Workshop - Particle On a Ring

Workshop: Particle on a ring Quantum aspects of physical chemistry A particle of mass m moving on a ring of radius r in the x y plane (q = p ê 2, 0 § f § 2 p) is an important model quantum system. It also provides nice examples of working with operators, the properties of their eigenfunctions and eigenvalues, and time dependence of wave functions. à Angular momentum 1. The operator for the angular momentum of a particle moving on a ring in the x y plane (q = p ê 2, 0 § f § 2 p) is bz = -Â Ñ „ ê „ f. Show that this operator is hermitian, by showing that ‡ 0 2p f HfL* bz gHfL „ f = ‡ 2p 8bz f HfL<* gHfL „ f, 0 where f and g are smooth, single-valued functions of f. è!!!!!!! 2. Show that F j HfL = ‰Â j f ë 2 p , where j is an integer (including zero and negative values), is normalized over the ring, 0 § f § 2 p. 3. Show that F j HfL is an eigenfunction of angular momentum and find the expression for the eigenvalue. 4. Show, by explicit integration, that different F j HfL are orthogonal. That is, show that 2p * Ÿ0 F j HfL Fk HfL „ f = d j k . where djk = : 1, j = k 0, j ∫ k is called the Kronecker delta. 5. Show, without doing explicit integration, that wave functions F j HfL with different values 2p of j must be orthogonal. That is, show that Ÿ0 F j HfL* Fk HfL „ f = d j k . à Kinetic energy 6. Show that if an operator, q, is hermitian, then the operator q2 = q q is also hermitian. 7. The operator for the kinetic energy of a particle of mass m moving on a ring in the x y plane is bz 2 ê 2 I = -H Ñ2 ê 2 IL „2 ê „ f2 , where I = m r2 is the moment of inertia and r is the radius of the ring. Show that this operator is hermitian. 8. Show that F j HfL is an eigenfunction of kinetic energy and find the expression for the eigenvalue. 9. In classical physics, kinetic energy of a particle on a ring is bz 2 ê H2 IL. Reasoning by analogy, what values of angular momentum can a particle on a ring with moment of inertia I have? 10. Discuss whether you can use the hermiticity of the kinetic energy operator to predict 2p the value of Ÿ0 F-» j» HfL* F» j» HfL „ f. à Average values 11. Show that if the wave function of a system is an eigenfunction of an operator q, then the average value Xq\ is equal to the eigenvalue of q. 12. Obtain an expression for the average value of bz , for the wave function F j HfL. 13. Obtain an expression for the average value of bz 2 , for the wave function F j HfL. 14. Obtain an expression for the rms uncertainty, dbz , for the wave function F j HfL. 15. Discuss whether your result for dbz for the wave function F j HfL, makes sense. à Combination wave functions 16. Construct a normalized wave function, Fc,» j» HfL, proportional to F» j» HfL + F-» j» HfL. 17. Show whether Fc,» j» HfL is an eigenfunction of angular momentum and, if so, find the expression for its value. 18. Show whether Fc,» j» HfL is an eigenfunction of kinetic energy and, if so, find the expression for its value. 19. Discuss whether you can use the hermiticity of the kinetic energy operator to predict 2p the value of Ÿ0 Fc,»k» HfL* Fc,» j» HfL „ f. à Motion on the ring A time-independent wave function, F j HfL, with energy E j , has a corresponding time-dependent wave function Y j Hx, tL = F j HxL ‰- E j têÑ . 20. Write down the explicit expression for the time-dependent wave function Y j Hf, tL = F j HfL ‰- E j têÑ . 21. Construct the time-dependent wave function, Yc,» j» Hf, tL, corresponding to Fc,» j» HfL. 22. Use the stationary phase condition to show that the wave function Y j Hf, tL for j > 0 corresponds to a particle moving around the ring in the x y plane in a counter clockwise direction. 23. Use the stationary phase condition to show that the wave function Y j Hf, tL for j < 0 corresponds to a particle moving around the ring in the x y plane in a clockwise direction. 24. Show that the wave function Yc,» j» Hf, tL corresponds to a particle that is stationary on the ring, that is, that has no net motion around the ring. 25. Sketch the probability density corresponding to Y1 Hf, tL. Does the probability density depend on time? 26. Sketch the probability density corresponding to Y-1 Hf, tL. Does the probability density depend on time? 27. Sketch the probability density corresponding to Yc,1 Hf, tL. Does the probability density depend on time? 28. Discuss your answers to questions 20–22 by comparing the probability densities of F1 Hf, tL and Fc,1 Hf, tL.