1. 15_01fig_PChem.jpg Particle in a Box

2. 15_01fig_PChem.jpg Particle in a Box

3. 15_01fig_PChem.jpg Particle in a Box

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

5. 15_02fig_PChem.jpg Wavefunctions are Orthonormal

6. 15_02fig_PChem.jpg Wavefunctions are Orthonormal

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

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 odd even

12. 15_02fig_PChem.jpg Expectation Values

13. Uncertainty Principle

14. Free Particle 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,  and  are also continuous. This is exactly the same as the classical free particle. Two travelling waves moving in the opposite direction with velocity v.

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

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

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

20. Particle in a Square Box 1 1 2 3 1 3 2 2 5 1 1 2 03 2 2 41 213 108 265 Quantum Numbers Number of Nodes Energy

21. Particle in a Three Dimensional Box

23. Free Electron Models R R L 6  electrons HOMO LUMO EE

24. 16_01tbl_PChem.jpg Free Electron Models n H = 2 345 nm 375 nm 390 nm max n H = 3 n H = 4

25. Particle in a Finite Well

26. Classically forbidden region as KE E n Limited number of bound states. WF penetrates deeper into barrier with increasing n. A,B, A’ B’ and C are determined by V o, m, a, and by the boundary and normalization conditions.

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

28. 16_05fig_PChem.jpg Conduction Bound States (localized) Unbound states Occupied Valence States- Band Unoccupied Valence States - Band electrons flow to + increased occupation of val. states on + side Consider a sodium crystal sides 1 cm long. Each side is 2X10 7 atoms long. Sodium atoms Energy spacing is very small w.r.t, thermal energy, kT. Energy levels form a continuum Valence States (delocalized)

29. 16_08fig_PChem.jpg Tunneling Decay Length = 1/  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 Tip Surface work functions no contact Contact Contact with Applied Bias Tunneling occurs from tip to surface

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

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

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