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Unformatted text preview: 3. The Hamiltonian for a harmonic oscillator is H = 1 2 ( p 2 + q 2 ) , where p and q satisfy [ q,p ] = i . The creation and destruction operators are dened by a = 1 2 ( qip ) , a = 1 2 ( q + ip ) . Now consider the construction F ( t,q ) = h q  e a t  i , where  i is the ground state of H . Obviously (prove it!) F ( t,q ) = X n t n n ! n ( q ) , where n ( q ) is the eigenfunction with energy eigenvalue n + 1 / 2. Show that F ( t,q ) can be directly evaluated from its denition to be F ( t,q ) = 1 1 / 4 exp 1 2 q 2 + 2 tq1 2 t 2 , and therefore F ( t,q ) is a generating function for all n ( q ). Find the eigenfunctions for n = 1 and n = 2. Compare your answers with the well known results....
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This note was uploaded on 03/27/2011 for the course PHYS 6572 at Cornell University (Engineering School).
 '08
 ELSER, V
 mechanics, Work

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