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Chapter1_1

# Chapter1_1 - t t x i t x x V x t x m ∂ Ψ ∂ = Ψ ∂ Ψ...

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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 ∂ Ψ ∂ = Ψ + ∂ Ψ ∂ − ) , ( ) , ( ) ( ) , ( 2 2 2 2 h h . where 1 − = i and s J h ⋅ × = = − 34 10 054572 . 1 2 π h . The wave function is a complex function and Schrödinger’s equation is analogous to Newton’s equation. Given suitable initial conditions (typically Ψ (x,0) ), one can solve Schrödinger’s equation for Ψ (x,t) for all future times, just as in classical physics, Newton’s equation determines x(t) for all future times. Schrödinger 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). Newton’s Equation! Schrödinger’s Equation!...
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