Dirac_4 - k dx x t x t x p x h h = >=...

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PHY4604 R. D. Field Department of Physics Dirac_4.doc University of Florida Klein-Gordon Equation Time-Independent Equation: Solutions of the form t E i e x t x h m ) ( ) , ( ψ = Ψ ± with ) ( ) ( ) ( ) ( 2 2 2 2 2 2 2 x E x mc dx x d c = + h correspond to states with definite energy ±E. Probability Density: For these “stationary state” solutions the probability density is independent of time as follows: ) ( | ) ( | | ) , ( | ) , ( 2 2 x x t x t x ρ = = Ψ = ± . Free Particle Solutions: A free particle ( i.e. V(x) = 0) with energy E must satisfy () ) ( ) ( 1 ) ( 2 2 2 2 2 2 2 x mc E c dx x d = h and hence ikx ikx Be Ae x + = ) ( where A and B are constants and 2 2 2 2 2 2 ) ( mc E k c = h . There are the following four solutions: ) / ( ) , ( h m Et kx i R Ae t x = Ψ ± and ) / ( ) , ( h Et kx i L Be t x ± ± = Ψ . All four solutions have definite energy 2 2 2 2 2 2 ) ( mc k c E + = h and k dx x t x t x i p R R x h h = Ψ Ψ >= < ± ± ) , ( ) , ( k dx x t x t x i p L L x h h = Ψ Ψ >= < ± ± ) , ( ) , ( 2 2 2 2 * 2 2 ) , ( ) , (
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Unformatted text preview: k dx x t x t x p x h h = &gt;= &lt; Hence 2 2 = &gt; &lt; &gt; &lt; = x x x p p p . Hence we have the following: Solution Momentum Energy Interpretation ) / ( ) , ( h Et kx i R Ae t x + = k p x h = 2 2 2 ) ( ) ( mc cp E x + = Right moving particle ) / ( ) , ( h Et kx i L Ae t x + + = k p x h = 2 2 2 ) ( ) ( mc cp E x + = Left moving particle ) / ( ) , ( h Et kx i R Ae t x + = k p x h = 2 2 2 ) ( ) ( mc cp E x + = Unknown ) / ( ) , ( h Et kx i L Ae t x = k p x h = 2 2 2 ) ( ) ( mc cp E x + = Unknown Not possible!...
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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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