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Physics 451

Physics 451. Quantum mechanics I Fall 2012. Nov 9, 2012 Karine Chesnel. Phys 451. Announcements. HW #18 today Nov 9 by 7pm Homework next week: HW #19 Tuesday Nov 13 by 7pm HW #20 Thursday Nov 15 by 7pm. Step1 : determine the principal quantum number n.

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Physics 451

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  1. Physics 451 Quantum mechanics I Fall 2012 Nov 9, 2012 Karine Chesnel

  2. Phys 451 Announcements • HW #18 today Nov 9 by 7pm • Homework next week: • HW #19 Tuesday Nov 13 by 7pm • HW #20 Thursday Nov 15 by 7pm

  3. Step1: determine the principal quantum number n Step 5: Multiply by the spherical harmonics (tables) and obtain 2l +1 functions Ynlm for given (n,l) Phys 451 The hydrogen atom How to find the stationary states? Step 2: set the azimuthal quantum number l (0, 1, …n-1) Step 3: Calculate the coefficients cj in terms of c0 (from the recursion formula, at a given l and n) Step 4: Build the radial function Rnl(r) and normalize it (value of c0) (Step 6): Eventually, include the time factor:

  4. Phys 451 The hydrogen atom Representation of Bohr radius

  5. Pb 4.13 Most probable values Pb 4.14 Quantum mechanics The hydrogen atom Expectation values

  6. Pb 4.15 Quantum mechanics The hydrogen atom Expectation values for potential

  7. Phys 451 The angular momentum Pb 4.19

  8. Anisotropy along Z axis Phys 451 The hydrogen atom Representation of

  9. If eigenvector of L2, then eigenvector of L2, same eigenvalue • If eigenvector of Lz with eigen value m • then eigenvector of Lz, new eigenvalue Phys 451 The angular momentum Ladder operator

  10. Top Value =+l Eigenstates Bottom Value = -l Phys 451 The angular momentum Ladder operator Pb 4.18

  11. A. 0 • B. • D. • E. Phys 451 Quiz 25 When measuring the vertical component of the angular momentum (Lz ) of the state , what will we get?

  12. z r y x Phys 451 The angular momentum in spherical coordinates

  13. z r y x Phys 451 The angular momentum In spherical coordinates Pb 4.21, 4.22

  14. z r y x and Phys 451 The angular momentum eigenvectors were the two angular equations for the spherical harmonics! Spherical harmonics are the eigenfunctions

  15. z r y x Phys 451 The angular momentum and Schrödinger equation 3 quantum numbers (n,l,m) • Principal quantum number n: integer • Azimutal and magnetic quantum numbers (l,m) • can also be half-integers.

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