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Chem 430 Particle on a ring 09/22/2011

Chem 430 Particle on a ring 09/22/2011. Richard Feynman. I think I can safely say that nobody understands quantum mechanics. Quantum mechanics is based on assumptions and the wave-particle duality. The nature of wave-particle duality is not known.

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Chem 430 Particle on a ring 09/22/2011

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

  2. Richard Feynman I think I can safely say that nobody understands quantum mechanics 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.

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

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

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

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

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

  10. Need to eliminate The chain Rule Apply it twice

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

  12. Moment of inertia

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

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

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

  16. Consequence of arbitrary position No zero point energy

  17. Double degeneracy Particle can rotate clockwise or counterclockwise

  18. The Circular Square Well y x

  19. The angular part is known from the above

  20. Divided by angular Radial

  21. Bessel’s Equation

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

  23. 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? • 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

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

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

  26. Angular Momentum

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

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