PHYS4115 Quantum MechanicsII Fall 2014 Assignment 5: Variational Principle and Time-Dependent Perturbation Theory The due date for this assignment is Monday 1 December, 2014. 1. Variational Priciple applied to the 1D Harmonic Oscillator: Let’s pretend h2 d2 that we are unable to solve H = 2m + 12 kx2 , and will try using the variational dx2 priciple to estimate the energy of the second excited state. (a) From class discussion, explain why the test function, 2e (a) = A cos 32a ; a < x < a; and 2e (a) = 0; x < a; x > a, would be a good candidate test wavefunction for the second excited state. Brie‡y explain the physical meaning of the variational parameter, a. (b) In part a) A is the normalization constant. By direct integration show that A = (c) Show that E2e (a) = h2 8ma2 (3 )2 + 1 m! 2 6 (3 )2 (3 )2 p1 a 6 (d) Minimize E2e (a) to …nd the best estimate of the second excited state. Compare your answer to the known value of second excited state, and comment brie‡y. 2. Problem 7.16 (see Solution of problem 7.15 done in class) 3. A hydrogen atom in the ground state is placed in a uniform electric …eld (along the ! y axis), E = E exp ( t= ) b j, which is turned on at t = 0. Calculate the probability that the atom is excited to the 2P state at time t . 4. A particle of charge q is in a 1D harmonic oscillator …eld with Hamiltonian h2 d2 H 0 = 2m + 12 m! 2 , with stationary state solutions, jni ; H 0 jni = En jni, dx2 En = h! n + 21 , n = 0; 1; 2::::. At t = 0, a constant electric …eld is turned on, so that the Hamiltonian becomes, H = H 0 qEx exp ( t= ). If the particle is initially in the …rst-excited state n = 1, calculate the probability that it will be in the second-excited state, n = 2,after t . 5. Sudden Perturbation: Radioactive tritium, H 3 (one proton, two neutrons, and one orbiting electron), decays to helium 3 ion, He3+ (two protons, one neutron and one orbiting electron), by emitting one electron (a beta decay) to transform a neutron into a proton: H 3 ! He3+ + e . The electron quickly leaves the system, and can be igonored in the following calculation. The e¤ect of this beta decay is to change the nuclear charge at t = 0 from +e (of H 3 ) to +2e (of He3+ ). If the electron of the H 3 is initially in the ground state, what is the probability that the electron of the He3+ remains in the ground state. If the H 3 ! He3+ + e process occur really slowly (adibatically), what state would the electron of He3+ occupy. 6. Problem 9.11 7. Selection Rule: Consider a particle in a one-dimensional in…nite square well described by the potential V =0 a x a 1 V =1 jxj > a The quantum eigenenergy is En = h2 2 n2 ; 8ma2 n = 1; 2; 3::: The corresponding eigensolution is for n = odd n =a n =0 1=2 cos (n x=2a) for n = even n =a n =0 1=2 sin (n x=2a) a x a jxj > a a x a jxj > a Supose that the system is an eigenstate n . It is then subjected to a time-dependent perturbation H = = V0 x cos !t. Show that transitions are only possible between states n and m , only if n + m is odd. 2