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What to recall about motion in a central potential

What to recall about motion in a central potential Example and solution of the radial equation for a particle trapped within radius “a” The spherical square well (Re-)Read Chapter 12 Section 12.3 and 12.4. m 1. m 2. O. Focus on this “H μ ”. Ignore center of mass motion.

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What to recall about motion in a central potential

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  1. What to recall about motion in a central potential Example and solution of the radial equation for a particle trapped within radius “a” The spherical square well (Re-)Read Chapter 12 Section 12.3 and 12.4

  2. m1 m2 O Focus on this “Hμ” Ignore center of mass motion

  3. the angular momentum operator

  4. Normalization not yet specified Spherical Neuman function, Irregular @ r=0, so get D=0

  5. Particle in 3-D spherical well (continued) Energies of a particle in a finite spherical well Read Chapter 13

  6. In spectroscopy the rows are labelled by:

  7. I. Energies of a particle in a finite spherical square well

  8. V(r) I II r 0 -V0 a 0

  9. V(r) I II +V0 r 0 -V0 0 a 0

  10. y 2π 5π/2 0 π/2 π 3π/2 3π 7π/2 4π LHS(-cot)

  11. How the value of λ affects the plot: y 2π 5π/2 0 π/2 π 3π/2 3π 7π/2 4π y3 y1 y2

  12. V(r) r 0 I II -V0 a 0

  13. To solve this define: and Then the radial equation becomes: The solution is:

  14. Energies of a particle in a finite spherical square well, continued The Hydrogen Atom

  15. Energies of a particle in a finite spherical square well (continued) The Hydrogen Atom

  16. energy levels # protons in nucleus Charge of electron

  17. Defined as E<0

  18. solution solution

  19. The Hydrogen Atom (continued) Facts about the principal quantum number n The wavefunction of an e- in hydrogen

  20. Facts about the principal quantum number n (continued) The wavefunction of an e- in hydrogen Probability current for an e- in hydrogen

  21. Normalization not determined yet

  22. Facts about Ψe in hydrogen Probability current for an e- in hydrogen Read 2 handouts Read Chapter 14

  23. E>0 (scattering) states (we will study these in Chapter 23) E<0 (bound) states

  24. 2p 2s 3p 3s etc. The n=2 shell The n=1 shell The n=3 shell

  25. Review Syllabus Read Goswami section 13.3 and chapter 14 plus the preceding chapter as necessary for reference

  26. Read Chapter 14 Probability current for an e- in hydrogen The effect of an EM field on the eigen functions and eigen values of a charged particle The Hamiltonian for the combined system of a charged particle in a general EM field

  27. A O

  28. quantum number “m” This factor is called the Bohr magneton Magnitude of the z-component of magnetic dipole moment of the e- angular momentum operator magnetic momentum operator

  29. The effect of an EM field on the eigen functions and eigen values of a charged particle The Hamiltonian for the combined system of a charged particle in a general EM field The Hamiltonian for a charged particle in a uniform, static B field Read Chapter 15

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