Homework8 - Physics 401 - Homework #8 1) Three-state system...

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Physics 401 - Homework #8 1) Three-state system (three points each). Suppose that we have a physical system for which there are only three states. For example, perhaps the system is a molecule where the atoms can take on three different arrangements. Regardless of how we interpret the physical meaning of the three states, we can write down the equations of quantum mechanics for this system using these states as a basis. As usual, we will assume that the states are orthonormal and complete. In other words, let's assume that any state can be thought of as a superposition of the three base states, and that different states have no overlap. Let's further suppose that the Hamiltonian for this system in our three-state basis is: 3 0 0 0 0 0 0 ˆ E E A A E H Then the time-independent Schroedinger Equation is: 3 2 1 3 2 1 3 0 0 0 0 0 0 a a a E a a a E E A A E n a) Find the three energy eigenvectors and energy eigenvalues (E n ) for this system. To answer this question, you may either use the methods of linear algebra, or you may guess the eigenvectors, show that your guesses are correct, and find the eigenvalues through direct substitution. (By the way, it's a good habit to always plug your eigenvectors back into the original equation to check your work.) b) Write down the fully time-dependent solutions for the three stationary states of this system. In other words, show how each eigenvector found in part (a) evolves in time. c) We have not specified the physical meaning of our three base states, but there is one question that we can answer about them. Are our base states the energy eigenstates? d) Sometimes people use Dirac notation to write down a Hamiltonian. If we label
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Homework8 - Physics 401 - Homework #8 1) Three-state system...

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