{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture37

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

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

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

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

View Full Document
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.
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 = + +

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

View Full Document
( 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 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern