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Understanding Finite Potential Wells and Barriers in Quantum Mechanics

This lecture covers key concepts in quantum mechanics, focusing on finite potential wells and barriers. Professor Virginia Ayres discusses electron energy states and tunneling phenomena, illustrated with mathematical expressions and visual aids. Key topics include the behavior of electrons when energy is greater or less than potential barriers, transport over barrier regions, and the implications of the Kronig-Penney model in energy band theory. Ideal for students in electrical and computer engineering looking to deepen their understanding of quantum mechanics principles.

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Understanding Finite Potential Wells and Barriers in Quantum Mechanics

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  1. ECE 874:Physical Electronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

  2. Lecture 13, 11 Oct 12 VM Ayres, ECE874, F12

  3. Finite Potential Well: (eV) Electron energy: E > U0 Electron energy: E < U0 (nm) Regions: -∞ to 0 0 to a a to +∞ VM Ayres, ECE874, F12

  4. Last section of Chp. 02 is about the Finite Barrier: VM Ayres, ECE874, F12

  5. VM Ayres, ECE874, F12

  6. A last look at the finite well, for E > U0 too: VM Ayres, ECE874, F12

  7. Finite barrier Anderson, Modern Physics andQuantum Mechanics VM Ayres, ECE874, F12

  8. E > Anderson V0  Pierret U0 VM Ayres, ECE874, F12

  9. E > Anderson V0  Pierret U0 VM Ayres, ECE874, F12

  10. E < Anderson V0  Pierret U0 This is the expression for T that Pr. 2.8 is referring to. VM Ayres, ECE874, F12

  11. Which situation is this: to start? When part (a) is finished? cosh sinh VM Ayres, ECE874, F12

  12. To start: situation is: tunnelling through the barrier cosh sinh VM Ayres, ECE874, F12

  13. When part (a) is finished, situation being described is: transport “over” the barrier region, by using Pr. 2.9’s mathematical manipulations Starting description: E < U0 Finish description for: E > U0 VM Ayres, ECE874, F12

  14. Which situation is this? VM Ayres, ECE874, F12

  15. Transport “over” the barrier region: E > U0 with transmission coefficient T given by: VM Ayres, ECE874, F12

  16. Chapter 03: Energy band theory VM Ayres, ECE874, F12

  17. e- VM Ayres, ECE874, F12

  18. Describe e- as a wave: Next Unit cell VM Ayres, ECE874, F12

  19. e- described as a wave fitting into a periodic U0 situation.What happens? The Block theorem is the end result of boundary condition matching over multiple Unit cells. Result is: Only a phase shift when you get back to a repeat situation. The repeat situation is not the lattice constant unless you are moving in <100> direction. Variable “a” = the distance between atomic cores in a particulates transport direction. VM Ayres, ECE874, F12

  20. Another useful way to describe the same wave function for e-: This emphasizes that the e- is described by a travelling wave expikx that is being modulated by a repetitive environment. VM Ayres, ECE874, F12

  21. The two descriptions are equivalent. Equation (3.3) p. 54 VM Ayres, ECE874, F12

  22. Next Unit cell VM Ayres, ECE874, F12

  23. Kronig-Penney model: approximate the real U(x) due to a row of atomic cores (top) by a series of wells and finite barriers (bottom). VM Ayres, ECE874, F12

  24. Kronig-Penney model; VM Ayres, ECE874, F12

  25. Kronig-Penney model allowed energy levels: where LHS = RHS VM Ayres, ECE874, F12

  26. Graphical solution of 2.18b: VM Ayres, ECE874, F12

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