Chapter1_12 - emitted and absorbed. 2. Assumes that all...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY4604 R. D. Field Department of Physics Chapter1_12.doc University of Florida Schrödinger’s Equation The Classical Hamiltonian: Classically the energy is the sum of the kinetic energy plus the potential energy as follows ( in one dimension ): ) ( 2 2 x V m p E x + = , and hence corresponding Quantum Mechanical Hamiltonian operator is ) ( 2 2 2 2 x V x m H op + = h and t i H op = h . Time-Dependent Schrödinger Equation: We now operate on the wave function Ψ (x,t) with both forms of H op yielding t t x i t x x V x t x m Ψ = Ψ + Ψ ) , ( ) , ( ) ( ) , ( 2 2 2 2 h h . Time-Independent Schrödinger Equation: Look for solutions of the form h / ) ( ) , ( iEt e x t x = Ψ ψ . Solutions of this form correspond to states with definite energy since H op | Ψ > = E| Ψ > . Substituting Ψ (x,t) into the time dependent equation yields ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V dx x d m = + h . Schrödinger Equation: 1. Ignores the creation and annihilation of particles, but photons may be
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: emitted and absorbed. 2. Assumes that all relevant velocities are much less than the speed of light ( i.e. non-relativistic). Free Particle: A free particle ( i.e. V(x) = 0) with energy E must satisfy ) ( ) ( 2 2 2 2 x E dx x d m = h and hence ikx Ae x = ) ( where A is a constant and E m k = ) 2 /( 2 2 h . The state (x) has k p x h >= < and 2 2 2 k p x h >= < which means p x = 0 ( i.e. no uncertainty in p x ). Thus, ) ( ) , ( t kx i Ae t x = corresponds to a free particle with definite momentum k p x h = and definite energy h = E , but the position of the particle is completely uncertain ( i.e. the particle is equally likely to be anywhere)....
View Full Document

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.

Ask a homework question - tutors are online