When dealing with the classical Hall effect and the quantum Hall effect, the magnetic field B, is often pointed along the z direction perpendicular to the sample plane. Now consider that the magnetic field points towards the x direction, parallel to the electron flow in the 2DEG (i.e. B). The 2DEG is confined with a quantum well potential of V (2) = m* wz = (a) Write the Schrdinger equation for this system using the vector potential A (r) eikteikyy (2) and we = (* remember that H = 2 (p q) + V(z)) - (b) Show that the Schrdinger equation can be expressed in the following form [k k w + 2m* qB m*. kywe m*w (B) where = + + 1 m w 2m* w(B) 2 (0, -zB, 0), plane wave ansatz *w (B)(z + c) V = EV and w (B) = w + w. (c) Sketch the confining potential along the z direction (i.e. E vs z(ky)) for B||| = 0 and B|| #0 (d) What can be said about the effective masses for x and y at B 0 ? (e) What can be said about the energy levels of the new system? (2)