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c120a_lecture1 - Chem 120A Spring 2007 Elementary Classical...

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Chem 120A Elementary Classical Mechanics and Probability Theory 01/17/07 Spring 2007 Lecture 1 READING: Read Feynmann Lectures in Physics, vol III: Chapter 1(on the web) Review complex numbers! Motion of a single particle in 1D Consider a particle with mass m in 1D (one dimensional) space. At any given time it is at a position x ( t ) and has velocity v ( t ) = dx ( t ) dt . The functions x ( t ) and v ( t ) are obtained from Newton’s second law: m d 2 x ( t ) dt 2 = F ( x ( t )) , (1) or F = ma . The actual form of F depends on the specific circumstances. Due to the fact that Equation 1 is a second-order differential equation in time, if we are given a specific set of initial conditions, x ( t 0 ) and v ( t 0 ) , the functions x ( t ) and v ( t ) are uniquely determined for all t . This is called a ”‘trajectory.”’ x(t 1 ) v(t 1 ) x'(t 1 ) v'(t 1 ) time x(t 2 ) v(t 2 ) x'(t 2 ) v'(t 2 ) The trajectories for different initial conditions are unique; they never cross. Figure 1: An example of trajectories for different initial conditions The energy for this 1D system is given by E ( t ) 1 2 mv 2 ( t )+ U ( x ( t )) , (2) where U ( x ( t )) is defined in terms of F ( x ( t )) : - dU ( x ( t )) dx ( t ) = F ( x ( t )) (3) U potential energy and 1 2 mv 2 ( t ) T kinetic energy. In quantum mechanics, F plays little role, however, U plays a crucial role. Chem 120A, Spring 2007, Lecture 1 1
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Conservation of energy Note: in the following section we will use the ‘’dot” notation for derivatives WRT time, · ≡ d dt We remember from physics that the total energy is a conserved quantity, meaning that E must be constant: dE ( t ) dt = dT dt + dU dt 0 . (4)
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