Chapter1_all - PHY4604 R D Field Classical Mechanics vs...

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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: 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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PHY4604 R. D. Field Department of Physics Chapter1_2.doc University of Florida De Broglie’s Pilot Waves Bohr’s Model of the Hydrogen Atom: One way to arrive at Bohr’s hypothesis is to think of the electron not as a particle but as a standing wave at radius r around the proton. Thus, r n π λ 2 = and π λ 2 n r = with n = 1, 2, 3, … The orbital angular momentum is h n h n p n rp L = = = = π π λ 2 2 , which is Bohr’s hypothesis provided that p = h/ λ ! Pilot Waves: In his doctoral dissertation (1924) Louis De Broglie suggested that if waves can act like particles ( i.e. the photon), why not particles acting like waves? He called these waves “pilot waves” (wave of what?) and assigned them the following wavelength and frequency: h E f p h = = λ or k p E h h = = ω where k = 2 π / λ and ω = 2 π f . Phase Velocity of a Plane Wave: For a traveling plane ( i.e. monochromatic) wave given by ) sin( ) , ( t kx A t x ω = Ψ the position of the n th node ( i.e. zeros) is given by π ω n t kx n = and the speed of the n th node along the x-axis is f k dt dx v n phase λ ω = = = Phase Velocity of a Plane Pilot Wave: For a plane De Broglie pilot wave we get c cp c m cp c cp E p E k v phase 2 2 0 2 ) ( ) ( + = = = = ω , which is greater than c for particles with non-zero rest mass!
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