Why is association rule in angular momentum sum not valid?

In summary: In LS coupling, the electron-electron repulsion effect is far stronger than the spin-orbit effect and thus the latter is treated as a perturbation. However, in JJ coupling, the spin-orbit effect is much larger than the electron-electron repulsion effect and so the latter is not taken as a perturbation.
  • #1
goodphy
216
8
Hello.

The quantum mechanics textbook shows the relation of J1 + J2 + J3 ≠ J1 + (J2 + J3). I believe Ji is total angular momentum operator for ith group of electrons (but actually I have not seen J1 operator while I have seen J12 operator so far).

I don't know how to prove J1 + J2 + J3 ≠ J1 + (J2 + J3). I think this play essential role of explaining why there are many coupling schemes.

Please help me to do this:)
 
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  • #2
goodphy said:
The quantum mechanics textbook shows the relation of J1 + J2 + J3 ≠ J1 + (J2 + J3).
That's strange, which book did you find this from?
 
  • #3
blue_leaf77 said:
That's strange, which book did you find this from?

This is actually Korea book so you may not know this. Not famous book but it shows this relation with some proof which is...difficult for me to understand. Have you ever seen this relation? I think essence is this; total angular momentum quantum number J resulted from LS coupling is different to that of being from jj coupling.

Can you at least tell me it is right?
 
  • #4
goodphy said:
I think essence is this; total angular momentum quantum number J resulted from LS coupling is different to that of being from jj coupling.
The thing about LS and jj coupling schemes is that which effect is larger than which. In LS coupling, the electron-electron repulsion effect is far stronger than the spin-orbit effect and thus the latter is treated as a perturbation. In this case, the zeroth order state is determined by performing the sum ##\mathbf L = \sum_{i=1}^N \mathbf L_i## and ##\mathbf S = \sum_{i=1}^N \mathbf S_i## to obtain ##L## and ##S##. When the spin-orbit perturbation is taken into account, there might be states/terms which are degenerate, so one must switch into using the total angular momentum ##\mathbf J = \mathbf L+\mathbf S##. In JJ coupling case where spin-orbit is much larger than electron-electron repulsion, one calculate ##\mathbf J_i = \mathbf L_i + \mathbf S_i## first, then calculate ##\mathbf J = \sum_{i=1}^N\mathbf J_i##.

My point is, I think writing ##(\mathbf J_1 + \mathbf J_2) + \mathbf J_3 \neq \mathbf J_1 + (\mathbf J_2 + \mathbf J_3)## is not mathematically correct. The reason why the final state in LS and jj coupling differs lies in the question of which effect is taken as perturbation.
 
Last edited:

1. Why is the association rule not valid for angular momentum sum?

The association rule, which states that the sum of two quantities is equal to the sum of their individual values, is not valid for angular momentum sum because it is a vector quantity and follows the rules of vector addition. This means that the sum of two angular momenta is not simply the sum of their magnitudes, but also takes into account their direction, resulting in a different value.

2. Can you provide an example of why the association rule is not valid for angular momentum sum?

For example, if two objects with angular momenta of 3 units and 4 units are added together, the association rule would suggest that the total angular momentum should be 7 units. However, if the two objects are spinning in opposite directions, the resulting angular momentum would be 1 unit (3-4), illustrating how the association rule is not valid for angular momentum sum.

3. How does the association rule differ for scalar and vector quantities?

The association rule is valid for scalar quantities, which have magnitude but no direction, as the sum of two scalar quantities is simply the sum of their individual values. However, for vector quantities like angular momentum, the association rule does not hold as the direction of the vectors must also be taken into account when adding them together.

4. Are there any exceptions to the association rule for angular momentum sum?

Yes, there are exceptions to the association rule for angular momentum sum. One exception is when the two angular momenta being added are parallel to each other, in which case the resulting total angular momentum will be the sum of their individual magnitudes. Another exception is when the two angular momenta are perpendicular to each other, resulting in a total angular momentum that is equal to the square root of the sum of their squared magnitudes.

5. How is the vector sum of angular momenta calculated instead of using the association rule?

The vector sum of angular momenta is calculated using vector addition, which takes into account both the magnitude and direction of the vectors being added. This can be done geometrically by drawing the vectors to scale and adding them graphically, or mathematically using the Pythagorean theorem and trigonometric functions. This method ensures that the sum of angular momenta is valid and follows the rules of vector addition.

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