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: Green’s 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 | ψ ) = | ψ ) + GV | ψ ) , (1) to solve the one-dimensional problem of tunneling through delta potentials. Take ψ ( z ) = e ikz , and let V ( z ) = gδ ( z ) + gδ ( z − L ) . (2) (a) Express Eq. (1) as an integral equation for ψ ( z ), and then use the delta-functions to perform the integral. It might be helpful to introduce the dimensionless parameter α = 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 ψ ( z ) = ψ ( z ) + integraldisplay dz ′ G ( z,z ′ ) V ( z ′ ) ψ ( z ′ ) . (3) Use V ( z ′ ) = gδ ( z ′ ) + gδ ( z ′ − L ) to handle the integrals, giving: ψ ( z ) = ψ ( z ) + gG ( z, 0) ψ (0) + gG ( z,L ) ψ ( L ) . (4) To find the unknowns, ψ (0) and ψ ( L ), we set first z = 0, and then z = L , giving ψ (0) = ψ (0) + gG (0 , 0) ψ (0) + gG (0 ,L ) ψ ( L ) (5) ψ ( L ) = ψ ( L ) + gG ( L, 0) ψ (0) + gG ( L,L ) ψ ( L ) (6) Solving simultaneously for ψ (0) and ψ ( L ) and taking G ( z,z ′ ) → G ( | z − z ′ | ) gives ψ (0) = 1 + iα (1 − e i 2 kL ) 1 + 2 iα − α 2 (1 − e i 2 kL ) (7) ψ ( L ) = e ikL 1 + 2 iα − α 2 (1 − e i 2 kL ) (8) This gives as the solution: ψ ( z ) = e ikz − iα e ik ( L + | z − L | + e ik | z | ( 1 + iα (1 − e i 2 kL ) ) 1 + 2 iα − α 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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## This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.

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852_2010hw10_Solutions - PHYS852 Quantum Mechanics II...

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