Dirac_2 - Et x p i R x Ae t x − = Ψ corresponds to a...

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PHY4604 R. D. Field Department of Physics Dirac_2.doc University of Florida Schrödinger’s Equation (Non-Relativistic) Time-Independent Equation: Solutions of the form t E i e x t x h = Ψ ) ( ) , ( ψ with ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V dx x d m ψ ψ ψ = + h correspond to state 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 ρ = = Ψ = . Free Particle Solutions: A free particle ( i.e. V(x) = 0) with energy E must satisfy ) ( ) ( 2 2 2 2 x E dx x d m ψ ψ = h and hence ikx ikx Be Ae x + = ) ( ψ where A and B are constants and E m k = ) 2 /( 2 2 h . There are the following two solutions: ) / ( ) , ( h Et kx i R Ae t x = Ψ and ) / ( ) , ( h Et kx i L Be t x + = Ψ . Both solutions have definite energy 0 ) 2 /( 2 2 > = m k 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 ) , ( ) , ( k dx x t x t x p x h h = Ψ Ψ >= < Hence 0 2 2 = > < > < = Δ x x x
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Unformatted text preview: ) , ( Et x p i R x Ae t x − = Ψ corresponds to a particle with definite energy ) 2 /( 2 > = m p E x and definite momentum k p x h = ( right moving particle ) and h / ) ( ) , ( Et x p i L x Be t x + − = Ψ corresponds to a particle with definite energy ) 2 /( 2 > = m p E x and definite momentum k p x h − = − ( left moving particle ). Probability Flux: The probability flux is given by ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ Ψ ∂ Ψ − ∂ Ψ ∂ Ψ ≡ x t x t x x t x t x im t x j ) , ( ) , ( ) , ( ) , ( 2 ) , ( * * h and for the free particle we see that ) ( | | | | ) , ( 2 2 x v A m p A m k t x j x x = = = h ....
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