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qmhw5 - PHY4221 Homework 5 Due Oct 9 2008 1 In the 1D...

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PHY4221: Homework 5 Due Oct. 9, 2008 1. In the 1D infinite square well problem defined in the region | x | ≤ L , an extra potential V = kx is added. Express the Hamiltonian of the system as a form of matrix in the basis defined by the eigenvectors of the original square well Hamiltonian. 2. The Hamiltonian of a system is given by H = αp 2 + βx, where α and β are constants. Find and solve the Heisenberg equation of motion of the position and momentum operators. The initial state is a Gaussian wave packet, centered at zero speed and x = 0, with a position width given by σ . Find expectation value of the position and its variance as a function of time. 3. (a) Show that 1 A - B = 1 A + 1 A B 1 A + 1 A B 1 A B 1 A + 1 A B 1 A B 1 A B 1 A + ..... where A and B are two operators whose inverse exist. Then calculate the expectation value of 1 / ( A - B ) with respect to the state | 0 i , given that B | 0 i = | 1 i and B | 1 i = | 0 i , and
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