Question

3. A particle with mass m and charge q moves in a uniform magnetic filed of magnitude B that is oriented along the z axis. (a) Neglecting the effects of spin and using the so-called Landau gauge with the vector po- tential given by A = (-By,0,0), show that the Hamiltonian may be written as À = 2m 2 ++øp +29BD2y +(, 2] (1) с (b) Because Pa and Êz commute with Ĥ, the time-independent Schrödinger equation for (x, y, z) is separable, and hence its solution may be written as 4(x, y, z) = ((y) exp(ikzx + ik22), where kr and k, are wavenumbers for motion in the directions of the x and z axes that may be considered as free. Derive an eigenvalue problem for the wavefunction (y) describing the motion in the direction of the y axis. (c) Show that the eigenvalue problem found in part (b) for motion in the direction of the y axis corresponds to a simple harmonic oscillator with frequency w and with an appropriately shifted coordinate, y + yo. Find expressions for the frequency w and the coordinate shift Yo (d) Knowing that the eigenenergies of an oscillator with frequency w are given by ħw (n + 1). with n = 0,1, 2..., find the eigenenergies En(kx) of the Hamiltonian (1) (they are called Landau levels).

Answer 1

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