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Unformatted text preview: HOMEWORK ASSIGNMENT 9 PHYS852 Quantum Mechanics II, Spring 2008 1. Use the Lippman-Schwinger equation: | ) = | ) + GV | ) (1) to solve the one-dimensional problem of resonant tunneling through two delta-potentials. Take ( x ) = e ikx and V ( x ) = g [ ( x ) + ( x L )] . (2) a.) Express (1) as an integral equation for ( x ) and use it to find the general solution ( x ) for arbitrary k and g . Project onto the state ( x | to get: ( x ) = ( x ) + ( x | GV | ) . Insert the projector integraltext dx | x )( x | = 1 between G and V and use V | x ) = | x ) V ( x ) to arrive at ( x ) = e ikx + integraldisplay dx G ( x,x ) V ( x ) ( x ) . Insert the definitions of G ( x,x ) and V ( x ) to get ( x ) = e ikx i bracketleftBig e ik | x | (0) + e ik | x L | ( L ) bracketrightBig , where = Mg/ ( planckover2pi1 2 k ). Set x = 0 to find (0) = 1 i bracketleftBig (0) + e ikL ( L ) bracketrightBig . Set x = L to get ( L ) = e ikL i bracketleftBig e ikL (0) + ( L ) bracketrightBig . Solve these two equations for (0) and ( L ) to find (0) = 1 + i (1 e 2 ikL ) 1 + 2 i 2 (1 e 2 ikL ) . ( L ) = e ikL 1 + 2 i 2 (1 e 2 ikL ) . Plugging this into the equation for ( x ) gives ( x ) = e ikx i e ik ( L + | L x | ) + e ik | x | ( 1 + i (1 e 2 ikL ) ) 1 + 2 i 2 (1 e 2 ikL ) . b.) use your result from a.) to compute the transmission probability T = | t | 2 as a function of k . For x > L this becomes ( x ) = e ikx bracketleftbigg 1 i 2 + i (1 e 2 ikL ) 1 + 2 2 (1 e 2 ikL ) bracketrightbigg . 1 Which leads to t = 1 1 + 2 2 (1 e 2 ikL ) . So that the transmission probability is T = | t | 2 = 1 1 + 2( 2 + 4 ) 2( 4 2 )cos(2 kL ) + 4 3 sin(2 kL ) . c.) Consider an infinite square-well V ( x ) = 0 from x = 0 to x = L , and V ( x ) = otherwise. Show that the bound states are of the form n ( x ) sin( k n x ), where k n = n/L . The bound states need to be superpositions of e ikx and e ikx , with n (0) = n ( L ) = 0. This is satisfied by n ( x ) = radicalbig 2 /L sin( n/L ), so that k n = n/L ....
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