Constructing the wave functions requires choosing a gauge. Consider first the Landau gauge: A =Bx. In...

 

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Constructing the wave functions requires choosing a gauge. Consider first the Landau gauge: A =Bx. In this gauge, the dynamical momenta take the form II = Px = -ihx, IIy=py + eBx=-ihy+ 12 and the lowering operator a takes the form B a = 2 IB 2 (II - II,) = i 1/2 (Vz iVy + 1722). The Landau levels are generated by a and a, as is the case for the harmonic oscillator. (a) For the lowest Landau level we should have a|0) = 0, which will allow us to find the wave function. Show that this equation can be solved by choosing a wave function of the form vo(x, y) =ek(x), and obtain a differential equation for (a). Solve this equation for p(x). (b) Construct the wave functions (x, y) for Landau level index n. What is the quantum number corresponding to the degeneracy of each Landau level? (c) Note the peculiar structure of the Landau level wave functions: they are extended in y but localized in x. In particular, around what position are the Landau orbitals localized in the x direction? That is to say, where is their center? (d) Let us consider a system of size L by Ly, and let us impose periodic boundary conditions. What are the allowed values of ky if we require (x, y + Ly) = (x, y)? Constructing the wave functions requires choosing a gauge. Consider first the Landau gauge: A =Bx. In this gauge, the dynamical momenta take the form II = Px = -ihx, IIy=py + eBx=-ihy+ 12 and the lowering operator a takes the form B a = 2 IB 2 (II - II,) = i 1/2 (Vz iVy + 1722). The Landau levels are generated by a and a, as is the case for the harmonic oscillator. (a) For the lowest Landau level we should have a|0) = 0, which will allow us to find the wave function. Show that this equation can be solved by choosing a wave function of the form vo(x, y) =ek(x), and obtain a differential equation for (a). Solve this equation for p(x). (b) Construct the wave functions (x, y) for Landau level index n. What is the quantum number corresponding to the degeneracy of each Landau level? (c) Note the peculiar structure of the Landau level wave functions: they are extended in y but localized in x. In particular, around what position are the Landau orbitals localized in the x direction? That is to say, where is their center? (d) Let us consider a system of size L by Ly, and let us impose periodic boundary conditions. What are the allowed values of ky if we require (x, y + Ly) = (x, y)?