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Nanophotonics

Nanophotonics. Atilla Ozgur Cakmak , PhD. Unit 1. Lecture 6: Electron in complex potentials-Part2. Outline. Schrödinger Equation (SE) in Spherical Coordinates The Radial Equation. A couple of words….

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Nanophotonics

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  1. Nanophotonics AtillaOzgurCakmak, PhD

  2. Unit 1 Lecture 6: Electron in complex potentials-Part2

  3. Outline • Schrödinger Equation (SE) in Spherical Coordinates • The Radial Equation

  4. A couple of words… We have covered the most fundamental scenarios in the previous lecture and the electron confinement due to these potentials. Now, we can continue with more advanced potentials. This is the second part of the discussion that was initiated before. Suggested reading: David J. Griffiths, Introduction to Quantum Mechanics, 4th Chapter.

  5. SE Spherical Coordinates

  6. SE Spherical Coordinates

  7. SE Spherical Coordinates

  8. SE Spherical Coordinates(problem) • Find the fifth Legendre polynomial. a) P5=(63x5-70x3+15x)/8 b) P5=(64x5-72x3+16x)/8 c) P5=(64x4-72x2+16x)/8 d) P5=(63x4-70x2+15x)/8

  9. SE Spherical Coordinates(solution) • Find the fifth Legendre polynomial.

  10. SE Spherical Coordinates(problem) • Which one of the following is not a second Legendre function? a) 3cosϴ b)3sin2ϴ c) 3/2sin2ϴ d) ½(3cos2ϴ-1)

  11. SE Spherical Coordinates(solution) • Which one of the following is not a second Legendre function?

  12. SE Spherical Coordinates

  13. SE Spherical Coordinates(problem) • Which one is Y3-3? a) (35/(64π))1/2sin3ϴe-3iφ b)-(35/(64π))1/2sin3ϴe3iφ c) (5/(16π))1/2(3cos2ϴ-1) d) -(5/(16π))1/2(3cos2ϴ-1)

  14. SE Spherical Coordinates(solution) • Which one is Y3-3?

  15. The radial equation

  16. The radial equation

  17. The radial equation

  18. The radial equation(problem) • Use Taylor’s expansion and find Bessel and Neumann functions around x=0 for the zeroth orders. a) J0(x)=1, n0(x)=-1/x b) J0(x)=x, n0(x)=-1/x2 c) J0(x)=1, n0(x)=-x/cosx d) J0(x)=0, n0(x)=-1/x

  19. The radial equation(solution) • Use Taylor’s expansion and find Bessel and Neumann functions around x=0 for the zeroth orders.

  20. The radial equation

  21. The radial equation(problem) • Use either the full analytical expressions or MATLAB to calculate and compare an electron in a 1-D and 3-D infinite wells. Find the lowest 4 energy states for a width of 1nm for 3-D. a) E10, E11, E12, E20 b)E10, E21, E11, E22 c) E10, E11, E12, E13 d) E10, E21, E22, E23

  22. The radial equation(solution) • Use either the full analytical expressions or MATLAB to calculate and compare an electron in a 1-D and 3-D infinite wells. Find the lowest 4 energy states for a width of 1nm for 3-D.

  23. The radial equation(solution) • Use either the full analytical expressions or MATLAB to calculate and compare an electron in a 1-D and 3-D infinite wells. Find the lowest 4 energy states for a width of 1nm for 3-D.

  24. The radial equation(solution) • Use either the full analytical expressions or MATLAB to calculate and compare an electron in a 1-D and 3-D infinite wells. Find the lowest 4 energy states for a width of 1nm for 3-D.

  25. The radial equation

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