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Unformatted text preview: Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TA: Dan Hartzler Office: PHYS 7 dhartzle@purdue.edu Grader: Fan Chen Office: PHYS 222 chen926@purdue.edu Office Hours: If you have questions, just email us to make an appointment. We enjoy talking about physics! Help Session: Thursdays 2:00 4:00 in PHYS 154 Reading: Chapter 9 in the text. Reread sections 8.5 and 8.6 Final Exam: Monday, December 12 at 10:20am in MATH 175 l x l y l z 3D Quantum Systems Last lecture we used our experience with the idealized model of an electron metal film system (quanton in a 1d box) to construct a similarly idealized model of an electron bound within an ionic crystal defect. 2 2 2 2 2 2 2 ( , , ) ( , , ) ( , , ) ( , , ) 2 E x y z x y z V x y z x y z m x y z =  + + + Solving the 3d timeindependent Schrdinger equation (SE) determines the electron crystal systems energy eigenvectors and corresponding energy eigenvalues. As in our 1d examples, this sort of general form of the 3d SE encompasses the cases of models of single particles bound to macroscopic objects, like a crystal with a defect, as well as to the internal states of twoparticle systems, like a vibrating diatomic molecule. What distinguishes one of these models from another is the form of the potential energy function that defines the interaction between a systems components. l x l y l z We could determine approximate colorcenter energy eigenvectors and eigenvalues as easily as we did because of a twofold idealization we made regarding the interaction between the electron and the crystal: 1. We approximated the potential energy function as having a constant negative value, V , within the defect. 2. We approximated the electron as being so strongly bound within the defect that tunneling by the electron into the forbidden regions outside the box is negligible. x y z This former idealization means that the electron moves freely within the defect, 2 2 2 2 2 2 2 ( ) ( , , ) ( , , ) ( , , ) 2 E V x y z K x y z x y z m x y z  =  + + while the latter means that the wavefunction must vanish on the defects surfaces, 8 ( , , ) sin sin sin nlm x y z x y z n x l y m z x y z l l l l l l = where l x , l y and l z are the lengths of the boxs sides and where n , l and m are positive integers. The corresponding kineticenergy eigenvalues are 2 2 2 2 2 2 2 2 2 8 8 8 nlm x y z n h l h m h K ml ml ml = + + ( 29 2 2 2 2 2 2 2 2 2 2 2 2 2 2 8 8 8 8 nlm x y z n m l h n h l h m h K ml ml ml md + + = + + = l x l y l z In the case of a cubical Fcenter defect, l x = l y = l z = d , we encountered our first example of the phenomenon of degeneracy, The ground (lowest energy) state, n = m = l =...
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This note was uploaded on 12/09/2011 for the course PHYS 344 taught by Professor Garfinkel during the Fall '08 term at Purdue UniversityWest Lafayette.
 Fall '08
 Garfinkel
 Physics

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