1. Chem 430 Particle on a ring 09/22/2011

2. Richard Feynman Quantum mechanics is based on assumptions and the wave-particle duality The nature of wave-particle duality is not known To explain and predict experimental results: (A) A quantum system has many possible states; (B) Each state has a well defined energy ; (C) At anytime, the system can be in one or more states; (D) The probability in each state is determined by energy and other factors. I think I can safely say that nobody understands quantum mechanics

3. What is energy ? In many cases, define probability The energy of each state will not change The system energy can change Energy value (frequency) is obtained from the oscillation of the coefficients Oscillating dipole generates electromagnetic radiation

4. 30cm -1

5. Polar Coordinates (2D) y x P (x,y) y x O

6. Cylindrical Coordinates (3D) x y z P(x,y,z) r y x Why use the new coordinates rather than the Cartesian Coordinates? Fewer variables, easier to calculate Variables can be separated because of symmetry

7. Rotation a rotation is a rigid body movement which keeps a point fixed. a progressive radial orientation to a common point In Cartesian coordinates, two variables In polar coordinates, only one variable r is constant

8. General procedure Write down Hamiltonian Simplify Math with symmetry Use boundary conditions to define energy levels Particle on a ring a Particle mass : m Potential:0 Radius: r=a=constant Angle: the only variable

9. Particle on a ring Hamiltonian only contains the kinetic energy part In the polar coordinate system

10. The chain Rule Apply it twice Need to eliminate

12. http://en.wikibooks.org/wiki/Partial_Differential_Equations /The_Laplacian_and_Laplace's_Equation

13. Moment of inertia

14. On the ring, Points P = Q Cyclic boundary condition

15. Why only choose one portion for the wave function?

16. Normalized Constant for all angles Implications: (1) probability is same at any point (2) Position can’t be determined at all Why? Why is it different from in the 1D box?

17. Consequence of arbitrary position No zero point energy

18. Particle can rotate clockwise or counterclockwise Double degeneracy

19. y x The Circular Square Well

20. The angular part is known from the above

21. Divided by Radial angular

22. Bessel’s Equation

23. Chem 430 Particle in circular square well and 09/27/2011

24. If particle is confined in a ring, At 0K, what is the most probable location to find it? How about at high temperature? How to explain these in terms of QM? 1.Find out possible states 2. Find out the energy of each state 3. Find out the wave function of each state to obtain its spatial distribution of probability Nothing to do with rotation

25. Boundary condition The condition gives allowed k and therefore energy

26. normalized (A) (C) (B) (A) (B) (C) Many more states are possible if kr is bigger

29. Energy level energy 005.78 1114.68 2226.38 3030.47 4340.71 5149.22

30. Angular Momentum

31. Spherical Polar Coordinates P(x,y,z) r x y z x y z