Lecture37 - Physics 344 Foundations of 21 st Century...

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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. Re-read sections 8.5 and 8.6 Final Exam: Monday, December 12 at 10:20am in MATH 175 l x l y l z 3-D Quantum Systems Last lecture we used our experience with the idealized model of an electron metal film system (quanton in a 1-d 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 3-d time-independent Schrdinger equation (SE) determines the electron crystal systems energy eigenvectors and corresponding energy eigenvalues. As in our 1-d examples, this sort of general form of the 3-d 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 two-particle 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 color-center energy eigenvectors and eigenvalues as easily as we did because of a two-fold 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 kinetic-energy 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 F-center 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 University-West Lafayette.

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Lecture37 - Physics 344 Foundations of 21 st Century...

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