68111003-Midterm-2-Solved-Problems

68111003-Midterm-2-Solved-Problems - Solved Problems Dear...

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Unformatted text preview: Solved Problems Dear students, Second exam include Postulates Operators (some simple commutator algebra and application of ladder operators and . Quantum numbers and degeneracy Various one dimensional potentials (Harmonic oscillator problem, and the following potentials) Various one dimensional potentials Step Potential (Dashed lines are reflected wave and full lines are transmitted wave) 1. Consider a particle of mass m and energy interacting with the simple square barrier: Where Continuity of the wave function at give: Solution of the equations for and : Reflection and transmission coefficients can be calculated as follows: is the probability current density. Then 1.0 0.5 0.0 0.5 1.0 1.5 2.0 x V Step Potential Reflection and transmission coefficients Where subscripts and stand for transmission, reflection and incident respectively. Since the current is a conserved quantity then We check that For the step potential given in the figure, for the potentials 1, 5 and 9, Hartree, graph of the transmission and reflection coefficients are plotted. Analyze the graphs.(Maximum reflection points, maximum transmission points, critical points etc) (Dashed lines are reflected wave and full lines are transmitted wave) , , 2. Potential Barrier(Dashed Lines are transmitted) Consider a particle of mass m and energy E>0 interacting with the simple potential barrier. Write down equations and analyze the given results. 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 Energy T,R V , 1, L , 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 Energy T,R V , 5, L , 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Energy T,R V , 9, L , Wave function of the particle is given by Using continuity equations at x=0 and x=L In order to obtain transmission and reflection coefficients you must eliminate C and D and you must...
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This note was uploaded on 01/12/2012 for the course CHEM 133 taught by Professor Staff during the Spring '08 term at UCSD.

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68111003-Midterm-2-Solved-Problems - Solved Problems Dear...

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