1. Lecture 9 The Hydrogen Atom
Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004)
R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006)
R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974)

2. Topics Today Spherical Harmonics and Quantum Numbers
Probability of Finding Electron in a given volume
Most probable radius of hydrogen atom
Expectation value of radius of hydrogen atom
Angular Momentum
Angular Momentum Operators

3. Problem 1

5. Quantum Numbers

7. Problem 3

8. Probability of Finding Electron in a given Volume

9. The Most Probable Radius Hydrogen Ground State
                                                      
The radial probability density for the hydrogen ground state is obtained by multiplying the square of the wavefunction by a spherical shell volume element.
                                                                                             
It takes this comparatively simple form because the 1s state is spherically symmetric and no angular terms appear.

10. Dropping off the constant terms and taking the derivative with respect to r and setting it equal to zero gives the radius for maximum probability.
                                                                                       
which gives
               
where
                                                                 
The most probable radius is the ground state radius obtained from the Bohr theory. The Schrodinger equation confirms the first Bohr radius as the most probable radius but goes further to describe in detail the profile of probability for the electron radius.

11. Spherical Harmonics

12. Radial Wave Function of Hydrogen Atom

13. Hydrogen 2s Radial Probability
                                                                                                                                                                     

14. Hydrogen 2p Radial Probability
                                                                                                                                                                     

15. Hydrogen 3s Radial Probability
                                                                                                                                                                     

16. Hydrogen 3p Radial Probability
                                                                                                                                                                     

17. Hydrogen 3d Radial Probability
                                                                                                                                                                     

18. The Expectation Value for Radius Hydrogen Ground State
                                                                  
The average or "expectation value" of the radius for the electron in the ground state of hydrogen is obtained from the integral
                                                                    
This requires integration by parts. The solution is
                                                                                                               
All the terms containing r are zero, leaving
                        
It may seem a bit surprising that the average value of r is 1.5 x the first Bohr radius, which is the most probable value. The extended tail of the probability density accounts for the average being greater than the most probable value.

19. Probability for a Range of Radius Hydrogen Ground State
                                                                  
Finding the probability that the electron in the hydrogen ground state will be found in the range r=b to r=c requires the integration of the radial probability density.
                                          
This requires integration by parts. The form of the solution is
                                                                                   

20. Problem 2
Problem 4.44 (Griffith) – n, l and m values are changed.

21. Angular Momentum
Operators Lx and Ly do not commute:
Use: =0

22. Angular Momentum Operators
The non-commutativity of these operators means that in general no two components of L can be known simultaneously with infinite precision. (The only exception is that they can all be zero simultaneously.)

23. Generalized Uncertainty Principle