{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 [email protected] TA: Dan Hartzler Office: PHYS 7 [email protected] Grader: Fan Chen Office: PHYS 222 [email protected] 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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 ψ ψ ψ = - + + + Solving the 3-d time-independent Schrödinger equation (SE) determines the electron – crystal system’s 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 system’s components.
Image of page 2
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 ψ ψ ψ - = - + + while the latter means that the wavefunction must vanish on the defect’s 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 π π π ψ = where l x , l y and l z are the lengths of the box’s 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 = + +
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
( 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 = 1, is not degenerate. The is a unique energy eigenvector for this energy eigenvalue, ψ 111 .
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern