Spring 2011 Homework Assignments
Consider a two-level system with energies E1 and E2 so that H0 = |1> E1 <1| + |2> E2 <2| A time dependent potential of the form V(t) = γ exp (-α t) ( |1> <2| + |2> <1| ) acts on the system, starting at time t=0. The constants γ and α are real. It is given that the particle is in state 2 at time t=0. (a) Find the differential equation satisfied by c1(t), the interaction picture projection variable of the system state on state |1> . (b) Show that by transforming to the variable z = q exp(-α t), and setting c1(t) = zν Zν(z) , Zν(z) obeys the differential equation for Bessel functions. Determine the values for the constants q and ν . (c) Which dimensionless quantities enter into the equation? How do you think their values will influence the transition? (d) Write down the equations necessary to determine the contributions of the two independent Bessel function solutions to c1(t) , but do not attempt to solve them. (e) How would one obtain the final transition probability |c1(∞)|2 ?
1. Consider a particle in the ground state of a one dimensional quantum box of width a, with infinitely high walls. At time t=0, a time dependent potential V(x,t) = α x exp(-t/τ) starts to act on the sytem. The quantities α and τ are constants. Use first order perturbation theory to obtain an estimate of finding the particle in the first excited state at t=∞. 2. (This problem is a required assignment for students who have not attended any of the lectures on Tuesday March 22, extra credit for others.) Find the transition rates from the ground state of Hydrogen to all possible states in the second excited energy level when the atom is illuminated with an electromagnetic field. Your results should give the numerical value of this rate when the electric-field peak intensity of the EM field is 1V/m. What is the wavelength of the radiaton?
Do problem 6.1, part b in the textbook for a three dimensional harmonic oscillator.
(An exercise to remember the Green's function method.) The one-dimensional time-independent Schrodinger equation may be written as follows: (ħ2/2m) d2/dx2 u(x) + E u(x) = V(x) u(x) . Now, suppose one can find the solution to the following equation: (ħ2/2m) d2/dx2 g(x,x') + E g(x,x') = δ(x - x') . (1) (a) Show that multiplying both sides of the above equation by V(x') u(x') and integrating with respect to x' leads to the following alternative integral equation for u(x): ∫ g(x,x') V(x') u(x') dx' = u(x) . (b) Find the Fourier transform of equation (1) with respect to the variable x, so that an algebraic equation for G(k,x') is obtained, where G(k,x') = ∫ exp(-ikx) g(x,x') dx / (2 π)1/2 . (c) Solve this equation for G(k,x') and inverse transform it to find g(x,x'). (You will need to add an infinitesimal imaginary part to the energy E in order to evaluate the corresponding contour integral.) Note that the solution allows different forms for x > x' and x < x'. Choose solutions which correspond to "outgoing" waves, i.e. waves that are moving away from x'.
Find the scattering cross section for the spherically symmeric potential V(r) = α / (r2 + r02) where α and r0 are constants.
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.