Add. Angular Momentum: Finding j=2 States

[Zero86]

Homework Statement

There are two independent subsystems with angular momenta j₁ = j₂ = 1. States have to be found for the whole system with angular momentum j = 2.

Homework Equations

Basic procedure for addition of Angular Momentum in Quantum Mechanics

The Attempt at a Solution

Basically j = j₁ + j₂

I go through the states starting with 2 using the equations in the guide.
So basically we are limited to states
|2, 2>
|2, 1>, |1, 1>
|2, 0>, |1, 0>, |0, 0>

But I'm not sure if it's the right way to go through the states. Also, maybe there are better guides since I'm not even sure if this one applies to the problem. I'm looking into Clebsch–Gordan coefficients at the moment.

 

[vela]

What do you mean "the right way to go through the states"?

 

[Zero86]

Well I have found that there are 2*2+1 = 9 states which are |2, 2>, |2, 1>, |2, 0>, |2, -1>, |2, -2>, |1, 1>, |1, 0>, |1, -1>, |0, 0>.
In 'big O' notation,
1X1 = 2+1+0+|-1|+|-2|
3*3 = 9
So these seem to be all the possible states. I'm not sure if it means I have found all the states. I'm not sure how they have come up with
\ket{j, j-1} = \sqrt{j_1 \over j} \ket{j_1, j_1-1, j_2, j_2} + \sqrt{j_2 \over j} \ket{j_1, j_1, j_2, j_2 - 1}

 

[vela]

The states with total angular momentum j=2 are |2, 2>, |2, 1>, |2, 0>, |2, -1>, and |2, -2>. You're probably expected to find them in terms of the eigenstates of J₁ and J₂.

 

[paranormal]

Great job starting with the basic procedure for adding angular momentum in quantum mechanics! Your approach is on the right track. Yes, you are correct that the total angular momentum of the system, j, can take on values of 2, 1, or 0. And the states for the whole system will be a combination of the states for the individual subsystems.

To find the specific states for the whole system with j=2, you can use the Clebsch–Gordan coefficients. These coefficients relate the individual subsystem states to the total system states. You can use the Clebsch-Gordan table to determine the allowed combinations of j1 and j2 that will give you a total angular momentum of j=2.

For example, using the table, you can see that the combination of j1=1 and j2=1 will give you j=2. This means that the state |2,2> is allowed for the total system. Similarly, you can use the table to find the other allowed combinations of j1 and j2 that will give you a total angular momentum of j=2. This will give you a total of six states, as you have listed above.

Overall, your approach is correct and using the Clebsch-Gordan coefficients will help you find the specific states for the whole system with j=2. Keep in mind that there may be other guides or methods for solving this problem, but using the basic procedure and the Clebsch-Gordan coefficients is a good approach.