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Unformatted text preview: Chem. 540, Fall 2011 Instructor: Nancy Makri Computer Assignment 2 Eigenstates of Discrete Hamiltonians The Hamiltonian operator for a two-state system has the form ( 29 1 1 1 2 2 2 1 2 2 1 ˆ H ε φ φ ε φ φ φ φ φ φ = +- Ω + h where 1 2 , φ φ are orthonormal states that form a complete set. (a) Calculate the matrix of this Hamiltonian in the given basis. You may copy the results from Problem 22. (b) Using Mathematica (or any other similar program) calculate the eigenvalues and eigenvectors of the Hamiltonian matrix. (c) It's nicer to work with as few parameters as possible. Set 1 ε = , 2 ε ε ≡ . (Yes, you will lose an entire term in the Hamiltonian, but that's ok!) Again, calculate the eigenvalues and eigenvectors of the Hamiltonian. (d) Substitute values in the Hamiltonian. Set 1 Ω = h and keep this fixed. First, use ε = . What do the eigenvectors look like? Next, try 5 ε = . How do the eigenvectors change? To get the complete picture, vary ε between 0 and 5. What happens to the eigenvalues and the eigenvectors? (e) To investigate this more generally, expand the eigenvalues and eigenvectors from part (c) in the Taylor series in the ratio / ε Ω h (i.e., consider this ratio to be a small parameter) and examine the leading terms of the eigenvectors. What do you observe? What do you observe?...
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