hw4 - 3. The Hamiltonian for a harmonic oscillator is H = 1...

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HOMEWORK # 4 Physics 6572 Friday, 10/3/08; due 10/10/08 Assume the radii of the inner and outer cylinders are a and b, respectively. Also assume the two ends of the cylinder are at z = 0 and z = L . Find the energy eigenvalues for the general case, not just the ground state. Solve the problem with and without the magnetic flux, and compare the results in both cases. You will need to express your results in terms of the nth root k mn of the transcendental equation J m ( k mn b ) N m ( k mn a ) - N m ( k mn b ) J m ( k mn a ) = 0 , where J m ( x ) and N m ( x ) are Bessel functions of the first and second kind, respectively. This is another example that the behavior of a charged particle is influenced by a magnetic field which is not directly experienced by the particle. coherent states before working on this problem.
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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 ( q-ip ) , 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 tq-1 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).

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