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Chapter2_27

# Chapter2_27 - PHY4604 R D Field Quantum Mechanical...

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PHY4604 R. D. Field Department of Physics Chapter2_27.doc University of Florida Quantum Mechanical Tunneling (2) Boundary Conditions: There are four boundary conditions. At x = 0 we have ) 0 ( ) 0 ( 2 1 ψ ψ = which implies D C B A + = + 0 2 0 1 = = = x x dx d dx d ψ ψ which implies D C ikB ikA κ κ = and at x = L we have ) ( ) ( 3 2 L L ψ ψ = which implies ikL L L Fe De Ce + + = + κ κ L x L x dx d dx d = = = 3 2 ψ ψ which implies ikL L L ikFe De Ce + + = κ κ κ κ We now must solve for B , C , D , and F in terms of A . We have four equations and four unknowns: 0 0 0 0 0 0 1 1 1 = + = + + = + + = + + + + + + F ike D e C e B F e D e C e B ikA F D C ikB A F D C B ikL L L ikL L L κ κ κ κ κ κ κ κ I will use determinates to solve these simultaneous linear equations as follows: ( ) ( ) ( ) ( ) ( ) ) cosh( 2 ) sinh( ) ( 2 ) ( 2 ) )( ( ) ( ) ( ) ( ) ( ) ( ) ( 1 0 1 1 1 0 1 1 0 0 0 0 1 1 1 2 2 2 2 2 2 L ik L k e e e ik e e k e ik e ik e e ik e ik e ike ik e ik e e ik e e e e ike ik e
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