851HW7_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 7 Topics Covered: 1D scattering problems with delta- and/or step-functions, transfer matrix approach to multi-boundary 1D scattering problems, finding bound-states for combinations of delta- and/or step- functions in 1D. Some Key Concepts: Wave-vector, probability current, continuity equation, reflection and transmission amplitudes/probabilities, quantum tunneling, quantum reflection, continuity conditions, transfer matrix. 1. The continuity equation: The probability that a particle of mass m lies on the interval [ a,b ] at time t is P ( t | a,b ) = integraldisplay b a dx | ψ ( x,t ) | 2 (1) Differentiate (1) and use the definition of the probability current, j = i planckover2pi1 2 m ( ψ * d dx ψ ψ d dx ψ * ) , to show that d dt P ( t | a,b ) = j ( a,t ) j ( b,t ) . (2) Next, take the limit as b a 0 of both (1) and (2), and combine the results to derive the continuity equation: d dx j ( x,t ) = d dt ρ ( x,t ).
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