09a - Lecture 09 Recall from last time that for a Hilbert...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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:....
View Full Document

This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.

Page1 / 6

09a - Lecture 09 Recall from last time that for a Hilbert...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online