852_2010hw10_Solutions

852_2010hw10_Solutions - PHYS852 Quantum Mechanics II,...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 10 Topics covered: Greens function, Lippman-Schwinger Eq., T-matrix, Born Series. 1. T-matrix approach to one-dimensional scattering: In this problem, you will use the Lippman- Schwinger equation | ) = | ) + GV | ) , (1) to solve the one-dimensional problem of tunneling through delta potentials. Take ( z ) = e ikz , and let V ( z ) = g ( z ) + g ( z L ) . (2) (a) Express Eq. (1) as an integral equation for ( z ), and then use the delta-functions to perform the integral. It might be helpful to introduce the dimensionless parameter = Mg planckover2pi1 2 k . To solve for the two unknown constants, generate two equations by evaluating your solution at z = 0, and z = L . Hit with a ( z | from the left, and insert I = integraltext dz | z )( z | after the G to get the integral equation ( z ) = ( z ) + integraldisplay dz G ( z,z ) V ( z ) ( z ) . (3) Use V ( z ) = g ( z ) + g ( z L ) to handle the integrals, giving: ( z ) = ( z ) + gG ( z, 0) (0) + gG ( z,L ) ( L ) . (4) To find the unknowns, (0) and ( L ), we set first z = 0, and then z = L , giving (0) = (0) + gG (0 , 0) (0) + gG (0 ,L ) ( L ) (5) ( L ) = ( L ) + gG ( L, 0) (0) + gG ( L,L ) ( L ) (6) Solving simultaneously for (0) and ( L ) and taking G ( z,z ) G ( | z z | ) gives (0) = 1 + i (1 e i 2 kL ) 1 + 2 i 2 (1 e i 2 kL ) (7) ( L ) = e ikL 1 + 2 i 2 (1 e i 2 kL ) (8) This gives as the solution: ( z ) = e ikz i e ik ( L + | z L | + e ik | z | ( 1 + i (1 e i 2 kL ) ) 1 + 2 i 2 (1 e i 2 kL ) . (9) (b) Compute the transmission probability T = | t | 2 , with t defined via lim z...
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852_2010hw10_Solutions - PHYS852 Quantum Mechanics II,...

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