Introduction to Quantum Theory of Angular Momentum. Angular Momentum. AM begins to permeate QM when you move from 1-d to 3-d This discussion is based on postulating rules for the components of AM Discussion is independent of whether spin, orbital angular momenta, or total momentum.
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An angular momentum, J, is a linear operator with 3 components (Jx, Jy,Jz) whose commutation properties are defined as
Jz is diagonal
Where |jm> is an eigenket
h-bar m is an eigenvalue
For a electron with spin up
Or spin down
These Simple Definitions have some major consequences!
It is a lowering operator since it works on a state with an eigenvalue, m, and produces a new state with eigenvalue of m-1
Since Jx and Jy are Hermitian, they must have real eigenvalues so l-m2 must be positive!
l is both an upper and LOWER limit to m!
mlarge cannot any larger
So the eigenvalue is mlarge*(mlarge +1) for any value of m
Indicates a diagonal matrix
And we can make matrices of the eigenvalues, but these matrices are NOT diagonal
If j=0, then all elements are zero! B-O-R-I-N-G!
J+ look like?
Pauli Spin Matrices
Rotations about a given axis commute, so a finite rotation is a sequence of infinitesimal rotations
Now we need to define an operator for rotation that rotates by amount, q, in direction of q
Where n-hat points along the axis of rotation
Suppose we rotated through an angle f about the z-axis
The naïve expectation is that thru 2p and no change.
This is true only if j= integer. This is called symmetric
BUT for ½ integer, this is not true and is called anti-symmetric
And it should be no surprise, that a rotation of b around the y-axis is
Wigner’s Formula (without proof)
Is called the Clebsch-Gordan coefficient
Or Wigner coefficient
Or vector coupling coefficient
Some make the C-G coefficient look like an inner product, thus
Best approach: use a table!!!
And I’m afraid of the “simple” formula?
Well, there is another path… a 9-step path!
Adopt convention of Condon and Shortley,
if j1 > j2 and m1 > m2 then
Cm1 m2j1 j2 j3 > 0
(or if m1 =j1 then coefficient positive!)
Now isn’t that easier?
And much simpler…
You don’t believe me… I’m hurt.
I know! How about an example?
As low as we go
Now we have derived 3 symmetric states
Note these are also symmetric from the standpoint that we can permute space 1 and space 2
Which is 1? Which is 2?
“I am not a number; I am a free man!”
An orthogonal vector to this could be
Must obey Condon and Shortley: if m1=j1,, then positive value
j1=1/2 and |+> represents m= ½ , so only choice is
This state is anti-symmetric and is called the “singlet” state. If we permute space 1 and space 2, we get a wave function that is the negative of the original state.
These three symmetric states are called the “triplet” states. They are symmetric to any permutation of the spaces
Two particles of spin 1 are at rest in a configuration where the total spin is 1 and the m-component is 0. If you measure the z-component of the second particle, what values of might you get and what is the probability of each z-component?
It is understood that a “C” means square root of “C” (i.e. all radicals omitted)
An electron is spin up in a state, y5 2 1, where 5 is the principle quantum number, 2 is orbital angular momentum, and 1 is the z-component.
If you could measure the angular momentum of the electron alone, what values of j could you get and their probabilities?