15_01fig_PChem.jpg

1. Particle in a Box 15_01fig_PChem.jpg

2. 15_01fig_PChem.jpg Particle in a Box

3. 15_01fig_PChem.jpg Particle in a Box

4. Wavefunctions for the Particle in a Box 15_02fig_PChem.jpg

5. 15_02fig_PChem.jpg Wavefunctions are Orthonormal

6. 15_02fig_PChem.jpg Wavefunctions are Orthonormal

7. Particle in a Box Wavefunctions Normalized + + n=4 - Orthogonal Node # nodes= n-1 n > 0 + n=3 + Wavelength n=2 + n=1 Ground state 15_03fig_PChem.jpg

8. 15_02fig_PChem.jpg Probabilities For 0 <x < a/2 Independent of n

9. 15_02fig_PChem.jpg Expectation Values Average position Independent of n

10. 15_02fig_PChem.jpg Expectation Values

11. 15_02fig_PChem.jpg Expectation Values even odd

12. 15_02fig_PChem.jpg Expectation Values

13. Uncertainty Principle

14. Free Particle Two travelling waves moving in the opposite direction with velocity v. k is determined by the initial velocity of the particle, which can be any value as there are no constraints imposed on it. This implies that k is a continuous variable, which further implies that E , l and w are also continuous. This is exactly the same as the classical free particle.

15. Probability Distribution of a Free Particle Wavefunctions cannot be normalized over Let’s consider the interval The particle is equally likely to be found anywhere in the interval

16. Classical Limit Probability distribution becomes continuous in the limit of infinite n, and also with limited resolution of observation. 15_04fig_PChem.jpg

17. Particle in a Two Dimensional Box a,b 0,b y x a,0 0,0 15_p19_PChem.jpg

18. 15_p19_PChem.jpg Particle in a Two Dimensional Box

19. Particle in a Two Dimensional Box

20. 2 13 Particle in a Square Box 1 3 0 3 1 2 10 8 2 1 2 2 2 3 Quantum Numbers 26 5 Number of Nodes 1 5 Energy 4 1 1 2

21. Particle in a Three Dimensional Box

22. Particle in a Three Dimensional Box

23. Free Electron Models R 6 p electrons R L LUMO DE HOMO

24. 16_01tbl_PChem.jpg Free Electron Models lmax nH = 2 345 nm nH = 3 375 nm nH = 4 390 nm

25. Particle in a Finite Well

26. Particle in a Finite Well Limited number of bound states. WF penetrates deeper into barrier with increasing n. Classically forbidden region as KE < 0 when Vo > En A,B, A’B’ and C are determined by Vo, m, a, and by the boundary and normalization conditions.

27. 16_03fig_PChem.jpg Core and Valence Electrons Strongly bound states – W.Fns. are confined within the boundary - Localized. (core) - Have lower energy Weakly bound states - W.Fns. extend beyond boundary. - Delocalized (valence) - Have high energy. - Overlap with neighboring states of similar energy Two Free Sodium Atoms In the lattice xe-lattice spacing

28. 16_05fig_PChem.jpg Conduction Consider a sodium crystal sides 1 cm long. Each side is 2X107 atoms long. Unbound states Valence States (delocalized) Bound States (localized) Energy spacing is very small w.r.t, thermal energy, kT. Energy levels form a continuum Sodium atoms Unoccupied Valence States - Band increased occupation of val. states on + side Occupied Valence States- Band electrons flow to +

29. 16_08fig_PChem.jpg Tunneling Decay Length = 1/k The higher energy states have longer decay lengths The longer the decay length the more likely tunneling occurs The thinner the barrier the more likely tunneling occurs

30. 16_09fig_PChem.jpg Scanning Tunneling Microscopy Surface Tip work functions no contact Contact Tunneling occurs from tip to surface Contact with Applied Bias

31. 16_11fig_PChem.jpg Scanning Tunneling Microscopy

32. 16_13fig_PChem.jpg Tunneling in Chemical Reactions

33. 16_14fig_PChem.jpg Quantum Wells States are allowed Empty in Neutral X’tal. B. Gap Al doped GaAs > B.Gap GaAs No States allowed C. Band GaAs < C. Band Al Doped GaAs States Allowed Fully occupied e’s in CB of GaAS in energy well. 3D Box a = 1 to 10 nm thick b = 1000’s nm long & wide Alternating layers of Al doped GaAs with GaAs 1D Box along x !! Energy levels for y and z - Continuous Energy levels for x - Descrete

34. 16_14fig_PChem.jpg Quantum Wells finite barrier DE QW Devices can be manufactured to have specific frequencies for application in Lasers. DEex>Band Gap energy GaAS DEex<Band Gap energy Al doped GaAS

35. 16_16fig_PChem.jpg Quantum Dots Crystalline spherical particles1 to 10 nm in diameter. Band gap energy depends on diameter Easier and cheaper to manufacture 3D PIB

36. 16_18fig_PChem.jpg Quantum Dots