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 / Spinname
/ / / 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 nodes2s / / 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