Berkeley PHYSICS 137A - Homework (8 pages)

137A Homework 4 Solutions GSI Shannon McCurdy October 2 2006 Problem 2 14 Griffiths A particle is in the ground state of the harmonic oscillator with classical frequency when suddenly the spring constant quadruples so 0 2 without initially changing the wave function of course will now evolve differently because the Hamiltonian has changed The point is to expand our initial wave function x 0 which happens to be a state of definite energy under the 0 0 old Hamiltonian H a a 21 in terms of a linear combination of the new Hamiltonian s H 0 0 a a 1 0 2 energy wave functions n x Griffiths 2 16 X x 0 c n 0n x n a 0 n 0n x p 00 x n where 00 x is given by 00 x m 2 1 m 2 x 2 4 e 2 and the wave functions 0n x are states with definite energy E n0 1 1 E n0 n 0 2 n 3 5 2 2 in terms of the old classical frequency The coefficients c n measure the overlap between x 0 and 0n x We calculate c n using Fourier s Trick Griffiths 2 34 and find Z 0n x x 0 dx cn The probability P n of measuring energy E n0 is given by P n c n 2 Part a What is the probability that a measurement of the energy will still return the value 2 Since E 2 is not one of the allowed energies the probability of measuring 2 is zero 1 Part b What is the probability of getting Since we are interested in the probability of measuring the new ground state energy E 00 we only need to calculate Z c0 00 x x 0 dx c 0 2 1 4 r m Z e 3m x 2 2 dx 2 Therefore P 0 c 0 2 1 4 r m p 1 2 2 p 2 2 0 9428 3 2 s 1 2 2 4 3m r 2 3