15_01fig_PChem.jpg

1. Particle in a Box 15_01fig_PChem.jpg

2. 15_01fig_PChem.jpg Particle in a Box Recall

3. 15_01fig_PChem.jpg Particle in a Box Recall Initial conditions

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

5. 15_02fig_PChem.jpg Wavefunctions are Orthonormal Recall + + - - Odd Even + + - Odd Recall + - + Even

6. 15_02fig_PChem.jpg Wavefunctions are Orthonormal AND

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 Recall For 0 <x < a/2 Independent of n

9. 15_02fig_PChem.jpg Expectation Values Recall Average position Independent of n From a table of integrals as 2ca=2pn

10. 15_02fig_PChem.jpg Expectation Values From a table of integrals or from Maple.

11. 15_02fig_PChem.jpg Expectation Values odd even

12. 15_02fig_PChem.jpg Expectation Values Recall

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. Therefore k is a continuous variable, which implies thatE , 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 Product wavefunction 15_p19_PChem.jpg

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

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 pelectrons 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 Inside the box

26. Particle in a Finite Well Limited number of bound states. WF penetrates deeper into barrier with increasing n. Note: not ikx!!! Classically forbidden region as KE < 0 when Vo > En A,B, A’ B’ & 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 – Wavefunctioons are confined within the boundary - Localized. (core) - Have lower energy Weakly bound states - Wavefunctions 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 bias 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 The electrons tunnel to form the new bond Small tunnelling distance relatively large barrier

33. 16_14fig_PChem.jpg Quantum Wells Semi Conductor Band Gap of Al doped GaAs > Band Gap GaAs States are allowed Empty in Neutral X’tal. No States allowed Cond. Band GaAs < Cond. Band Al Doped GaAs States Allowed Fully occupied e’s in Cond. Band 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

37. Quantum Dot Solar Cells Dye Sensitized Solar Cell

38. Organic Polymer Solar Cells Fullerenes(Acceptor) Organic polymer (Donor) Fullerene(PCBM) Organic polymers Background