momentum quantum numberℓbut is written asS, P, D, F, ...forℓ= 0, 1 , 2 , 3 ,..., andjis the
total angular momentum quantum number.
A quick example is the single electron states, as we find in Hydrogen. These are:
12 S 12 22 S 12 22 P 32 22 P 12 32 S 12 32 P 32 32 P 12 32 D (^52)
32 D 3242 S 1242 P 3242 P 12 42 D 52 42 D 32 42 F 72 42 F 52 ...
All of these have the pre-superscript 2 because they are all spin one-half. There are twojvalues for
eachℓ.
For atoms with more than one electron, the total spin state has more possibilities and perhaps
several ways to make a state with the same quantum numbers.
21.5 General Addition of Angular Momentum: The Clebsch-Gordan Series
ries
We have already worked several examples of addition of angular momentum. Lets work one more.
- See Example 21.7.3:Addingℓ= 4 toℓ= 2.*
The result, in agreement with our classical vector model, is multipletswithj= 2, 3 , 4 , 5 ,6.
The vector model qualitatively explains the limits.
audio
1l 1 +l 2
l 2
l 2
l
l 1 l^2
l 2
In general,jtakes onevery value between the maximum an minimum in integer steps.
|ℓ 1 −ℓ 2 |≤j≤ℓ 1 +ℓ 2The maximum and minimum lengths of the sum of the vectors makes sense physically. Quantum
Mechanics tells up that the result is quantized and that, because ofthe uncertainty principle, the
two vectors can never quite achieve the maximum allowed classically. Just like the z component of
one vector can never be as great as the full vector length in QM.