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Unformatted text preview: Chem 120A Particle in a Potential Well 02/10/06 Spring 2006 Lecture 11 READING: QCS Chapter 2 and Chapter 4.2; FL3 Ch. 20, sections 1, 2, 4, and 5 We will now begin to solve the time-independent Schrodinger Equation for a variety of simple problems. Remember that the Hamiltonian gives energy as the eignenvalue when it acts on a eigenstate. H = E We will assume we know an orthonormal basis n with n = 1...N and m n = mn . We can then expand our state in that basis. = n n n H n n n = E n n n Taking the inner product m H n n n = E n m n n m H n n = E m n n This is an eigenvalue Matrix equation 1 H 1 ..... 1 H N . . . . . . . . . . N H 1 ..... N H N 1 . . . . . N = E 1 . . . . . N We first diagonalize the basis matrix and find the energy eigenvalues. We can then plug the eigenvalues in to get the eigenvectors. So in general if we can pick the right basis, we can solve for the eigenvalues (energies) and eigenvectors (states) using the Matrix equation above. We will deal with such problems later. However, first, we will look at a simpler system where our basis is composed a position eigenspace in one dimension (position representation). Position is continuous, so instead of summing over all basis states in our expansion, we will take the integral....
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