Chapter1_1 - t t x i t x x V x t x m = + ) , ( ) , ( ) ( )...

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PHY4604 R. D. Field Department of Physics Chapter1_1.doc University of Florida Classical Mechanics vs Quantum Mechanics Classical Mechanics: The goal of classical mechanics is to determine the position of a particle at any given time, x(t) . Once we know x(t) then we can compute the velocity v x = dx/dt , the momentum p x = mv x , the kinetic energy T = p x 2 /2m , or any other dynamical variable. Classical Equations of Motion: Newton’s Laws for a particle under the influence of the potential V(x) are as follows: dx x dV dt dp ma F x x x ) ( = = = with dt dx m p x = . To determine x(t) one must solve the Newton’s equation dx x dV dt t x d m ) ( ) ( 2 2 = with the appropriate initial conditions (typically the position and velocity at t = 0). Quantum Mechanics: In Quantum Mechanics the situation is much different. In this case we are looking for the particles wave function Ψ (x,t) which is the solution of Schrödinger’s equation:
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Unformatted text preview: t t x i t x x V x t x m = + ) , ( ) , ( ) ( ) , ( 2 2 2 2 h h . where 1 = i and s J h = = 34 10 054572 . 1 2 h . The wave function is a complex function and Schrdingers equation is analogous to Newtons equation. Given suitable initial conditions (typically (x,0) ), one can solve Schrdingers equation for (x,t) for all future times, just as in classical physics, Newtons equation determines x(t) for all future times. Schrdinger Equation: 1. Ignores the creation and annihilation of particles, but photons may be emitted and absorbed. 2. Assumes that all relevant velocities are much less than the speed of light ( i.e. non-relativistic). Newtons Equation! Schrdingers Equation!...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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