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# PS8 - Physics 3316 Spring 2010 Basics of Quantum Mechanics...

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Physics 3316, Spring 2010 Basics of Quantum Mechanics Problem Set 8 (Due in lecture, March 31, 2010) Required Readings : Griffiths Ch. 3 Key concepts: 1D potentials, Operator algebra 1 Resonant transmission U 0 x=0 x=a E U(x) x Fig. 1: Potential barrier 1.1 Quantum Amplitudes Compute as a function of the incoming wave vector k , the quantum amplitudes ( r ( k ), t ( k ), respectively) for scattering of left-incident particles from the potential of Figure 1. Hints: 1. Your algebra will be simpler if you choose the solutions of the form φ 1 ( x ) = A 1 cos( kx ) + B 1 sin( kx ), φ 2 ( x ) = A 2 cos( k ( x a )) + B 2 sin( k ( x a )), φ 3 ( x ) = A 3 e ik ( x a ) , and then match the boundary conditions first at x = a and then at x = 0. Making similar guesses “centering” all parts of the wave function on the right-most point in their interval, except for the right-most part of the wave function which should be centered on the left-most part of its interval, and then solving the matching conditions in right to left order will make the algebra in scattering problems. This approach is known as the transfer matrix approach and is a general method for rending all physical problems in one-dimension trivial in many advanced fields of physics. 1

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2 Hermitian operators 2 2. You should find t ( k ) = 1 cos( k a ) i 2 ( k k + k k ) sin( k a ) , for the transmission amplitude. 1.2 Probabilities for Transmission and Reflection From your result compute P r and P t
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PS8 - Physics 3316 Spring 2010 Basics of Quantum Mechanics...

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