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

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.

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 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.

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                                                                                    

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

Angular Momentum Operators Lx and Ly do not commute: 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