1. Physics 451 Quantum mechanics I Fall 2012 Nov 7, 2012 Karine Chesnel

2. Homework this week: HW #18 Friday Nov 9 by 7pm Pb 4.10, 4.11, 4.12, 4.13 Quantum mechanics

3. The hydrogen atom What is the density of probability of the electron?

4. Quantum mechanics The hydrogen atom Ground state: “binding energy” Quantization of the energy Bohr 1913 Principal quantum number

5. Quantum mechanics The hydrogen atom Bohr radius

6. Quantum mechanics The hydrogen atom Energies levels Stationary states n: principal quantum number l: azimuthal quantum number m: magnetic quantum number Degeneracy of n th energy level:

7. Quantum mechanics Quiz 24a A. 5 B. 9 C. 11 D. 25 E. 50 What is the degeneracy of the 5 th energy band of the hydrogen atom?

8. Quantum mechanics The hydrogen atom Energies levels Spectroscopy Energy transition E 0 E1E1 E2E2 E3E3 E4E4 Lyman Balmer Paschen Rydberg constant Pb 4.16 Pb 4.17

9. Quantum mechanics Quiz 24b A. 465 nm B. 87.5 x 10 -8 m C. 4.65  m D. 87.5 x10 -7 m E. 4.65 x 10 -8 m What is the wavelength of the electromagnetic radiation emitted by electrons transiting from the 7 th to the 5 th band in the hydrogen atom?

10. Quantum mechanics The hydrogen atom Coulomb’s law: Solution to the radial equation with Pb 4.10 4.11

11. Quantum mechanics The hydrogen atom Equivalent to associated Laguerre polynomials Pb 4.12

12. Quantum mechanics The hydrogen atom Spherical harmonics (table 4.3) Legendre polynomials Radial wave functions (table 4.7) Laguerre polynomials OR Power series expansion with recursion formula

13. French mathematicians Quantum mechanics Edmond Laguerre 1834 – 1886 Adrien-Marie Legendre 1752 – 1833

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

15. Quantum mechanics The hydrogen atom Representation of

16. Quantum mechanics The hydrogen atom Representation of Bohr radius

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

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