1. Lecture 11 Particle on a ring (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.

2. The particle on a ring The particle on a ring is the particle in a box curved with its both ends tied together. This is a model for molecular rotation, a part of an atom’s electronic wave function, and even a part of a crystal’s electronic wave function. We introduce a new operator, angular momentum operator, and a new boundary condition, cyclic boundary condition.

3. Rotational motion We review the relevant classical physics and mathematical concepts: Angular momentum Cylindrical coordinates We then switch to quantum mechanics: Angular momentum operator Hamiltonian in cylindrical coordinates Cyclic boundary condition The particle on a ring

4. The angular momentum Which is the most effective way of applying force to rotate the gear? (b); the right angle to the radius (a) and (c) are wasteful because the effective component is |F| sin θ; (d) is not using leverage. a b c θ d

5. The angular momentum For a given r (position) and p (momentum), its rotational motion is proportional to |r| and |p| sin θ. Angular momentum J (or l) is a vector vertical to the plane formed by r and p, with length |r||p|sinθ. θ r p

6. The angular momentum The mathematical definition of the angular momentum is a vector outer product: Without losing generality, we can assume r = (r x, 0, 0) and p = (p x, p y, 0). Then l = (0, 0, l z ). Note that the angular momentum is a vector parallel to z-axis

7. Cylindrical coordinates

8. The angular momentum operator Classical angular momentum The quantum mechanical operator can be obtained by the conversion Therefore

9. The Hamiltonian

10. The particle on a ring

11. Cyclic boundary condition In the cylindrical coordinates, the wave function is Ψ(r,φ,z) at fixed r and z. The wave function must satisfy the cyclic boundary condition: After a complete revolution, the function must have the same value and also must trace the function in the previous cycle. Otherwise it would not be single- valued.

12. The particle on a ring Promising This will have problems with the cyclic boundary condition Energy

13. The particle on a ring The cyclic boundary condition Unlike the particle in a box, we have m = 0. Unlike the particle in a box, we have negative m’s. Energy is doubly degenerate for m l ≥ 1. No zero-point energy; no violation of uncertainty.

14. The particle on a ring Box Ring Doubly degenerate No zero- point energy Energy differences in the microwave range for molecular rotations

15. Cyclic boundary condition Imagine bending the particle in a box and connect the both ends to make a ring. Notice the difference between the boundary conditions used in two problems.

16. Quantum in nature How does a microwave oven work? Rotation of H 2 O

17. Quantum in nature Why is grass green? Rotation of electrons in chlorophyll

18. The particle on a ring Normalization

19. The angular momentum operator Classical angular momentum The quantum mechanical operator can be obtained by the conversion Therefore

20. The angular momentum Let us act the angular momentum operator on the wave function The wave function is also an eigenfunction of angular momentum operator (H and l z commute and have the simultaneous eigenfunctions). Double degeneracy comes from the two senses of rotations.

21. The uncertainty principle The complementary observables The angle and angular moment are the complementary observables.

22. The particle on a ring For a state with a definite angular momentum, angle must be completely unknown. The probability density along the ring is, completely uniform regardless of m l.

23. Cyclic boundary condition and crystals The crystalline solids have repeated chemical units. We impose cyclic boundary condition on the wave functions of crystals because infinite periodic crystals are mathematically isomorphic to rings of large radius. Crystals’ wave functions as a result have particle on a ring eigenfunctions as their part.

24. Summary In the particle on a ring, we have learned a number of important concepts: angular momentum operator, cylindrical coordinates, cyclic boundary condition. Similarities and differences from the particle in a box: the ring has no zero-point energy, doubly degenerate (|m l | ≥ 1), uniform probability density.