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assignment7

# assignment7 - Problem Set 07 Note This problem set is due...

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Problem Set 07 Note: This problem set is due Nov 01 before midnight. Please leave it in my mailbox located at the first floor of Rutherford building. 1. A Hilbert space is parametrised by the basis vectors | λ + i i which are eigenstates of an operator A with eigenvalues λ + i . The Hilbert space is finite dimensional with i = 0 , ..., n , and allows a Hamiltonian operator H which satisfies the following commutation relation with A : [ H , A ] = α H (1) where α is an integer with 0 α n . Clearly this means that the basis vectors are not eigenstates of the Hamiltonian. Now at time t = 0 let us construct a state | ψ i = n X i =0 c i | λ + i i (2) with c i time independent constants. Determine the evolution of this state at a time t = t 0 > 0 using the variables specified in the problem. Under what condition is the system completely solvable? 2. If I provide you with a n -dimensional Hilbert space in which an operator A is given by an upper triangular matrix with elements 1. An upper triangular matrix is a n × n

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assignment7 - Problem Set 07 Note This problem set is due...

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