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Unformatted text preview: Lecture 09 Recall from last time that for a Hilbert space, H , a state is a unit vector H and an observable is a self-adjoint operator L : H H . If L is measured when the system is in state then the value (an eigenvalue of L ) is obtained with probability |h u | i| 2 where u is the unit eigenvector with Lu = u . Since we are assuming the eigenvectors of L form an orthonormal basis for H , we can write = h u | i u . Then the probability that L = when the system is in state is given by P ( L = ; ) = |h u | i| 2 . Note that |h u | i| 2 = k k 2 Parseval = 1. The expected value of L in is given by h | L i . Dynamics : i ~ = H where = d d time and H is the quantum Hamiltonian corresponding to the classical Hamiltonian h . By corresponding to we mean: first express the classical Hamiltonian h (total energy, potential + kinetic) in terms of position, q , and momentum, p ; then replace q by the operator X = t times and p by the operator Y =- i ~ d dt . We will use the convention that ~ = 1. Example 1. Harmonic Oscillator Let a mass, m , be attached to a spring of stiffness, k . We define q to be the 1 displacement of the mass from equilibrium and v to be its velocity; then the momentum is p = mv . Hence, the classical Hamiltonian is given by:....
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