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# hw1 - F = 9 4 | 1 aA 1 | √ 3 4 | 1 aA 2 | √ 3 4 | 2 aA...

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HOMEWORK ASSIGNMENT 1 PHYS852 Quantum Mechanics I, Spring 2008 1. Let the states | 1 ) and | 2 ) form an orthonormal basis spanning a 2-d Hilbert space. Let H = a | 1 )( 1 | + b | 2 )( 2 | + c | 1 )( 2 | + d | 2 )( 1 | be the Hamiltonian of the system. What condition on c and d does the requirement that H be hermitian impose? Use this to eliminate d and then find the eigenstates and normalized eigenvectors of the system in terms of the unspecified parameters a , b , and c . 2. Consider a Hilbert space spanned by the basis set {| 1 ) , | 2 ) , | 3 )} . Let the hamiltonian be H = planckover2pi1 ω ( | 1 )( 1 | + 2 | 1 )( 2 | + 2 | 2 )( 1 | + | 2 )( 2 | + 2 | 2 )( 3 | + 2 | 3 )( 2 | + | 3 )( 3 | ) , where ω = 1 . 5 kHz Find the energy eigenvalues and normalized eigenvectors of this Hamiltonian. 3. In the same Hilbert space as problem 2, consider the observable F , described by the operator
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Unformatted text preview: F = 9 4 | 1 aA 1 | + √ 3 4 | 1 aA 2 | + √ 3 4 | 2 aA 1 | + 11 4 | 2 aA 2 | + | 3 aA 3 | . What are the possible results of a measurement of F ? ±or the state | ψ a = 1 √ 14 (2 | 1 a + | 2 a + 3 | 3 a ), what are probabilities associated with each possible result. 4. Evolve the state | ψ a from the problem 3 in time, using the Hamiltonian from problem 2. What are the probabilities for the di²erent outcomes as a function of time? 5. Consider a spin-1 / 2 particle of mass M subject to the potential V = 1 2 Mω 2 X 2 + βX ⊗ S z . Determine the energy eigenfunctions and eigenvalues....
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