PHYCS 498CQM: Homework 5

1. Problem A2 in Thijssen. This problem asks you to carry out the proofs needed to establish the basic properties of the conjugate gradient algorithm.
2. Car-Parrinello simulation of simplest possible model, one quantum state in a two level system.
   0. Solve the problem exactly for the hamiltonian H₁₁ = e₁, H₂₂ = e₂, H₁₂ = H₂₁ = t, and give the wavefunctions psi in terms of the basis functions phi
   psi = c₁ phi₁ + c₂phi₂.
   1. Write down the equations of motion for the two state problem (analogous the problem 9.1 in Thijssen).
   2. Write the explicit Verlet equation.
   3. Derive and the expression for the Lagrange multipliers. Hint: Assume c₁ and c₂ describe a normalized function at time t. Then find the change in the coefficients c₁' and c₂' of the wavefunction at time t + delta t without constraints. Finally, find a quadratic equation the gives the proper normalized c₁ and c₂ at time t + delta t.
   4. Show that if the starting point is c₁ = 1 and c₂=0, then the system oscillates conserving the energy. You may show this analytically or with a computer program.
   5. For small oscillations around the ground state, show that the angular oscillation frequency is the square root of the energy gap (the difference of the eigenvalues) divided by the fictitious mass mu. Show this analytically or with a computer program.