the trajectory collides with zero, one, or both channel walls. In (X,E) space these three types of trajectories are separated by the two parabolas (X ±W/2)2 = 2mE(eB)−2 (Fig. 39). The quantum mechanical dispersion relation En(k) can be drawn into this classical “phase diagram” because of the correspondence k = −XeB/h.This correspondence exists because both k and X are constants of the motion and it follows from the fact that the component ¯hk along the channel of the canonical momentum p = mv − eA equals公式3.5in the Landau gauge.In Fig. 40 we show En(k) both in weak and in strong magnetic fields, calculated266 for typical parameter values from the Bohr-Sommerfeld quantization rule discussed here. The regions in phase space occupied by classical skipping orbits are shaded. The unshaded regions contain cyclotron orbits (at small E) and traversing trajectories (at larger E) (cf. Fig. 39). The cyclotron orbits correspond quantum mechanically to states in Landau levels. These are the flat portions of the dispersion relation at energies En = (n − 12 )¯hωc. The group velocity (3.3) is zero in a Landau level, as one would expect from the correspondence with a circular orbit. The traversing trajectories correspond to states in magnetoelectric subbands, which interact with both the opposite channel boundaries and have a nonzero group velocity. The skipping orbits correspond to edge states, which interact with a single boundary only. The two sets of edge states (one for each boundary) are disjunct in (k,E) space. Edge states at opposite boundaries move in opposite directions, as is evident from the correspondence with skipping orbits or by inspection of the slope of En(k) in the two shaded regions in Fig. 40.