Spring 2019 Homework Assignments
Consider the unit vector n = (i + j + k)/√3 where i, j and k are the unit vectors in the x, y and z directions. (a) Construct the spin operator Sn for the component of the spin in the n direction. (b) Construct the rotation operator R(φ) for rotations with respect to the n axis, as a 2x2 matrix. (c) Rotate the | 1/2 1/2 > spin state with respect to the n axis by 2π/3, 4π/3 and 2π. Interpret your results.
Four non-interacting electrons are placed into the energy states of a one dimensional box with infinitely high potential walls such that their total enegry is a minimum. (They are in the ground state of the four-particle system.) Write down their fully antisymmetric wavefunction including the space and spin parts.
A particle of mass m is confined to a spherical cavity with radius R, with zero potential inside and infinite potential outside. (a) Find the wavefunction and the energy of the lowest energy state. Now a spherical object of radius a is placed at the center of the cavity, as shown in the figure, so that the potential in this region is Vo. (b) Find the first order change in the energy of the particle due to this perturbation.
Consider the three-fold degenerate energy states for the first excited state of a particle in a cubical box with a side of length a. An additional field acts on the particle in the box so that the potential energy has an additional term of the form H(1) = V(x) = -qEoxy. Use degenerate perturbation theory to find the first order change in the energy and the corresponding eigenfunctions.
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.