Winter 2013 / Chem 356: Introductory Quantum Mechanics

Chapter 7a – Hydrogen Atom and Angular Momentum

Angular Momentum Theory

The Radial Equation for the Hydrogen Atom

Chapter 7a – Hydrogen Atom and Angular Momentum

Angular Momentum Theory

Definitions:

We know:

Derive fundamental commutation relations, for angular momentum

More definitions

Derive:

Before we discuss further let me list a few useful tricks when dealing with commutators.

1)

2)a. (pull in front of everything)

b.if commutes with and then

3)a.

b.

4)

Proofs are easy:

e.g.

Derive commutation relations easily:

No deviatives, not acting on a function. Just use tricks!

Next we will derive the spectrum (eigenvalues and eigenfunctions) using only commutation relations. have a set of common eigenfunctions

Because

Proof: for any (use Hermiticity )

Next: it is eigenfunction of , , then so is

Proof:

same eigenvalue of, raised eigenvalue of (or: )

: ladder operators

Same eigenvalue

When do the ladder operators terminate?
use ;

At the other end:

Hence from purely the commutation relations we find

states associated with

There are 2 possibilities for such a structure:

A: is integer

B: is a half integer

/ / / Degen / Spatial name / Spin
name
/ / / 1 / s / Singlet
1 / 2 / -1,0,1 / 3 / p / Triplet
2 / 6 / -2,-1,0,1,2 / 5 / d / Quintet
3 / 12 / -3,-2,-1,0,1,2,3 / 7 / f / Septet
/ / / 2 / - / Doublet
/ / / 4 / - / Quartet
/ / / 6 / - / Sextet

When do spin-multiples show up?

-Electron in spin particle

-Nuclear spin; many nuclear spin states are possible

-Many electron states: triplet, quartet, doublet excited states.

We have used only the commutation relations and to derive the eigenvalues for and operators. Any set of 3 operators that satisfy the commutation relations will yield the same ‘spectrum’

We can do more, if we go back to our original problem, using the representation of operators.

One can also derive expressions for and

It is easy to find eigenfunctions for

Boundary Condition is an integer

The periodic boundary condition is the reason that we do not know how to represent spin functions:

eigenvalue , but rotating over , the function changes ‘sign’!

Let us return to eigenfunctions of and . These eigenfunctions are denoted

are called the ‘associated Legendre’ polynomials. They will be seen to be polynomials in , .We can easily generate them using the ladder operators.

The highest function in the multiplet has to satisfy

The solution to this equation is

since

Hence we have found the highest function , for any

All the other functions can be generated by differentiation: acting with

yields other functions in multiplet

Eg. To generate the p-functions ()

This procedure works for all . I did not worry about normalization.

In general one finds that only depends on , hence the -part of and are the same. Then one can combine the functions

And

Using the new linear combination we get

The are then related to the usual Cartesian angular momentum functions.

In the lecture notes on angular momentum I work out the d-functions in this way. You are asked to do it on the assignment.

The Radial Equation for the Hydrogen Atom

In spherical coordinates:

try solution

Or

Multiply through by

Define

Where is the Bohr radius, also define

These type of 1d differential equations are easy to solve on a computer. I give a few examples of solutions in my lecture notes on the Hydrogen atom.

The general solution to the Hydrogen atom is:

where is a polynomial in having radial nodes

The energies are exactly the same as for the Bohr atom. The theory is very different. This strange coincidence may have held back physics for 10 years!

The themselves have angular nodes, and so the total number of nodes associated with wavefunctions with is always

1s / / No nodes
2s / / 1 radial, 0 angular
2p / / No radial, 1 angular
3s / / 2 radial nodes
3p / / 1 radial, 1 angular
3d / / 0 radial, 2 angular

To understand radial distribution

You have been shown before constant, no dependence

spherically symmetric

justification to look at radial part only, multiplied by volume element.

usual radial distribution, see figure 7.2 in McQuarrie

Chapter 7a – Hydrogen Atom and Angular Momentum / 1