hw1solutions - HOMEWORK ASSIGNMENT 1 PHYS852 Quantum...

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Unformatted text preview: 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 eigenvalues and normalized eigenvectors of the system in terms of the unspecified parameters a , b , and c . Answer: H = H requires d = c , and also a = a and b = b . The leads to the eigenvalue equation ( a )( b ) | c | 2 = 2 ( a + b ) + ab | c | 2 = 0. The solutions are 1 = 1 2 bracketleftBig a + b radicalbig ( b a ) 2 + 4 | c | 2 bracketrightBig , 2 = 1 2 bracketleftBig a + b + radicalbig ( b a ) 2 + 4 | c | 2 bracketrightBig . with the normalized eigenvectors | 1 ) = ( b 1 ) | 1 ) c | 2 ) radicalbig ( b 1 ) 2 + | c | 2 | 2 ) = c | 1 ) + ( 2 a ) | 2 ) radicalbig ( 2 a ) 2 + | c | 2 , 2. Consider a Hilbert space spanned by the basis set {| 1 ) , | 2 ) , | 3 )} . Let the hamiltonian be H = planckover2pi1 ( | 1 )( 1 | + 2 | 1 )( 2 | + 2 | 2 )( 1 | + | 2 )( 2 | + 2 | 2 )( 3 | + 2 | 3 )( 2 | + | 3 )( 3 | ) , where = 1 . 5kHz Find the energy eigenvalues and normalized eigenvectors of this Hamiltonian....
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This note was uploaded on 11/26/2010 for the course PHYSICS PHYS 852 taught by Professor Michaelmoore during the Spring '10 term at Michigan State University.

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hw1solutions - HOMEWORK ASSIGNMENT 1 PHYS852 Quantum...

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