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)

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

Problem 1

Quantum Numbers

Problem 3

Probability of Finding Electron in a given Volume

φdθdθsindrrφ,θ,rΨP 22

V

φdθdθsindrrφ,θ,rΨdrrP 22π

0θ

π2

0φ lnlmnl

drrπ4φ,θ,rΨdrrP 22nl

The Most Probable RadiusHydrogen 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.

mlnlnlm YrR,,r

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.

Spherical Harmonics

Radial Wave Function of

Hydrogen Atom

Hydrogen 2s Radial Probability

                                                                                                                                                                     

Hydrogen 2p Radial Probability

                                                                                                                                   

                                  

Hydrogen 3s Radial Probability

                                                                                                                                                           

          

Hydrogen 3p Radial Probability

                                                                                                                                                                     

Hydrogen 3d Radial Probability

                                                                                                        

                                                             

The Expectation Value for RadiusHydrogen 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.

Probability for a Range of RadiusHydrogen 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

                                                                                   

Problem 2

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

Angular Momentum

Operators Lx and Ly do not commute:

C,AB,ACB,A

ijijji ir,pp,r

Use:

=0

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.)

Generalized Uncertainty Principle

2

2B

2A B,A

i21σσ