PHY4221: Homework 5Due Oct. 9, 20081. In the 1D infinite square well problem defined in the region|x| ≤L, an extra potentialV=kxis added. Express the Hamiltonian of the system as a form of matrix in the basisdefined by the eigenvectors of the original square well Hamiltonian.2. The Hamiltonian of a system is given byH=αp2+βx,whereαandβare constants.Find and solve the Heisenberg equation of motion of theposition and momentum operators. The initial state is a Gaussian wave packet, centeredat zero speed andx= 0, with a position width given byσ. Find expectation value of theposition and its variance as a function of time.3. (a) Show that1A-B=1A+1AB1A+1AB1AB1A+1AB1AB1AB1A+.....whereAandBare two operators whose inverse exist. Then calculate the expectation valueof 1/(A-B) with respect to the state|0i, given thatB|0i=|1iandB|1i=|0i, and
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