Quantum Theory II

1. Quantum Theory II Schrödinger's Equation Particle in a box or well Tunneling Hydrogen Atom Spin

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4. Quantum Theory II Schrödinger's Equation Particle in a box or well Tunneling Hydrogen Atom Spin

5. Schedule Change Wednesday, April 26 Quantum Theory I Read: Chapter 39 Friday, April 28 Quantum Theory II Read: Chapter 40, Sections 1-4, 10 Chapter 41, Sections 1-5 Monday, May 1 Exam #3 Wednesday, May 3 Nuclear Physics

6. Class Question This is the wave function of a neutron. At what value of x is the neutron most likely to be found? 1. x =xA 2. x =xB 3. x =xC 4. x = 0

7. Class Question This is the wave function of a neutron. At what value of x is the neutron most likely to be found? 1. x =xA 2. x =xB 3. x =xC 4. x = 0

8. Reading Question The quantity is called the 1. wave function. 2. probability. 3. probability density. 4. amplitude density. 5. Schrödinger function.

9. Reading Question The quantity is called the 1. wave function. 2. probability. 3. probability density. 4. amplitude density. 5. Schrödinger function.

10. Quantum Theory • Matter Waves

11. imaginary function Quantum Theory • This is one of the most important equations in physics imaginary numbers

12. Quantum Theory • Schrödinger's Equation in One-dimension

13. Quantum Theory • Below are the wave functions for two different particles. Which particle has the largest momentum? Explain. In one-dimension and no potential energy particle 1 particle 2 So matter wave with the largest curvature (smallest wavelength) has the largest momentum.

14. The largest probability is where the function is a maximum. Quantum Theory • The wave function has no physical meaning, but Y*Ydx does. What is the physical meaning of Y*Ydx? Discuss this in your group. • Below is a plot of Y*Y for an electron in a electrical potential. Where is the electron most likely to be found? Explain. Probability of finding the particle in dx

15. Quantum Theory • What should the total area under the curve be? • Estimate the probability of finding the electron with a position between 0 and 2? Explain how you found the probability. one

16. Student Workbook

17. Student Workbook

18. Student Workbook

19. Quantum Theory • Particle in a one-dimensional box

20. Boundary conditions Quantum Theory • Particle in a one-dimensional box x = 0 x = L

21. Boundary condition Quantum Theory • Particle in a one-dimensional box x = 0 x = L

22. Quantum Theory

23. Quantum Theory The Infinite Square Well or Particle in a Box • What is the equation that describes the energy for a particle in a box? • What is the ground state energy for an electron in a box 5.0 nm in length? • What is the energy of the first excited state for an electron in a box 1.0 nm in length?

24. P2(As2) TMIn 50nm 0.6 eV Ec InP InP InAs InAs InAs Quantum Theory Growth via Chemical Beam Epitaxy (CBE) A contacted InAs/InP nanowire TEM photo of InAs/InP nanowire Energy landscape

25. + Quantum Theory Energy levels Energy Fermi Sea Fermi Sea • Gate voltage (Vg)moves energy levels up and down • Conductance only when top of Fermi sea aligns with a dot state

26. Student Workbook

27. Student Workbook

28. Student Workbook

29. Student Workbook

30. Student Workbook

31. Class Question A particle in the ground state of a potential energy well • 1. is at rest. • 2. has a zero-point motion. • 3. is equally probable to be found at any point inside the well. • 4. has a wave function that is zero at all points. • 5. has zero energy.

32. Class Question A particle in the ground state of a potential energy well • 1. is at rest. • 2. has a zero-point motion. • 3. is equally probable to be found at any point inside the well. • 4. has a wave function that is zero at all points. • 5. has zero energy.

33. Class Question A particle in a rigid box in the n = 2 stationary state is most likely to be found 1. One-quarter of the way from either end. 2. One-third of the way from either end. 3. In the center of the box. 4. It is equally likely to be found at any point in the box.

34. Class Question A particle in a rigid box in the n = 2 stationary state is most likely to be found 1. One-quarter of the way from either end. 2. One-third of the way from either end. 3. In the center of the box. 4. It is equally likely to be found at any point in the box.

35. Class Question For which potential energy is this an appropriate n = 4 wave function? (1) (2) (3) (4)

36. Class Question For which potential energy is this an appropriate n = 4 wave function? (1) (2) (3) (4)

37. n = 3 n = 2 n = 1 n = infinity n = 3 n = 2 n = 1 Quantum Theory • Plot the energy levels for a particle in a box (n = 1 to 5) and the energy levels we found for the hydrogen atom (n = 1 to 4).

38. Quantum Theory • Tunneling

39. Quantum Theory • Tunneling Microscope

40. Student Workbook

41. Class Question A quantum particle can pass through a region of space that would be forbidden to a classical particle. What is the name of this process? 1. Teleportation 2. Vacuum decay 3. Lasing 4. Trapping 5. Tunneling

42. Class Question A particle with energy E approaches an energy barrier with height U0 > E. If U0 is slowly decreased, the probability that the particle reflects from the barrier 1. Decreases. 2. Increases. 3. Does not change.

43. Class Question A particle with energy E approaches an energy barrier with height U0 > E. If U0 is slowly decreased, the probability that the particle reflects from the barrier 1. Decreases. 2. Increases. 3. Does not change.